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 principles essentially the same as 
 those here adopted. But the mathematical reasoning, as I 
 afterwards found, was inadequate to the purpose and incor- 
 rect. The repulsive actipn was attributed in part to the mo- 
 
 mentum of the fluid due to a velocity expressed as *- cos ^> 
 
 which was supposed to be transmitted instantaneously in the 
 directions of the radii produced, and to be unaccompanied by 
 condensation. The present more complete investigation has 
 
462 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 shewn that the part of the velocity due to the reaction of the 
 sphere, so far as it is the same as that for an incompressible 
 fluid, varies inversely as the cube of the distance from the 
 centre of the sphere, and, moreover, is not in the direction of 
 a produced radius (see p. 284). The parts of such velocities 
 resolved transversely to the directions of the radii may be 
 supposed to be neutralized by the composition of undulations 
 from different sources, so that there remain only resulting 
 velocities in the directions of the radii varying inversely as 
 the cube of the distance. But the action of the momentum of 
 the fluid due to velocity of this kind, besides being extremely 
 small, would only produce (since there is no propagation of 
 accompanying condensation) aperiodic effect on the surround- 
 ing atoms, and may, therefore, be left out of consideration. 
 In the mathematical theory now proposed the repulsive force 
 is wholly due to the condensation which, being generated by 
 the reaction that takes place at the surface of the sphere, is 
 thence propagated in the directions of the prolongations of 
 the radii (the law of rectilinear transmission (p. "188) being 
 here applicable), and varies with the propagation inversely as 
 the square of the distance (p. 193), Hence at the distance r 
 
 from the centre it may be expressed as - 3 cos ^> so ^ ar 
 
 as it is due to a single series of incident waves. The conden- 
 sation resulting from an unlimited number of series incident 
 from all quarters may be represented by the expression 
 
 , , 6 disappearing. Also the relation between the 
 
 propagated condensation and the corresponding velocity, 
 whether in the composite resultant, or in each component, 
 is, to the second approximation, that given by the equation 
 (28) in page 246. 
 
 The theory of the repulsive force of heat having thus 
 been sufficiently exhibited, it is now required to shew in 
 what manner the repulsion is counteracted by a molecular 
 attraction. For this purpose let us suppose that waves of 
 
THE THEORY OF MOLECULAR ATTRACTION. 463 
 
 condensation of the kind above considered are propagated 
 from all the atoms of any visible or tangible substance that 
 are contained within a spherical surface of radius r, and that 
 the amount of the condensation is the same in all directions 
 from each atom. Then the condensation resulting from the 
 composition of the waves emitted from the atoms may be 
 assumed to vary at any distance (E) from the centre of the 
 spherical space, very large compared to r, as the number of 
 atoms in the space, that is, as r 3 , directly, and as JR Z inversely, 
 the mean interval between adjacent atoms being given. If 
 we now take another space of larger radius r, and another 
 position whose distance (R) from the centre is such that 
 
 r r 
 
 "# = ~W ' ^ e con densation a * * ne fi rs * point being to that at 
 
 r 3 /3 
 
 the second as == to -^ > will also be as r to /, or as R to E. 
 
 Hence, however small may be the condensation propagated 
 from a single atom, the resulting condensation from an ag- 
 gregation of atoms contained in a spherical space will be of 
 sensible magnitude at distances from the centre of the space 
 very large compared to its radius, provided the space be not 
 less than a certain finite magnitude, and the atoms contained 
 in it be not fewer than a certain finite number. It will now 
 be assumed that this condition is always satisfied in liquid 
 and solid bodies. Further, it may be argued that as the 
 motion at the distance R is composed of vibratory motions, 
 it must itself be vibratory. Hence, since the original small 
 motions are unlimited in number, and may within certain 
 limits have values of X and c of all gradations, it must be 
 admitted that there are combinations of these quantities 
 which make the resultant velocity and condensation zero for 
 certain values of R at a given time, and maxima, positive 
 or negative, for certain other values at the same time; for 
 otherwise the resultant motion would not be vibratory. It 
 is evident that the intervals between the points of zero and 
 maximum values of the velocity and condensation must be 
 
464 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 very much larger than the values of X for the original waves. 
 Hence, the compound motion being still such as results 
 from the laws of spontaneous vibrations, if we express, as is 
 required by those laws, the vibratory character of the motion 
 by the equations of the first order, 
 
 rt 
 
 V=a'S=2.m sin ' (B-a't + C), 
 
 X 
 
 X having gradations of value between certain limits, it must 
 be supposed that all these values of X, as well as the value 
 of w, which may be supposed to be common to all the waves 
 of this class, are quantities of much greater magnitude than 
 the analogous quantities for the original waves. In other 
 respects the two classes of waves are exactly similar, both 
 being equally independent of particular modes of disturb- 
 ance. And it may reasonably be admitted that in their ac- 
 tion upon the atoms of substances they produce contempo- 
 raneously dissimilar and independent effects. This, in fact, 
 would be the case, according to the reasoning in page 457, 
 if, while the values of X for the first class are so small as to 
 
 Jc k 
 
 make the factor fc--r|-f r| positive, those for the second 
 X X 
 
 class are large enough to make it negative. 
 
 Thus the theory, at the same time that it accounts for 
 the repulsion of heat, shews how the very waves that have 
 that effect, may by their combination subsequently assume a 
 form proper for producing the controlling effect of attrac- 
 tion. I call the latter molecular attraction because it exists 
 only under the condition of the emanation of waves from 
 the individuals of a mass of atoms. 
 
 It is true that according to the same theory there might 
 be produced composite waves having such larger values of X 
 
 k k 
 
 as would make the quantity \ * + A again become posi- 
 
 X X 
 
 tive. We have seen that the theory of the dispersion of 
 light (p. 428) renders it probable that this is actually the 
 
THE THEORY OF MOLECULAR ATTRACTION. 465 
 
 case. Such waves may possibly account for certain effects 
 of radiant heat, as distinguishable from internal caloric re- 
 pulsion. May not the suspension of a drop of water close 
 to a metal plate heated to a white heat be attributable to a 
 repulsive action of this second class*, and is not the same 
 force concerned in the enormous expansion under the sun's 
 heat of the tails of some comets? The manner in which 
 the repulsion of this order is controlled by another order of 
 attraction will be hereafter considered in treating of the force 
 of gravity. 
 
 I proceed now to another important part of the theory, 
 namely, that relating to the conditions under which bodies 
 are solid, liquid, or aeriform. Considering that the values of 
 \ for light are about the fifty thousandth part of an inch, 
 and that from the fact cited in page 456 they must still be 
 very large compared to the sphere of activity of the atomic 
 and molecular forces, it will be seen that the dimensions of 
 the molecules themselves from which the molecular attrac- 
 tions proceed must be comparatively of extremely small 
 dimensions. For the sake of precision of idea I supposed 
 (p. 463) the form of the molecules to be spherical; but the 
 argument would remain the same if it were supposed to be 
 cubical, and in that case a molecule might be regarded as an 
 elementary portion of the liquid or solid substance. Taking, 
 then, the centre of each such element to be a centre of attrac- 
 tive force, the effect of molecular attraction might be calcu- 
 lated by the process usually applied to forces acting sensibly 
 only at immeasurably small distances from their centres. The 
 same process is evidently applicable to the atomic forces. It 
 results also from the account the theory gives of the origi- 
 nation of these forces and their limited spheres of activity, 
 that 1 in the interior of any substance of uniform density, the 
 atomic forces acting on any atom are equal in opposite di- 
 rections, and that the same is the case with respect to the 
 
 * Report of the British Association for 1845, Transactions of the Sections, 
 p. 27. 
 
 30 
 
466 THE MATHEMATICAL PKINCIPLES OP PHYSICS. 
 
 molecular forces. But if the atom be situated at or very 
 near the boundary of the substance, the circumstances of its 
 equilibrium are no longer the same, and it becomes neces- 
 sary to inquire particularly what in that case are the condi- 
 tions under which the equilibrium is maintained. * 
 
 Conceive, at first, the atom to occupy a position exactly 
 at the boundary of the substance, and suppose the boundary 
 through a small extent to be a plane. It is evident that the 
 resulting attractive and repulsive forces will both act in di- 
 rections perpendicular to the surface, the former towards the 
 interior, and the other in the opposite direction, and that 
 these forces will be equal. Now since caloric repulsion is 
 in general a much more energetic force than the attraction 
 of aggregation, that equality between the forces cannot sub- 
 sist if the density of the substance be the same throughout. 
 We must, therefore, suppose that there is a gradual diminu- 
 tion of density from the interior to the boundary through a 
 very small superficial stratum, till the repulsive action is so 
 reduced by reason of the increment of the mean interval 
 between adjacent atoms, that it becomes equal to the attrac- 
 tive force. It is evident that equilibrium of the atom must 
 result under these conditions, because the amount of repul- 
 sive force is much more diminished by the superficial dimi- 
 nution of density than that of the attractive force, on ac- 
 count of the much larger sphere of activity of the latter. 
 
 Selecting, now, an atom within the boundary situated at 
 a very small distance from it, it is clear that the repulsive 
 force urging the atom towards the boundary will be the dif- 
 ference of two opposing repulsions, while the attractive force 
 will be very nearly the same as before, being very little 
 altered by the comparatively small difference between the 
 positions of this atom and the former one. Hence in order 
 that the difference of the repulsions may be so nearly equal 
 to the total repulsion in the former case, there must be an 
 increment of the density of the substance from the boundary 
 towards the interior. This increment of density will go on 
 
THE THEORY OF MOLECULAR ATTRACTION. 467 
 
 to a depth below the surface equal quam proxime to the 
 radius of the sphere of molecular attraction, where both the 
 opposite attractions and the opposite repulsions neutralize 
 each other. The gradation of density thus produced is ana- 
 logous to the variation of the density of the earth's atmo- 
 sphere caused by the earth's attraction*. 
 
 This gradual increment of density in a very thin super- 
 ficial stratum is a necessary condition both of the liquid and 
 the solid state of a substance. The difference between the 
 two states is partly accounted for by supposing the control- 
 ling attraction to be feebler in the liquid than in the solid. 
 But besides this, the mobility of a liquid appears to depend 
 essentially on the circumstance that a disturbance of the 
 superficial atoms does not call into play any atomic or mo- 
 lecular force in directions parallel to the surface. When any 
 substance, as water, passes from the liquid to the solid state, 
 its atoms may be supposed to assume a crystalline arrange- 
 ment, and probably it is a consequence of such arrangement 
 that the atomic and molecular forces resist any disturbance 
 given to a superficial atom, both in directions perpendicular 
 to the surface, and also in any direction parallel to it. From 
 the fact that water expands while its temperature is descend- 
 ing towards that of ice, it would appear that coincident with 
 the tendency of the atoms towards greater stability of position 
 by a crystalline arrangement there is an increment of atomic 
 repulsion of such amount that within certain limits it ex- 
 ceeds the diminution due to the fall of temperature. 
 
 In the case of aeriform bodies the waves of atomic re- 
 pulsion do not, as in liquid and solid bodies, pass by compo- 
 sition into those of molecular attraction, apparently because 
 the mean interval between the atoms is so large that the 
 value of X for the composite waves of the second order ex- 
 
 k k 
 
 ceeds the greatest value for which \ ~ z + ~ 4 is negative. 
 
 A* A 
 
 * I proposed a theory of superficial molecular action very similar to this, in an 
 Article on " Capillary Attraction and the Molecular Forces of Fluids," contained 
 in the Philosophical Magazine for February, 1836. 
 
 302 
 
468 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 The atomic repulsion of such bodies is controlled by an 
 extraneous force as gravity. 
 
 The explanation that the theory gives of caloric repulsion 
 readily accounts for the loss of heat which is said to occur 
 when a body passes from the liquid, or solid, to the gaseous 
 state. The loss is simply caused by the sudden reduction of 
 the number of atoms in a given space, and the consequent 
 diminution of the number of centres in that space at which 
 the repulsive force is generated. As soon as the number of 
 atoms becomes the same as before by the restoration of the 
 gas to the liquid or solid state, the generation of atomic re- 
 pulsion recommences, and heat becomes sensible. In this 
 way the theory accounts for what is called latent heat. 
 
 The loss of heat arising from the continual radiation of 
 heat-waves from all bodies into space, is supplied, according 
 to this theory, from the secondary waves generated by the 
 reaction at the surfaces of the atoms. It is true that radia- 
 tion from the sun, and in some degree also from the planets 
 and stars, is the original source of heat-waves ; but the 
 supply from this source is conserved and multiplied in the 
 interior of bodies by the effect of atomic reaction. If the 
 external supply of heat be increased, the atomic repulsion is 
 increased also, and the body is thereby caused to expand, 
 till by such expansion the repulsive force is so much dimi- 
 nished on account of the augmentation of the mean atomic 
 interval, as to be just counteracted by molecular attraction, 
 or the force of gravity. And on the other hand, if the supply 
 falls short, the body is caused to contract by the superiority 
 of the attractive forces, till in consequence of the decrement 
 of the mean interval between the atoms, the atomic repul- 
 sions become adequate to counteract the attractions. 
 
 It may here be remarked that according to these views 
 there are no circumstances under which the forces of nature 
 can act differentially on two neighbouring atoms to such a 
 degree as to overcome their mutual repulsion, and that conse- 
 quently the collision of atoms is an impossibility. 
 
THE THEORY OF MOLECULAR ATTRACTION. 469 
 
 The foregoing theory of atomic and molecular forces gives 
 the means of investigating the relation between the pressure 
 and the density of substances, as depending on the gaseous, 
 liquid, or solid state. With respect to gaseous, or aeriform, 
 bodies, we have only to take account of atomic repulsions, 
 because, as already stated, the molecular attraction ema- 
 nating from a mass of atoms of such bodies contained within 
 a space not extremely large is of insensible magnitude. 
 When a portion of a gas is in an enclosure, the boundaries 
 of which are solid, an atom at the boundary, whether it be- 
 long to the gas or the solid, is kept in equilibrium by the 
 resultant of the atomic and molecular forces of the solid 
 acting in the direction towards the gas, and that of the 
 atomic repulsion of the gas in the opposite direction. In 
 order that the former resultant may act towards the gas, 
 it is necessary that the atomic repulsion should exceed the 
 molecular attraction. But since the contrary would be the 
 case if the gas had no effect, it follows that the atomic re- 
 pulsion of the gas so far presses together the extreme super- 
 ficial atoms of the solid (the density of which, as was shewn 
 in page 466, is very small), that the atomic repulsion be- 
 comes greater than the molecular attraction. Now this effect 
 is precisely what is measured by the experimental means 
 which determine the pressure of a fluid against a solid. And 
 as action and reaction are equal, the pressure so measured 
 must be in exact proportion to the resultant of the atomic 
 repulsion of the gas. It may, in short, be called the mecha- 
 nical equivalent of the caloric repulsion. Suppose now an 
 additional quantity of the gas to be put into the same space 
 so as to increase the density by Ap. Then assuming that on 
 account of the rarity of the atoms, the emanation of waves 
 from each atom, the temperature remaining the same, is not 
 sensibly increased by this addition to the number of atoms 
 in a given space, it will follow that the increment of resultant 
 atomic repulsion is in the same proportion as the increment 
 of density. But the former increment is measured by the 
 
470 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 increment Ap of the pressure. Hence, ~k being a certain 
 constant, Ap = &Ap ; and by integration p = Jcp + c. 
 
 The same result may be obtained by the following con- 
 siderations applied to the earth's atmosphere as acted upon 
 by terrestrial gravity. Conceive the atmosphere to be di- 
 vided into strata by surfaces concentric with the earth's sur- 
 face, and let the density throughout any stratum be uniform, 
 and equal to the mean of the actual densities in that stratum. 
 Then if we suppose the thickness of each stratum to be a 
 quantity As greater than the radius of the sphere of activity 
 of the atomic repulsions acting on a given atom, this thick- 
 ness will still be so small that the difference of the densities 
 of two contiguous strata may be treated as an infinitesimal 
 quantity. Now let F represent the mean repulsive action 
 upwards at the upper surface of a given stratum, and F' that 
 at the upper surface of the next higher stratum. Also, as- 
 suming that the two strata have the same temperature, let us 
 suppose, in order to fulfil this condition, that the emanation 
 of waves from individual atoms is to the same amount in 
 both. Hence F and F' will differ from each other only on 
 account of difference in the quantity of the atoms in a given 
 space, and will be proportional to the respective quantities, 
 that is, to the densities of the strata ; so that, p and p being 
 the densities, we may suppose that F= Gp, and F' = Gp'. 
 Since Az is equal to the interval between the middle points 
 
 of the successive strata, it follows that p' = p+-^Az, and 
 
 cLz 
 
 that FF' = G-f-kz. Now since the forces F counter - 
 dz 
 
 act the effect of gravity acting on the superincumbent column 
 of air, their mechanical equivalent must be the weight of the 
 column as measured by the pressure at the upper surface of 
 the stratum. Hence if p be this pressure, and p' the pres- 
 sure at the upper surface of the next superior stratum, F F' 
 
 will vary as p-p t r ~~ ~ ^ z - Consequently, k being a 
 
THE THEORY OF MOLECULAR ATTRACTION. 471 
 
 certain constant, -j- kz = Jc -~ As. Hence, if the temperature 
 
 be given, whatever function p or p may be of z we shall have 
 dp = kdp, and by integration p = kp + c. 
 
 It is to be observed that in obtaining this law for the 
 relation between the pressure and density of a gas, it was 
 assumed that the emanation of waves from a given atom 
 which might be called its proper caloric is not dependent 
 on the number of surrounding atoms of the gas in a given 
 space, but is due exclusively to the incidence on the atom 
 of waves coming from extraneous sources. The waves pro- 
 ducing this effect are of the class to which, as explained 
 in page 465, radiant heat is attributable. A thermometer 
 placed within the receiver of an air-pump will indicate, after 
 a portion of the air has been extracted and the equilibrium 
 of heat is restored, the same temperature as before ; and this 
 will be the case however far the exhaustion be carried. There 
 is, therefore, what may be called a temperature of position, 
 depending upon the radiation of heat from all bodies sur- 
 rounding the position within large distances. The waves of 
 this radiant heat comport themselves very nearly in the same 
 manner as waves of light (which, in fact, may be classed 
 among them), being similarly reflected and refracted, and 
 passing freely through certain substances whilst by others 
 they are stopped and absorbed. In the case of the substances 
 that are diathermous, these waves produce little or no effect 
 on the amount of the proper caloric of the atoms; but in 
 other substances they are converted in greater or less degree, 
 by transmutation of the vibrations, into waves of the order 
 proper for atomic repulsion. This conversion may possibly 
 be only a resolution of the waves into their components, 
 which, according to the theory in page 463, mainly consist of 
 waves of that order. 
 
 But it is evident that among the radiations from sur- 
 rounding substances which in this manner contribute to pro- 
 duce the temperature of the position, those proceeding from 
 
472 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the gas itself ought to be taken into account, and that their 
 effect will be greater as the density of the gas is greater. 
 Thus the factor k in the foregoing formula is not wholly 
 independent of density. If, for instance, we take a position 
 in the higher regions of the atmosphere, the proper caloric 
 of an atom, since it must depend very largely on the radia- 
 tions from the surrounding fluid, will, on account of the 
 diminished density, be considerably less than that in the 
 lower regions, and the temperature of the position and value 
 of Jc will consequently be less also. It has been found ex- 
 perimentally that the relation between pressure and density, 
 inclusive of the effect of variation of temperature, is expressed, 
 at least very approximately, by the formula p = kp + c, if 
 Jc = K (1 -f oc0), being the number of degrees of tempera- 
 ture above that for which k K, and oLZTthe increment of k 
 for every 1 increment of temperature. Since, according to 
 the theory, if the density be given, the pressure varies as 
 the proper caloric of the atoms, or the atomic repulsion, it 
 follows from the above expression for Jc that the increments 
 of proper caloric are in the same proportion as the increments 
 of temperature. Also, since it results from the theory that 
 in large gaseous masses, such as the earth's atmosphere, the 
 pressure varies, cceteris paribus, not simply as the density, 
 but in a somewhat higher ratio, the relation between pressure 
 and density might be approximately expressed in such cases 
 by the formula p = K'p l+m + c, m being a small fraction, and 
 the factor K' varying with any changes of temperature that 
 may occur from other causes than that of change of density. 
 
 When the quantity of air in an enclosed space is suddenly 
 added to, sensible heat is produced by the sudden increase of 
 the number of atoms in the space, and the consequent increase 
 of the number of centres of the secondary waves of atomic 
 repulsion, these waves forming by their subsequent combina- 
 tion (see p. 465) waves of the order of those of radiant heat, 
 by which, as was argued above, the temperature of a given 
 position is determined. In the present instance the effect is 
 
THE THEORY OF MOLECULAR ATTRACTION. 473 
 
 augmented by the circumstance that the additional waves of 
 that order cannot at once escape from the enclosure, being par- 
 tially reflected by its solid boundaries, so as to be made to tra- 
 verse the space probably a very large number of times. The 
 increment of the temperature of the space produced by this 
 means is dissipated in a very short time after the exciting cause 
 ceases to act, and thus the equilibrium of the atoms situated 
 at the boundaries is quickly restored. In like manner, if the 
 quantity of the air in the space be suddenly diminished, there 
 will be a sudden decrease of the number of centres of secon- 
 dary waves, and a consequent abstraction of some portion of 
 the larger composite waves by which the temperature of the 
 space is maintained. As these abstracted waves contributed 
 towards the temperature of the position by repeated reflections 
 in the manner just described, it follows, as in the former case, 
 that the amount of the depression of temperature is increased 
 by the circumstance that the extent of the fluid is limited by 
 solid boundaries. It would seem, therefore, that changes of 
 temperature consequent upon sudden condensations and rare- 
 factions might be much more sensible in closed spaces than 
 in unenclosed, and that from the changes in the former those 
 that take place in the latter cannot be legitimately inferred. 
 When, for instance, waves of sound are propagated in the 
 lower regions of the atmosphere, where the temperature of 
 the position depends much more on the radiations from the 
 earth than on those from the atmosphere, it does not seem 
 possible that the changes of density which accompany the 
 aerial vibrations can sensibly affect the amount of the caloric 
 of individual atoms, inasmuch as these condensations arid 
 rarefactions are very small compared to the total density, 
 and, as appears from the argument by which the equation 
 p kp-\-c was obtained (p. 469), the total density itself de- 
 termines only to a small amount the proper caloric of the 
 atoms. Supposing, therefore, that caloric to be sensibly the 
 same at different points of the aerial waves, the temperature 
 will also be the same at different points, and there will be no 
 
474 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 appreciable increase of the rate of propagation such as would 
 result if, as has been supposed, momentary changes of tem- 
 perature were produced at all points of the waves in exact 
 proportion to the changes of density. (See p. 225). 
 
 The above conclusion from the antecedent theory of heat 
 receives confirmation from the comparison of the rate of 
 propagation in an elastic fluid, as mathematically determined, 
 with the observed velocity of sound. (See pages 224 and 317). 
 The relation between the pressure, density, and temperature 
 of the air being expressed generally by the equation 
 
 the mathematical reasoning gives for the rate of propagation 
 > K being a numerical factor, the value of which 
 
 is obtained on purely hydro dynamical principles by the 
 reasoning concluded in page 224. Since this rate is found 
 to differ very little from the velocity of sound as determined 
 by observation (exceeding it by a few feet possibly because the 
 fluid was supposed in the theory to be perfect), we may infer 
 that the effect of temperature is fully taken into account by 
 the factor 1 -f a0, which depends exclusively on the tem- 
 perature of the position, and that, as the above theory in- 
 dicates, no effect is attributable to momentary changes of the 
 temperature of the fluid due to the changes of density. 
 
 It is now required to investigate the relation between the 
 pressure and density in liquid, and solid substances. In the 
 first place, I remark that the difference between a liquid and 
 a solid depends, according to the foregoing theory, mainly on 
 the mode of action of the atomic and molecular forces by 
 which the atoms at and very near the boundaries of the 
 substance are held in equilibrium. In liquids, as already 
 said (p. 467), there appears to be no action in directions 
 parallel to the boundary, and the resultant molecular attrac- 
 tion in the perpendicular directions is comparatively feeble. 
 In solids, on the contrary, the superficial atomic and mole- 
 cular forces act so as to resist both extension and compression 
 
THE THEORY OF MOLECULAR ATTRACTION. 475 
 
 in directions perpendicular to the surfaces, and at the same 
 time are opposed to any separation of parts in the transverse 
 directions. The latter resistance, which most probably de- 
 pends on crystalline arrangement of the atoms, is overcome 
 by cutting or breaking the substance, but immediately re- 
 sumes its sway in the new surfaces which these operations 
 produce. It has already been explained (p. 465) that in the 
 interior of bodies, both liquid and solid, so long as there 
 is no gradation of density, the opposite atomic repulsions 
 acting on a giving atom, as well as the opposite molecular 
 attractions, are equal to each other, so that as far as regards 
 the equilibrium of an internal atom, there is no difference 
 between a liquid and a solid. In the solid bodies, however, 
 that are crystalline, the arrangement of the atoms may have 
 the effect of making the forces differ in magnitude in different 
 directions, at the same time that the opposite forces, both 
 atomic and molecular, neutralize each other. 
 
 Both solids and fluids offer great resistance to being com- 
 pressed within a smaller space. This resistance is due to the 
 repulsions of the superficial atoms, and to the sudden in- 
 crement of repulsive force which any compression of these 
 atoms brings into action. 
 
 It is a matter of experience that after a solid substance 
 has been separated into parts, in general the parts strongly 
 resist being joined together so as to form the same continuous 
 mass as before. This fact is theoretically accounted for by 
 the consideration that the gradation of density which exists 
 at the boundaries of the parts must be destroyed before the 
 separated portions can be perfectly united. It is^ however, 
 conceivable that the mechanical compression of fragments, 
 whether or not they have the same relative positions as before 
 the fractures, may, by acting in aid of the molecular at- 
 tractions, suffice to get rid entirely of the gradations of 
 density, and thus effect a perfect union. In fact, the experi- 
 ments of Professors Tyndall and Huxley detailed in the 
 Transactions of the Eoyal Society (Vol. cxlvii., pp. 329331), 
 
476 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 are an actual instance of the production of this effect by 
 crushing together fragments of ice*. 
 
 The foregoing theory of the conditions of fluidity and 
 solidity accounts for a fact relating to the Figure of the Earth 
 which does not seem to admit of any other explanation. 
 What is the reason that being apparently solid, the earth 
 takes the form which allows of its being covered to a great 
 extent, and to a comparatively small depth, with fluid the 
 spheroidal surface of which has the same axes and ellipticity 
 as the mean spheroidal surface of the solid part? The 
 answer which the theory gives to this question is, that the 
 mass of the earth, taken as a whole, must be regarded as 
 fluid, and that the rigidity of the superficial crust, while it 
 has the effect of maintaining local elevations and depressions, 
 has no sensible influence on the general figure. By the effect 
 of internal pressure, the distinction between solid and fluid 
 parts would be caused to disappear probably at no very great 
 depth below the surface ; for the vast amount of the pressure 
 of the superincumbent mass at considerable depths would 
 crush out, as in the experiment just referred to, the gradation 
 of density which is a necessary condition of the distinction of 
 parts, and thus all difference between contiguous portions in 
 respect to fluidity and solidity would be obliterated. 
 
 According to these views if the crust of the earth be a 
 rigid envelope surrounding a fluid nucleus, its thickness must 
 be considerably within the depth at which the distinction be- 
 tween liquid and solid parts ceases. Or, if this be not the case, 
 the crust merges gradually into a plastic interior, from which 
 the same form of the earth results as from a fluid nucleus. 
 The same theory of atomic and molecular forces, considered 
 with reference to terrestrial gravity, leads to the inference 
 that there must be limits to the heights and acclivities of 
 mountains, and to the depths of ocean-basins, depending on 
 
 * Like considerations might be brought to bear on the theory of the motion of 
 glaciers. I have briefly adverted to this question in the Philosophical Magazine 
 for February, 1860, page 90. 
 
, THE THEORY OF MOLECULAR ATTRACTION. 477 
 
 the energy of the superficial molecular attraction. The divi- 
 sion of large masses into parts by faults and fissures, by 
 increasing the quantity of containing surfaces, appears to 
 render a greater amount of superficial irregularity possible. 
 If these irregularities and the effect of centrifugal force be 
 disregarded, large bodies, like the sun and planets, would 
 necessarily, according to this theory, take the form of a 
 sphere. The form of a very thin plate, like that of Saturn's 
 Rings, would also be compatible with the theory; but it 
 does not appear that any form very unlike these could be 
 maintained by the molecular forces. 
 
 The preceding considerations relative to the condition of 
 the interior of the earth have been gone into on account of 
 their bearing on the question of the relation between pressure 
 and density in liquid and solid bodies. It is only in masses 
 of large magnitude that the pressures and densities are suffi- 
 ciently different at different parts to serve for verifying a theo- 
 retical determination of the law of their mutual relation. For 
 this reason reference is made to the earth's mass in the subse- 
 quent investigation, and considerations are employed analo- 
 gous to those which were applied to the atmosphere in obtain- 
 ing the relation of pressure to density in gaseous bodies. The 
 earth being supposed, under the conditions mentioned above, 
 to be spherical, and its density to be a function of the distance 
 from the centre, let us conceive it to be divided into concentric 
 strata of uniform and mean density, the densities of successive 
 strata varying gradatim, and let the thickness of each be 
 greater than the radius of the sphere of sensible activity of the 
 atomic and molecular forces. Then the upward resultant 
 actions F and F' of these forces at the upper surfaces of two 
 contiguous strata will vary not only as the number of atoms 
 in a given space, that is, as the densities of the respective 
 strata, but also as the proper caloric of each atom. In the 
 case of a gas the latter quantity was considered to be de- 
 rived from extraneous sources, and to be very nearly inde- 
 pendent of the emanations of caloric from the atoms of the 
 
478 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 gas. But in a liquid or solid mass such as the earth, this can 
 no longer be the case, the proper caloric of each atom depend- 
 ing almost exclusively on the emanation of waves from the 
 surrounding atoms. In proof of this law it suffices to state 
 that if the supply of caloric to the earth from extraneous 
 sources were entirely cut off, the heat of the interior would 
 for ages remain very nearly the same, dissipating very gradu- 
 ally into external space. Hence it may be supposed that the 
 proper caloric of each atom, which in this case is the excess of 
 repulsive above attractive emanations, depends wholly on the 
 waves emanating from the surrounding atoms, and is cceteris 
 paribus proportional to the number of the atoms in a given 
 space, or to the density of the medium at the position of the 
 given atom. Consequently if p and p be the densities at the 
 middle points of the two strata, we may suppose that 
 
 F= Gp x p, and F' = Gp x p'. 
 Hence since, if As be the common thickness of the strata 
 
 + ~ 
 
 , ,, it follows that 
 dz 
 
 As. 
 
 But if p and p be the pressures where the upward result- 
 ants of the atomic and molecular forces are F and F', they 
 will be the mechanical equivalents of these forces. Hence 
 
 F-F' will vary as p - p', or - As. Consequently 
 
 A., .,;,; 
 
 Ic being a certain unknown constant; and by integration 
 p = Jcp 2 + c . 
 
 It should be observed that according to this investigation 
 the proper caloric of the atoms increases with the depth below 
 the earth's surface in the same proportion as the density in- 
 creases, and that, consequently, the above relation between p 
 
THE THEORY OF MOLECULAR ATTRACTION. 479 
 
 and p includes the effect of the internal temperature of the 
 earth, it being assumed that such temperature is the exact 
 measure of the proper caloric of the atoms. 
 
 Let the relation between p and p obtained by this 
 theory be expressed relatively to the earth by the equation 
 p = fc (p 2 S z )j S being the mean density of the substances at 
 the earth's surface, a limit to which density is imposed by the 
 action, proper to each substance, of the superficial atomic and 
 molecular forces. (See p. 467). The constant &, as is shewn 
 above, takes account of the effect of temperature, and there- 
 fore has the same value for the whole interior mass, supposing 
 it to consist of homogeneous material throughout. 
 
 If, now, the density p at any distance r from the centre be 
 < (r), the attraction (A) at a given distance R will be 
 
 ~ 
 
 taken from r to r = E. This integral being expressed as 
 i|r (E) -\IT (0), if M be the Earth's mass, c its radius, and g 
 the usual measure of gravity at the surface, and if the attrac- 
 tion be referred to the same unit of measure as g, we shall have 
 
 Hence since dp = 2kpdp = Ap dB, and d . t|r (R) = 
 
 it will be readily found by differentiating, that if q 2 be sub- 
 
 stituted for -- 
 
 The integral of this equation gives the law qrp = D sin qr, 
 D being the density at the centre. After calculating the mass 
 of the earth by means of this relation between p and r, and 
 putting A for the mean density, it is found that 
 
 A 3 
 
480 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 The numerical value of ihe right-hand side of this equa- 
 tion has been ascertained by calculations in the theory of 
 the Figure of the Earth which rest on the hypothesis that 
 the relation between the internal pressure and density is 
 that which has resulted from the foregoing d priori inves- 
 tigation. The observed values of the Earth's ellipticity, and 
 of Precession and Nutation, are found to be very approxi- 
 
 mately satisfied by the theory if qc = - ; whence it follows 
 
 6 
 
 that -K- = 2,4225*. Now if the value of 8 be assumed to be 
 o 
 
 that which Mr Airy has adopted in the calculation of the 
 Earth's mean density from his experiments in the Harton 
 collieryt, viz. 2,50, the resulting value of A is 6,056. The 
 experimental determinations of the same quantity by Caven- 
 dish, Reich, Baily, and Airy, are respectively 5,448, 5,444, 
 5,675, and 6,566, the mean being 5,783. The close approxi- 
 mation of this to the theoretical value may, I think, be 
 regarded as confirmatory of the principles of the theory of 
 heat and molecular attraction from which the relation between 
 the pressure and density was deduced. 
 
 It being assumed that the equation p = k (p 2 S 2 ) gives the 
 general relation between p and p in the Earth's mass, the ve- 
 locity (V) of propagation in the material at the surface, where 
 the mean density is S, will, by a known formula, be 
 
 Hence, since Jc = -Jj^- = f , it follows that 
 
 l/3 
 ~qc( 
 
 By calculating with the foregoing values of qc and T , and 
 
 o 
 
 * See Pratt's Mechanical Philosophy, Art. 545 and 549. Also a paper on the 
 Ellipticities of the Planets which I communicated to the Philosophical Magazine 
 for September 1831 (p. 200). 
 
 t Transactions of the Royal Society for 18G6, Vol. CXLVI. p. 342. 
 
THE THEORY OF MOLECULAR ATTRACTION. 481 
 
 taking # = 32,1908 feet, and c = 3596 miles, this velocity is 
 found to be 11022 feet in a second, which is nearly the rate 
 which experiments give for propagation of vibrations in cast- 
 iron and brass. It may be remarked that this result depends 
 essentially on the determination by theory of the relation be- 
 tween^? and p. 
 
 It has been assumed in the foregoing investigations that 
 the setherial forces emanating from the ultimate atoms of 
 bodies have a mechanical equivalent, that is, admit of being 
 counteracted by forces measured by their action on masses. 
 The following considerations are intended to elucidate and 
 justify this assumption. Conceive a substance such as the 
 atmosphere, or the nucleus of the earth, to be divided into very 
 thin parallel strata, and let the density of each stratum be 
 throughout the same, and equal to the actual density at its 
 middle points, or the mean density. Then, taking an atom at 
 the common boundary of two contiguous strata, let it be kept 
 in equilibrium by the upward resultant (F) of the forces ema- 
 nating from the atoms below the boundary, the downward 
 resultant (F') of those emanating from the atoms above the 
 same, and an extraneous force (6r) acting downwards perpen- 
 dicularly to the boundary ; so that F F' = G. Further, let 
 us assume that the thickness of any stratum is equal to, or not 
 less than, the radius of the sphere of sensible activity of the 
 atomic and molecular forces which act on an atom either at 
 the upper or the lower boundary. From a previous argu- 
 ment (p. 470), it appears that such thickness may be treated 
 as an indefinitely small quantity relatively to the dimensions 
 of the earth or its atmosphere. Also, z being the distance 
 of any boundary from a fixed concentric surface, it will be 
 assumed that F=<j)(z) and F' ty (2), because F and F' are 
 functions of the density, and the density is a function of z. 
 Although on the above supposition of strata of uniform density 
 F and F' will vary gradatim, the reasoning will be ultimately 
 true, if <p (z) and ^ (z) be treated as continuous functions of z. 
 Now it is evident that the action upwards of a uniform stratum 
 
 31 
 
482 THE MATHEMATICAL PRINCIPLES OP PHYSICS. 
 
 on an atom at its upper boundary is the same as its action 
 downwards on an atom at the lower boundary. Consequently, 
 A^ being the thickness of the stratum, 
 
 ^ (3) = $ (z + As) = <f> (z} -\ ' J As. 
 
 dF 
 Hence F F' = Gf = -j- As ; 
 
 ,, , dF Gpdz dp 
 
 so that j- p Az = -^ = j 
 dz dz dz 
 
 Hence the law of equivalence between the resultants F of the 
 forces of heat and molecular attraction, and the mechanical 
 pressure p, (the existence of which law has been proved by 
 experiments) requires that /oAz should be a constant quantity. 
 This condition seems to indicate that the forces F must have, 
 as well as p, relation to a unit of mass, and that one unit has 
 to the other a fixed ratio. Since As, if it exceeds the before 
 mentioned limit, is in other respects arbitrary, it may be 
 presumed that the required condition can always be fulfilled, 
 and that each substance will have a characteristic value of 
 this constant. Let, therefore , p Az = K. Then by integra- 
 tion, K(FF }=p, it being supposed that F Q is the value 
 of F due to the action of the atomic and molecular forces at 
 the boundaries of the substance, and that p is due exclusively 
 to the action of extraneous moving forces, such as weights, 
 blows, &c. \ 
 
 I propose to conclude this department of physics with 
 some considerations respecting the different degrees of elas- 
 ticity of different gases. Taking, first, the case of a simple 
 gas, it is known from experiment that if there be a certain 
 quantity Q in an enclosed space of given temperature, and 
 the pressure be P, when an additional quantity q is intro- 
 duced into the same space, the pressure becomes P + . 
 
 Also -j would be the pressure if the space were occupied 
 
THE THEORY OP MOLECULAR ATTRACTION. 483 
 
 solely by the quantity q. It appears, therefore, that a given 
 quantity of gas enclosed in a given space produces the same 
 pressure, the temperature being given, whether or not other 
 gas of the same kind be enclosed in the same space. Expe- 
 rience also shews that if gases of different kinds occupy the 
 same space, each produces the same pressure as if the others 
 were not present, provided they do not act chemically on each 
 other. The explanation which the present theory gives of 
 these facts is such as follows. The temperature of the posi- 
 tion being given, the quantity of the repulsion-waves ema- 
 nating from an atom of a particular sort is given. Hence the 
 resulting action, on any one atom, of the repulsions from the 
 atoms of that sort, will be proportional to the number in a 
 given space, or to the density of such atoms, it having been 
 proved that each set of emanating waves produces independent 
 translatory effects. Now by what is shewn in the preceding 
 paragraph there is a certain amount of pressure which is the 
 mechanical equivalent of these atomic repulsions, which, there- 
 fore, will always be the same for the same number of atoms 
 of the same sort in a given space if the temperature be given. 
 There will, however, be a certain limitation to this law, be- 
 cause when a given number of atoms are pressed very closely 
 together the relation between the pressure and the density 
 becomes modified, approximating to that for a liquid, or a 
 solid. It has been ascertained by experiment that the pres- 
 sure of air varies proportionally to the density without per- 
 ceptible change of the law up to the pressure of 36 atmo- 
 spheres. 
 
 The quantity of the waves emanating from a given atom 
 will depend, the temperature of the position being given, on 
 its magnitude, and will therefore not be the same for different 
 simple gases, which in this theory can be supposed to differ 
 only in respect to the magnitudes of the atoms. According 
 to the mathematical reasoning the condensation of the second- 
 ary waves at the surface of an atom of radius r varies as 
 
 312 
 
484 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 r x 2 . . Therefore at a given distance R from the centre 
 
 A 
 
 7* "77? 
 
 of the atom the condensation will vary as -^ x 2 . . Hence 
 
 if two equal enclosed spaces of the same temperature contain 
 the same number of atoms of simple gases, and the radius of 
 the atoms in one be r and that of the atoms in the other r', 
 the resulting atomic repulsions, and consequently the pres- 
 sures, will be in the ratio of r 3 to r' 3 . This is also the ratio 
 of the quantities of matter, or weights, of the gases in the 
 two spaces. If, therefore, the weights are so adjusted as to 
 be the same, the pressures will also be the same, and we 
 shall thus arrive at the conclusion that the pressures of equal 
 weights of two gases contained in equal spaces are equal. 
 Thus all gases, although differing as to the magnitudes of 
 the atoms, would have the same elasticity. But this, as we 
 know, is not a law of nature, and it is, therefore, necessary to 
 enquire what other circumstances there are which may deter- 
 mine differences of elasticity. 
 
 To this enquiry I reply, in the first place, that besides 
 taking account of the waves emanating from the atoms, we 
 must also consider what their action will be on a particular 
 atom, and how it may be affected by the magnitude of the 
 atom. The complete answer to the question requires, there- 
 fore, an exact determination of the form and composition of 
 the function which has been called H^. In page 457 this 
 
 Jc k 
 quantity was assumed to be equal to \ ~ + -f , and reasons 
 
 A* A, 
 
 were added for supposing this to be its proper form. By 
 recurring to the series for 1 / in page 370, it will be seen, 
 since H^ applies only to points at the surface of the atom, 
 that the only line involved in its expression besides X is the 
 radius (c) of the atom. Consequently, as H v is of no dimen- 
 sions relatively to space, it may be put under the form 
 
THE THEORY OF MOLECULAR ATTRACTION. 485 
 
 Hence it may be inferred that the repulsive action of the 
 waves on any atom depends on the magnitude c of its radius, 
 and accordingly that the elasticities of equal weights of dif- 
 ferent gases in the same space differ in consequence of the dif- 
 ferent magnitudes of the component atoms. 
 
 Since 8^ = - -^ (&,' -^- ) , H v is greater, and by con- 
 
 A. \ A / 
 
 sequence the elasticity of a simple gas is greater, as the mag- 
 
 2k 'c 2 
 nitude of the atoms is less, if & 2 ' exceed A- , which most 
 
 A, 
 
 probably will generally be the case. 
 
 This theory seems to indicate that the forces which pro- 
 duce chemical combination and analysis, depend for the most 
 part on differences in the atomic and molecular forces as to 
 intensity and effect, due to differences in the magnitudes of 
 the atoms of the substances concerned. 
 
 The facts and laws relating to crystals, as pointing to 
 definite arrangements of the ultimate constituents of sub- 
 stances, may, I think, be regarded as being consistent with 
 the atomic hypothesis of this theory. I should refer the 
 forms of crystals and the existence of planes of cleavage, first, 
 to particular arrangements of the atoms, and then to the 
 maintenance of the atoms in equilibrium, in the positions 
 suitable to such arrangements, by the action of the atomic 
 and molecular forces; these forces differing in amount in 
 consequence of the different magnitudes of the atoms, and 
 having also different resultants in different directions on ac- 
 count of the arrangement of the atoms. 
 
 Having thus very briefly touched upon the connection of 
 the proposed theory of heat and molecular attraction with 
 chemical and crystallographical facts, I forbear to pursue the 
 subject farther, because there are various questions relating 
 to the laws that govern the action of the sether within sub- 
 stances, which require to be mathematically settled before 
 this extensive field of research can be securely entered upon*. 
 
 * I have given a " Theory of Molecular Forces" in thQ^Philosophical Magazine 
 
486 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 The Theory of the Force of Gravity. 
 
 There are the same reasons for theoretical investigations 
 respecting the quality and laws of gravity, as for theories of 
 light and heat, all three being physical agencies known to 
 us by their effects, and alike capable of transmission from 
 distant bodies; and so far as evidence has been given that 
 light and heat are transmitted by an intervening medium, 
 a presumption is established that gravity is transmitted by 
 the same medium. In short, the general theory of physical 
 force which I have thus far advocated, would have no foun- 
 dation to rest upon if in the nature of things it were inappli- 
 cable to the force of gravity. (See the preliminary discus- 
 sion in pages 356 362.) It is, therefore, essential to the 
 general argument to prove that it applies to this as to other 
 forces. With this view I proceed at once to refer the action 
 of gravity to the motion and pressure of the getherial medium, 
 employing for the purpose results obtained by the previous 
 solutions of hydrodynamical problems. The extension given 
 in the course of this work to the mathematical reasoning 
 appropriate to this question, will give the means of discussing 
 it more accurately and completely than I did in my first 
 attempt contained in the Philosophical Magazine for Decem- 
 ber 1859. 
 
 Before, however, entering upon this investigation I pro- 
 pose to advert to a step of the mathematical reasoning which 
 without farther explanation might seem to have been taken 
 arbitrarily. I refer to the circumstance that instead of the 
 equation (35) in page 260, which from the mode of its derivaT 
 tion might be thought to be generally applicable to motion 
 symmetrical about an axis, I have employed for the case of 
 
 for February 1860, which differs from that now produced chiefly in what relates 
 to the details of the mathematical reasoning. The conclusion towards the end of 
 the Article, that " the elasticity of a simple gas is greater the smaller the atoms," 
 depends on assuming that condensation propagated from a centre varies inversely 
 as the distance from the centre, instead of which law I have here adopted (for the 
 reasons stated in pages 199 and 253) that of the inverse square. 
 
THE THEORY OF THE FORCE OF GRAVITY. 487 
 
 the incidence of waves on a sphere, whether fixed or move- 
 able, the equation (36) in page 279, obtained from the other 
 by differentiating with respect to 0. And similarly in the 
 solutions of the same problems to terms of the second order, 
 I use the equation (e) in page 443, derived, by differentiating 
 with respect to 0, from an equation (at the bottom of page 
 442) which might be supposed to be of general application. 
 The reason that was assigned for this step is, that the equa- 
 tions (36) and (e) give the means of satisfying certain con- 
 ditions which do not admit of being satisfied by the other 
 equations. But this is a reason a posteriori, and if it be 
 valid, we ought to be able to give an antecedent reason for 
 these differentiations with respect to 0. This, I conceive, may 
 be done as follows. 
 
 Taking account, at first, only of terms of the first order, 
 we have, from the equations at the bottom of page 442, 
 
 . ._ 
 
 W t0 ) a'W~ 
 
 If now it be required to form the equation of which the un- 
 known function W is the principal variable, it is only neces- 
 sary to differentiate the first of these equations with respect 
 
 /i TXT i 11 i d.rd> d.r*W 
 
 to 6. We shall thus get, since ^f > 
 
 au do 
 
 which equation is identical with (e) in page 443 deprived of 
 its last term, P being equal to r*W. Now since the inte- 
 gration of this equation determines generally the function 
 that W is of r t 0, and , the equation is one that is necessary 
 for the general solution of problems in which the motion is 
 symmetrical about an axis, and consequently the differentiation 
 with respect to by which it was obtained was a necessary 
 operation. 
 
488 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 The same argument holds good, in the cases of disturb- 
 ances of the fluid by a fixed or moving sphere, when terms 
 of the second order are included, provided the equations be 
 applied only to the parts of the fluid contiguous to the sphere. 
 For under this restriction U' 2 in the last term of (e) may be 
 
 omitted, because when the sphere is fixed, 17=0 and -y- =0, 
 
 d 2 . U 2 
 
 so that ' 7 vanishes, and when the sphere has a small 
 auat 
 
 vibratory movement, the same quantity is of the third order. 
 Thus the equation contains only the variables W, r, 6, and t, 
 the principal variable being W. 
 
 The equation of which a is the principal variable is formed 
 by differentiating the first of the foregoing equations with 
 
 respect to t, and substituting for from the third. These 
 operations give 
 
 d*. a-r 1 fd z . or . d . 
 
 The arbitrary function in the last term may be got rid of 
 in two ways. First, by differentiating with respect to 6, 
 
 whereby an equation is produced in which if for ' its 
 
 d.r*W 
 
 equivalent -- '-^-^ be substituted, the result is the same as 
 a at 
 
 that obtained by differentiating the above equation (77) with 
 respect to t. Hence by integrating this differentiated equa- 
 tion, the final result is identical with the equation (rf), no 
 arbitrary function of r and 6 being required to be added, 
 because by hypothesis no part of W is independent of t. 
 
 Again, the arbitrary function in the above equation is 
 
 eliminated by substituting a + / for <r, the equation then 
 becoming 
 
 d\<r'r 1 d\a'r d.a'r A d\ a'r 
 
THE THEORY OF THE FORCE OF GRAVITY. 489 
 
 which is identical in form with the equation (35) in page 260. 
 It has already been stated (p. 284) that the integration of 
 the equation (35) only gives information respecting those 
 motions of a compressible fluid which, to the first approxi- 
 mation, are the same as if it were incompressible, in which 
 case the term containing a' 2 in the denominator becomes 
 insignificant, and the results of the integration apply to all 
 parts of the fluid. In this case also, the equation above 
 remains the same if cr be put in the place of cr'. But in order 
 to include with the motions of that class such motions of the 
 first order as are peculiar to a compressible fluid, the more 
 comprehensive equation (?;) is indispensable, although, as not 
 taking account of transverse action, it strictly applies only to 
 parts of the fluid contiguous to the sphere. For analogous 
 reasons the equation (e) is indispensable when the approxi- 
 mation is extended to terms of the second order. 
 
 The foregoing argument having shewn that the equation 
 (e), which was employed in the investigation (contained in 
 pages 443 452) for determining the acceleration of a sphere 
 acted upon by the vibrations of an elastic fluid, is the only one 
 proper for this purpose, we may now proceed to consider in 
 what way the results of that investigation are applicable in 
 the theory of gravitating force. 
 
 We have hitherto had under consideration the waves of 
 atomic repulsion and the waves of molecular attraction, and 
 it was argued that the latter might result from compositions 
 of the former, and that in that case the values of X would be 
 much larger for the composite waves than for the components. 
 (See pages 463 and 464). It may be assumed that there are 
 by original generation gradations of the values of X for the 
 primary waves, and, as a consequence, gradations of the values 
 of X for the compound waves, but the mean of the second set 
 of values will be much larger than the mean of the first. On 
 considering what must be the effect of the composition of all. 
 the secondary waves in the interior of a large fluid or solid 
 mass, it will be seen that besides these two sets of primary 
 
490 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 and composite waves, there will be successive composite sets 
 each of much greater breadth than the preceding one. For 
 as the waves emanating from the individual atoms of a cubical 
 molecule form by their composition a regular series of broader 
 waves, which may be regarded as emanating from the centre 
 of the molecule, so this new class of waves emanating from 
 the centres of a vast number of molecules may by composition 
 form a regular series of waves of a third order, characterized 
 by a set of larger values of X. The collection of the whole 
 number of cubical molecules required for producing this effect 
 may be a cubical molecule of the second order, of much larger 
 dimensions than that of the first order : and so on. These 
 successive orders of a3therial waves are analogous to the dif- 
 ferent orders of waves at the surfaces of oceans, which are 
 seen to coexist in every gradation from the least ripple to the 
 ocean-swell ; and very probably also the modes of generation 
 in the two cases are strictly analogous. 
 
 Now it is shewn by the reasoning concluded in page 453 
 that the translation of an atom by the action of the waves is 
 in the direction of their propagation or the contrary direction, 
 that is, the action is repulsive or attractive, according as 
 
 -- 2 J is positive or negative. Assuming for the 
 
 2H 
 
 present that 1 -- * is positive, and, as before, that 
 
 o 
 
 atomic repulsion is to be referred to the first class of waves, 
 for which the values of X are very small and H^ is conse- 
 quently positive, and molecular attraction to waves of the 
 second class having values of X which make H^ negative. 
 Reasons having been given in pages 428 432 for the sup- 
 position that this factor changes sign, and the action of the 
 waves becomes neutral, when X has some value included among 
 those of the waves of light, it may be inferred that for the 
 third and higher classes of waves /// is always positive. So 
 
THE THEORY OF THE FORCE OF GRAVITY. 491 
 
 far, therefore, as this factor indicates the larger classes of 
 composite waves in the interior of a mass of great extent 
 would have a repulsive effect. We have next to consider 
 how this inference may be modified by taking account of the 
 
 factor 1 ~- 2 . 
 
 o 
 
 It is evident from the expression for the acceleration of 
 an atom at the top of page 455, viz., 
 
 2A + 1 V 3 / 4a" 
 
 that if 1 becomes negative for large values of X, the 
 
 3 
 
 above-mentioned repulsion is changed to attraction. It is, 
 therefore, important to enquire whether the quantity H z may 
 have such values as would make that factor negative. It 
 may be remarked that H^ unlike H^ occurs only in terms of 
 the second order, and that its value, so far as it depends on 
 the parts of the motion and condensation which are given to 
 terms of the first order by the expressions for W and <T O in 
 page 286, is unity, which may be regarded as the first ap- 
 proximation to its general value. 
 
 As the analytical expression for H z has not been ascer- 
 tained, we are not prepared to give a direct and complete 
 answer to the above-mentioned enquiry. But certain dyna- 
 mical considerations, which I am about to adduce, will, I 
 think, furnish the means, without the guidance of an ana- 
 lytical formula, of arriving at some conclusions applicable 
 to the theory of gravitating force. To make the . following 
 argument clear, it will be necessary to advert, in the first 
 place, to results obtained in page 264, which shew that when 
 a sphere oscillates in fluid which is regarded as incompressible, 
 the velocity with which the fluid in contact with the sphere 
 crosses a plane through the centre perpendicular to the direc- 
 tion of its oscillation, is just half that of the sphere and in 
 the opposite direction, and that the quantity of fluid which 
 passes that plane is just as much as the sphere displaces. 
 
492 THE MATHEMATICAL PKINCIPLES OF PHYSICS. 
 
 As already stated in page 272, I formerly conceived that 
 under these circumstances a sphere of radius b moving with 
 the velocity V would impress instantaneously on any fluid 
 particle whose polar co-ordinates are r and the velocity 
 
 2- cos 6. I now admit that this velocity, although im- 
 pressed, is not that which effectively takes place, the actual 
 velocity being due to a cause which till lately I had not 
 recognised, and which I have not seen noticed in any writer 
 on hydrodynamics. This cause and the mode of its operation 
 may be understood by referring to a result obtained in page 
 256, from which it appears that if central motion could be 
 such as I supposed, an infinite amount of momentum might 
 be generated in a finite interval of time. As this cannot be 
 the case, the fluid mass itself by its inertia forms a barrier 
 to the impressed movement, producing the same kind of effect 
 as that which would result from enclosing the fluid in im- 
 passable solid boundaries. The consequence is that just as 
 much fluid is caused to flow backwards as the sphere dis- 
 places by a forward movement. In like manner, if the sphere 
 be fixed and the fluid be incident upon it, since the velocity 
 impressed by the reaction of the sphere would in exactly the 
 same way be counteracted by the inertia of the fluid, the 
 motion is actually such as it would be if the fluid were con- 
 tained within distant solid boundaries partaking of the motion 
 of incidence; and, consequently, as much fluid passes the 
 plane through the centre of the sphere perpendicular to the 
 direction of the incidence, as would have passed the same 
 plane if the sphere had not been encountered*. (See in pages 
 264 and 284.) 
 
 It is now to be observed that this reaction of the fluid 
 mass is necessarily accompanied by condensation and pres- 
 sure, and the acceleration of the sphere depends partly on 
 
 * By attributing the effects here described to reaction due to the inertia of the 
 fluid mass, an important principle is introduced, which, as I shall have occasion to 
 shew, admits of other applications. 
 
THE THEORY OF THE FORCE OF GRAVITY. 493 
 
 such pressure. In fact, the term containing H z is the ex- 
 ponent of the part of the acceleration due to this cause, as 
 may appear from the following considerations. When a series 
 of waves is incident on a fixed or moving sphere, there will 
 be an accelerative action on the sphere due to that distribu- 
 tion of condensation about its surface which results from the 
 propagation of condensation and the accompanying transverse 
 action. It is this part of the accelerative force, whether pro- 
 ducing vibratory or translatory motion, which is expressed 
 by the formula in page 452 when ff z is supposed to vanish. 
 (See page 455.) Assuming that the remaining part is due 
 to the pressure arising from the counteraction, by the inertia 
 of the fluid mass, of the momentum of the fluid which crosses 
 the plane that passes transversely through the centre of the 
 sphere, it may be asserted generally that the sum of the 
 pressures thus produced on the surface of the sphere in a 
 given time is proportional to the sum of the momenta of the 
 fluid which passes that plane in the same time: also, that 
 the direction of the resultant of the pressures is at each in- 
 stant contrary to that of the movement of the fluid across the 
 plane. But if during the transit of the condensed portion of 
 a wave the sum of the momenta be X . M W(l + cr), during the 
 transit of the rarefied portion the sum will be 2 . MW(1 cr), 
 taking effect in the contrary direction, M(\ + cr) and M (1 cr) 
 being elementary masses of the same size, which cross the 
 plane in opposite directions with the same velocity W. 
 Hence, V being the velocity of the incident waves and <7 
 
 V 
 the corresponding condensation, since W=V+ -5-5- (p. 284), 
 
 and cr may be taken with sufficient approximation to be equal 
 to <r , it follows that the former sum of momenta exceeds the 
 other by a quantity which, being equal to 2 2 . MWv, varies 
 as Fcr . that is, as F 2 , because <T O varies as V. Consequently 
 there will be a residual pressure, varying as the square of the 
 velocity of the fluid, and producing a residual acceleration 
 always in the direction contrary to that of the propagation of 
 
494 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the waves. It may, therefore, be concluded that so much of 
 the acceleration of the motion of translation as is due to this 
 cause is expressed by that term of the formula 
 
 3#; / 2H 2 \m' 2 
 2 A + 1 V 3 ) 4o' 
 
 which contains H 2 , and which in the case, now supposed, of 
 HI being positive, is always negative. (See p. 455.) 
 
 It will facilitate the remainder of the enquiry respecting 
 the composition and value of the function H 2 , if we now re- 
 capitulate the antecedent steps of the argument. In page 286 
 the following expression to the first approximation is obtained 
 for the condensation (<j) at any point of the surface of a sphere 
 on which a series of waves is incident : 
 
 3b dV 5* 
 
 the values of V and <r being, as usual, such that 
 V= acr Q = m sin q (at + c ). 
 
 This expression does not include the effect of transverse action. 
 By the reasoning in page 295, and the additional considera- 
 tions in page 457, it appears that transverse action produces 
 at the surface of the sphere the same, effect as that which 
 would result from the incidence on the sphere in the opposite 
 direction of another series of waves having the same phase 
 and the same value of X as the first, but a different value of m. 
 Under these circumstances the foregoing equation will be 
 changed, on the principle of the coexistence of small vibra- 
 tions, to the following : 
 
 cr being still the value of cr where cos 6 = 0. The reasoning 
 does not determine the values of H^ and H 2 ; but it will 
 readily be seen, since the term containing cos 6 has different 
 signs, and that containing cos 2 9 the same sign, for the two 
 opposite hemispherical surfaces, that H^ must be less than 
 
THE THEORY OF THE FORCE OF GRAVITY. 495 
 
 unity, and may even be negative, and that H 2 will always be 
 positive and greater than unity. 
 
 Again, on obtaining the expression for the accelerative 
 action on the moving sphere (p. 452), it was found that H^ 
 occurs only in terms of the second order, while H^ (which 
 differs insensibly from H^) is a factor of terms both of the 
 first and second orders. By integrating the expression for 
 the accelerative force of the sphere in page 452, we get 
 
 velocity of sphere = + ^^ j V+ ( 1 - 2 -f) g} 
 
 , , ^ 
 
 = C+ 2ATT ( l ~~ f ) 17+ P enodlc terms ' 
 
 This result may be interpreted as meaning that whatever 
 arbitrary constant velocity G the sphere may have in ad- 
 dition to its vibratory motion, it has also the constant velo- 
 city expressed by the term containing ra' 2 . In fact, this velo- 
 city exactly accounts for the excess of the excursion of the 
 sphere in the positive direction above that in the negative 
 direction (see p. 453). In page 454 I have argued, on the 
 ground that the action of the vibrations of the fluid on the 
 sphere is not altered by any uniform motion the sphere may 
 have (and the same argument applies to slowly accelerated 
 motion), that the translatory action of the waves is continu- 
 ous, being exactly like that of a uniform accelerative force 
 the amount of which is the above-mentioned constant velocity 
 referred to an appropriate unit. 
 
 In the application I now propose to make of that expres- 
 sion for the translatory accelerative force, X is supposed to be 
 so large that H l is reduced to its first term & 1? which is always 
 positive and differs little from unity. Consequently the ac- 
 celeration would be the same for waves of all breadths, if m 
 be the same, unless the case should be that H 2 is a function 
 of X. It will, I think, appear from the following considera- 
 tions that H 2 is such a function. 
 
 By reverting to the foregoing argument (p. 493) from 
 
496 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 which it was concluded that the value of H z is determined by 
 the excess of the sum of the momenta of the condensed por- 
 tion of a wave above the sum of the momenta of the 
 rarefied portion, it will be seen that the reasoning includes 
 the case of such motion as that which resulted from the solu- 
 tion of Example VI. (p. 279), in which transverse action is 
 not taken into account. The expression for or CT O obtained 
 in page 286, shews that for this case H 2 = I. Consequently 
 as that solution may be regarded as the first approximation 
 to the more complete one embracing transverse action, we 
 may infer that the first approximation to the value of H 2 is 
 unity, and that the remainder of its value is wholly due to 
 transverse action. These conclusions are in accordance with 
 the supposition made in page 457, namely, that the effect of 
 transverse action may be represented by that of an additional 
 series of waves having a certain ratio to the original waves, 
 but propagated in the opposite direction. In short, it is not 
 difficult to see that any transverse action tending to increase 
 the convergency of the lines of motion, on the further side of 
 the sphere, towards the axis about which the motion is sym- 
 metrical, would tend also to increase both the condensation 
 and the rarefaction of the fluid in contact with the sphere. 
 Thus the eifect due, as above stated, to the difference of the 
 momenta of the condensed and rarefied portions of a wave is 
 increased by the transverse action, and for this reason the 
 value of H z is greater than unity. Also, as we have had 
 reason to conclude that the part of H^ which takes account of 
 transverse action is a function of X, it may be presumed that 
 the part of H 2 which relates to the same action is a function 
 of the same quantity. But in the present state of the mathe- 
 matical theory of the motion of fluids we are not prepared to 
 determine the exact form of this function, nor can we have 
 recourse to the same kind of guidance as that which was de- 
 rived relatively to H^ from the theory of the dispersion of light*. 
 
 * I take occasion here to remark that the mathematical processes emploj'ed in 
 this work for the solution of hydrodynamical problems are limited in their applica- 
 
THE THEORY OF THE FORCE OF GRAVITY. 497 
 
 For the same reason we have not the means of directly 
 determining whether or not the value of H 9 depends on the 
 magnitude of the sphere. There are, however, certain con- 
 siderations which seem to indicate that the radius of the 
 sphere is not involved in the complete expression for that 
 quantity. The formula for W in page 442 shews that at 
 points for which cos 6 = the term containing H 2 vanishes, 
 and the formula for cr in the preceding page shews that for 
 the same points cr = <T O . Since, therefore, H t = 1 in the case 
 of large values of X, which is that now under consideration, 
 it follows that the motion of the fluid in contact with the 
 sphere, across the transverse plane through the sphere's centre, 
 is the same that it would be if there were no transverse action. 
 Since also, for a reason which is independent of transverse 
 action (see p. 492), the total quantity of fluid which passes that 
 plane must be the same that it would have been if the sphere 
 had not been an obstacle to the movement, it would seem 
 that at all points the motion across the plane is the same 
 whether or not there be transverse action. Hence, since for 
 that motion the value of ZT 2 , in the case of no transverse 
 action, is independent of the radius of the sphere, it may be 
 presumed that this is true also for the same motion when the 
 value of that quantity is inclusive of transverse action. It 
 this reasoning be good, H 9 may be supposed to be a function 
 
 of X only, or rather of the interval of time during which the 
 
 accelerative action of the condensed and rarefied portions of a 
 given wave is in operation. The formula for the acceleration 
 of the sphere (at the top of page 455) would in the case of 
 
 tion because they are restricted either to cases in which udx + vdy + wdz is an 
 exact differential, or to those in which the fluid is compelled to move in given 
 courses. In order to solve problems by a general process, and to carry the solu- 
 tions to terms of the second order, it would be necessary, I think, to suppose that 
 differential function to be integrable by a factor, and to include among the un. 
 known quantities to be determined by integration the factor contained in the 
 equation (1) in page 175. This process might be adequate to take account of what 
 1 have called transverse action. See the remarks at the end of page 179. 
 
 32 
 
498 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 large waves be altogether independent of X, unless ff 2 be a func- 
 tion of this quantity. It seems, therefore, reasonable to admit 
 that the acceleration, since it is continuous, is some function of 
 the interval occupied by the translatory action of individual 
 waves, and, consequently, that H 2 is a function of X. Accord- 
 ing to these views, if r be a certain fixed interval, we have 
 
 It may be supposed that for the forces of heat and mole- 
 cular attraction this unknown function is inappreciably small 
 on account of the small values of X, and that it increases in 
 value as X increases : not, ' however, to an indefinite extent, 
 because when X is extremely large the motion of waves 
 approximates to that of a stream, which, as we have seen 
 (p. 454), would produce no acceleration of the sphere. In 
 order that the acceleration may be in the direction contrary 
 to that of the propagation of the waves, it is necessary that 
 
 3 
 
 1 -- - should be negative, or H z greater than - , and there- 
 3 A 
 
 fore F(-T\ greater than -. 
 
 The foregoing inferences may now be brought into com- 
 parison with the known laws of the force of gravity. Although 
 I do not profess to have given by the preceding reasoning 
 & priori demonstrations of those laws, I may claim to have 
 succeeded in obtaining theoretical results which are not in- 
 compatible with them, and to have adduced primd facie evi- 
 dence that by means of more perfect and more comprehensive 
 mathematical reasoning they might, on the same principles, 
 admit of being demonstrated. 
 
 (1) The acceleration of the spherical atom by the action 
 of given setherial undulations was proved to be uniform, and 
 independent of acquired velocity. The force of gravity, as 
 is known by observation, accelerates according to the same 
 law. 
 
THE THEORY OF THE FORCE OF GRAVITY. 499 
 
 (2) If we assume that F(-J-) is greater than -, the 
 
 waves produce acceleration towards their origin, just as the 
 force of gravity attracts towards the body from which it 
 emanates. 
 
 (3) If it was rightly argued that that function is inde- 
 pendent of the radius of the atom, all atoms would be equally 
 accelerated, because, ex hypothesi, atoms differ from each other 
 only in respect to magnitude. This result is in accordance 
 with the observed law, that gravity acts in the same manner 
 and Degree on bodies whatever be their atomic constitution. 
 
 (4) The independent action of different series of undu- 
 lations from different sources, which is demonstrated by the 
 reasoning concluded in page 239, corresponds to the fact that 
 different bodies acting by gravity on the same body at the 
 same time produce independent effects. 
 
 (5) The acceleration of the atom having been shewn to 
 be a function of m and X only, the same law of variation 
 with distance from an origin as that obeyed by gravity, 
 namely, the law of the inverse square of the distance, results 
 from the reasoning contained in Corollaries II. and III. in 
 pages 308 and 309. 
 
 (6) The action of gravity is independent not only of the 
 atomic constitution of bodies, but also of the state of aggre- 
 gation of their atoms, being exactly the same whether the 
 bodies be in the aeriform, or liquid, or solid state. The 
 following supplementary considerations are required for shew- 
 ing the accordance of the theory with this law. 
 
 We found (p. 454) that a moveable sphere submitted to 
 the action of a uniform stream receives no acceleration. Con- 
 versely, a uniform stream incident on a fixed sphere is not 
 retarded by it, as much fluid flowing in a given time in the 
 direction of incidence as would have flowed if the course had 
 not been interrupted by the sphere. The result would be 
 the same if the stream were incident upon a collection of any 
 number of small spheres, provided the spaces occupied by the 
 
 322 
 
500 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 spheres be very small compared to the intervening spaces. 
 We have had reason to suppose (p. 410), that this condition 
 is fulfilled by the constituent atoms of all bodies. Thus a 
 stream of the gether might pass through an aggregation of 
 atoms constituting an aeriform, liquid, or solid substance, 
 without sensible diminution of the average rate of flowing*. 
 
 Hence as the setherial undulations to which the force of 
 gravity is in this theory ascribed, are assumed to be of very 
 large breadth, and as the motion of the particles throughout 
 the greater part of such undulations is nearly the same as 
 motion in a stream, it may be inferred that the gravity-un- 
 dulations suffer very little retardation in their passage through 
 bodies, and that it is possible they may not be sensibly re- 
 fracted or reflected on entering them, or issuing from them. 
 
 Having remarked that this conclusion, which seems to be 
 legitimately arrived at by the above argument, is not dedu- 
 cible from the formula (ft) for refraction in page 426, I was 
 led to reconsider the reasoning by which that formula was 
 obtained, and I found that as far as regards the determination 
 of the constant K in pages 365 and 366, it requires to be 
 supplemented by other considerations. In that investigation 
 a circumstance was overlooked which has now been fully 
 recognised, namely, that in proportion as the velocity of the 
 fluid tends either to be uniform, or to be like that of incom- 
 pressible fluid, the mean reaction of the atoms tends to vanish. 
 In fact, in those cases the mean flow of the fluid is not im- 
 peded by the atoms, so that the integral /< (a) da. in page 366 
 is zero. 
 
 This may be more distinctly shewn as follows. Taking 
 'the case of a series of waves incident on a fixed sphere, and 
 supposing that there is no transverse action, or that the 
 
 * Conversely, the bodies of the Solar System, their motions being nearly uni- 
 form, suffer no sensible retardation from the aether ; nor do they carry along with 
 them any considerable portion of it, very nearly as much flowing backwards as they 
 displace by their forward movements. Comets that are rapidly accelerated, as, for 
 instance, Encke's Comet, might be sensibly retarded. 
 
THE THEORY OF THE FORCE OF GRAVITY. 501 
 
 motion is the same as that of incompressible fluid, it may be 
 found (as in page 364), that the velocity due to the reaction 
 of the sphere, resolved in a direction parallel to that of the 
 propagation, is 
 
 Vb* f 3sin 2 0\* 
 
 If with this velocity the quantity of fluid be calculated 
 which in the small time $t flows across a plane perpendicular 
 to the axis of the motion at any given distance from the centre 
 of the sphere, it will be found to be zero at all distances 
 greater than b, and at any distance b cos /3 to be Trb 2 sin 2 /3F&; 
 that is, an amount which just makes up for what is inter- 
 cepted by the sphere. The result would be the same for any 
 number of atoms constituting a thin slice of any substance, 
 the spaces occupied by the atoms being supposed to be very 
 small compared to those which intervene. Consequently, in 
 the cases now considered, the mean reactionary velocity of all 
 the atoms of the slice would be zero, and we should thus have 
 K 0. By taking account of the values of U and W in page 
 287, given by the second part of the integration, I find that 
 by these velocities there is a residual flow of the fluid across 
 a transverse plane, but that the amounts are equal in opposite 
 directions at equal distances from the centre of the sphere, so 
 that the mean effect of all the atoms of a slice would be zero. 
 Thus we can come to no other conclusion than that the di- 
 minished rate of propagation in the interior of media is due to 
 transverse action, (which the foregoing reasoning does not take 
 into account,) and that for this reason the factor K is a func- 
 tion of X. Without an exact mathematical theory of trans- 
 verse action, the form of this function cannot be known ; but 
 we may presume that its value is less as \ is greater, since it 
 vanishes when X is indefinitely great. Also, without reference 
 
 * In the formula at the top of page 365 the expression in the brackets is 
 1 - * t through mistake : on which account the inferences that follow respect- 
 ing the sign of the formula, and the mean reactionary velocity, are erroneous. 
 
502 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 to the exact form of that function, the preceding argument 
 justifies the conclusion that for large waves, such as those to 
 which the theory attributes the force of gravity, K may be 
 assumed to be so small that $ I very nearly, and refraction 
 is, consequently, of very small amount*. 
 
 It is, however, possible that gravity -undulations may be 
 subject in some degree to refraction. For supposing the 
 undulations proceeding from the earth's interior to be slightly 
 refracted on issuing from it, either on account of the form of 
 the surface, or the different densities of the materials of which 
 the superficial crust is composed, the only effect of the refrac- 
 tion would be to cause local irregularities of the gravitating 
 force; and it is well known that such irregularities are found 
 by observation actually to be of measurable amount at the 
 earth's surface. Ivory has called attention to the fact that in 
 the instances of six small islands, the force of gravity, as 
 measured by pendulum experiments, exceeds the normal 
 amount by quantities far too large to be attributed to errors of 
 observation, being in every case considerably greater than the 
 anomalies exhibited by observations at thirty-four other sta- 
 tions, which for the most part were continental. With re- 
 spect to one instance he remarks as follows : " At the Isle of 
 France, the excess of gravity is no less than 10 vibrations a 
 day ; yet this is a small island, surrounded on all sides by an 
 extensive sea; and we might infer, a priori, that a pendulum 
 placed upon it very little above the level of the sea, instead of 
 being accelerated, would be retarded by the great defect of 
 density in the waters of the ocean." (Philosophical Maga- 
 zine, Vol. ill. for January June, 1828, p. 171 173). It is 
 evident that, on the hypothesis of some degree of refraction of 
 gravity-undulations, there would be produced by refraction 
 
 * If the theories of refraction and dispersion which I have proposed hold good, 
 K must be such a function of \ that its changes of value are not considerable for the 
 values of A. within the range of those that belong to phenomena of light. This 
 being admitted, the reasoning on which those theories rest remains unaffected 
 by the above considerations, or, at least, is very approximate. 
 
THE THEORY OF THE FORCE OF GRAVITY. 503 
 
 under the circumstances above described, an augmentation of 
 the condensation of the waves towards the centre of the 
 island, and that the increment of gravity might thus be ac- 
 counted for. 
 
 It has been urged (p. 490) that the ultimate gravity- 
 molecules are composed of a very large number of the mole- 
 cules of the attraction of aggregation, and must consequently 
 very much exceed the latter in magnitude. Yet they can 
 only be regarded as elementary parts of the attracting bodies; 
 otherwise no definite results could have been obtained from 
 experiments made with the comparatively small masses used 
 by Cavendish, Reich, and Baily for determining the mean 
 density of the earth. If the diameters of the spheres they 
 experimented with were very much less than the breadths of 
 the gravity-undulations, so far as this .theory indicates the 
 results could not have been considerably influenced by this 
 circumstance; but it might still be a question, considering the 
 magnitudes of the molecules, whether the small size of the 
 masses may not have had the effect of diminishing to some 
 amount the determinations of the density. 
 
 According to the foregoing theory of gravity, the quantities 
 of matter in masses differing in size and in kind are correctly 
 measured by their weights, because given gravity-undulations 
 accelerate all atoms equally under all circumstances. Hence 
 the comparative densities of substances of different kinds 
 are correctly given by dividing the weights of masses by 
 their magnitudes. It appears also from the theory that dif- 
 ferent masses of the same substance would attract propor- 
 tionally to the quantities of matter they contain, and such 
 attraction may consequently be taken as a measure of quan- 
 tity of matter. But the constant ratio of this measure to the 
 other can only be known by experiment; neither is it possible 
 to decide without experiment whether that constant be the 
 same for substances of different kinds ; that is, whether the 
 same weights of different substances attract the same body 
 equally. 
 
504 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 There is one other point relating to the force of gravity 
 which remains for consideration. If large masses act upon 
 each other solely by a force of attraction which varies through 
 all space according to the law of the inverse square, there 
 would be no security for the stability of a system of bodies 
 such as the stars constituting the Milky Way, or the compo- 
 nents of a resolvable cluster of stars. However small the 
 mutual attractions between the constituent bodies might be 
 rendered by distance, in the course of ages there must be a 
 general movement tending to produce a central region of 
 maximum density. But the forms of the Milky Way and of 
 certain resolved nebulas, together with the extremely irre- 
 gular distribution of the component stars, stand in direct con- 
 tradiction to any such tendency. It would appear to be abso- 
 lutely necessary for the stability of these systems that there 
 should be a repulsive action conjoined with one that is attrac- 
 tive. According to the principles of the present theory, the 
 solution of this difficulty may be supposed to be such as 
 follows. 
 
 It has already been argued (pages 489 and 490) that the 
 broader ae-therial waves are composed of a vast number of 
 others of much inferior breadth, and that besides the waves of 
 atomic repulsion and molecular attraction, there may be suc- 
 cessive composite sets each of much greater breadth than the 
 preceding one. Let us, therefore, suppose that between the 
 waves of molecular attraction and those of the attraction of 
 gravity, there is an intermediate set constituting the more 
 immediate components of the latter. It is conceivable that 
 the values of A, for this set may be so large that the factor 
 H l is reduced to k lt or to unity, and yet so much below the 
 
 values for gravity-undulations that F( ) becomes less than 
 
 %H 
 
 J, and consequently 1 ^positive. (See pages 490 and 498). 
 
 o 
 
 In that case these intermediate waves would act repulsively. 
 Now it scenes allowable to suppose that the composite waves 
 
THE THEORY OF ELECTRIC FORCE. 505 
 
 emanating from the Sun, (regarded as one of the components 
 of the Milky Way), while they act as gravity-waves within 
 limits comprehending all the bodies of the Solar System, are 
 resolved at the distance of the nearest star into the above- 
 mentioned order of repulsive waves by the mere effect of 
 the divergence of the axes of propagation. Thus if these 
 components be of such magnitude that they can traverse 
 substances without undergoing retardation, it will follow that 
 the Sun may by that means act repulsively on neighbouring 
 stars. According to these views it would be a general law, 
 that neighbouring stars are mutually repellent. This repul- 
 sion might then be counteracted by another order of gravity- 
 undulations formed by the composition of waves from all the 
 individuals of the whole stellar system. The stability of 
 the heavens would thus be effected in a manner analogous 
 to that in which, according to the theory of the combined 
 action of the atomic and molecular forces, the constituent 
 atoms of bodies are held in positions of equilibrium. It may 
 be presumed that the conjoint action of the cosmical repul- 
 sions and attractions would produce periodic motions of the 
 stars, so that the theory would receive much confirmation 
 if, as there is reason to expect, the existence of stellar proper 
 motions that are vibratory or periodic should be fully esta- 
 blished by astronomical observation. 
 
 The Theory of Electric Force. 
 
 The application of the term "Electricity" has of late 
 years received great extension, insomuch that Treatises on 
 Electricity, as, for instance, that of M. De La Rive, are made 
 to embrace, together with the phenomena of frictional electri- 
 city, those of galvanism, electro-magnetism, magnetism and 
 diamagnetism. The latter classes of phenomena are in no 
 manner dependent on friction, so that in calling them " elec- 
 tric" a new sense is given to the word altogether different 
 from that which it derived from the effects of rubbing a piece 
 of amber. This extension of its meaning may be traced to 
 
506 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the practice, usually adopted by experimentalists, of deducing, 
 as far as possible, general inferences or laws, from the results 
 of their experiments. It was too obvious to be overlooked 
 that the different kinds of phenomena above mentioned had 
 something in common. The existence, as matter of experi- 
 ence, of this common quality, or point of relation, was 
 expressed in the use thus made of the term electricity; but 
 those who employed the word in this comprehensive sense 
 did not profess to explain the exact modus of the correlation, 
 or to indicate any common fact by which the various phenomena 
 are connected. As far as I am aware no general conception 
 of this kind had been published when I first proposed 
 the theory which I still advocate, namely, that these physical 
 forces, (as well as light, heat, and gravity), all result from 
 motions and pressures of the setherial medium, differing in 
 their effects under different circumstances, according to laws 
 ascertainable by the application of hydrodynamical theorems. 
 This view I endeavoured to maintain in four Articles commu- 
 nicated to the Philosophical Magazine ; the first, on the Force 
 of Electricity, in the Number for October, 1860 ; the second, 
 on Galvanic Force, in the number for December, 1860 ; and 
 the two others, on Magnetic Force, in the numbers for Janu- 
 ary and February, 1861. The theories of these forces which 
 I am now about to propose, are fundamentally the same as 
 those which are there advanced, but I shall have occasion to 
 point out certain corrections and modifications by which I 
 hope to render the general physical theory more complete. 
 
 The first requisite of a general theory of the above-named 
 phenomena is to account for the qualities which they have in 
 common, and which have induced experimentalists to place 
 them all under the head of Electricity. This demand is, 
 I conceive, met by the present theory, in so far as it ascribes 
 all the phenomena that have been called electric to the motions 
 and pressures of a single fluid, the sether. But in order to 
 shew specifically how the phenomena may be thus accounted 
 for, it is important to distinguish those that are different in 
 
THE THEORY OF ELECTRIC FORCE. 507 
 
 kind by different appellations. For this reason I propose to 
 include in the theory of Electric Force only those phenomena 
 in the production of which friction is a necessary condition. 
 The theory of Galvanic Force will embrace phenomena that 
 are due to simple contact, or chemical action, or the action of 
 heat; and the theory of Magnetic Force will be limited to 
 phenomena depending on natural, or uncommunicated, pro- 
 perties of matter, such as those exhibited by the load-stone 
 and bismuth. These terms will answer the purpose of making 
 the required distinctions, and all the better because, being 
 merely historical, they do not involve the theoretical con- 
 siderations implied by such terms as statical and dynamical 
 electricity, thermo-electricity, and electro-magnetism. 
 
 The conditions of the equilibrium of the atoms at the 
 boundaries of substances have already been considered (p. 466), 
 and it was found that through a very thin superficial stratum 
 there will be a gradual increase of density from the surface 
 towards the interior. The atoms at the surface itself are as 
 much urged outwards by the atomic repulsion as they are 
 drawn inwards by the molecular attraction. But this balancing 
 of the forces, and the normal state of the superficial gradation 
 of density, would be disturbed by any such extraneous action 
 as the friction of one substance against another. The kind 
 of effect produced by the friction will perhaps be understood 
 from the following argument. 
 
 Suppose a rod to be held by the hand and to be moved 
 in the direction of its length. Then, admitting that it is 
 composed of discrete atoms, at whatever points the moving 
 force be applied the disturbance immediately impressed gives 
 rise to an action of atomic and molecular forces which extends 
 throughout the substance of the rod, causing each atom to 
 move in the same direction and through the same space as 
 the rod itself. So long as the motion is uniform, the atomic 
 and molecular forces are in the same state of action as when 
 the rod is at rest. But if we suppose it to be accelerated, and 
 consider the case of a discrete atom at its preceding extremity, 
 
508 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the acceleration will evidently be due to an excess of action 
 from within to without above that in the contrary direction. 
 At the following extremity there will in like manner be an 
 excess of action from without to within. At first sight it 
 might seem that these effects would be explained by sup- 
 posing that the superficial stratum at the preceding end re- 
 ceives, in consequence of the impressed force, an increment 
 of density which causes the atomic repulsion to exceed the 
 molecular attraction, and that at the other end a decrement 
 of density similarly produced causes an excess of molecular 
 attraction. But the true explanation will, I think, be found 
 to be such as follows. 
 
 When the motion of a mass is accelerated by the applica- 
 tion of a pushing or pulling force at any point or points, cer- 
 tain interior forces are called into action, by means of which 
 all the atoms are equally accelerated at each instant. These 
 forces are not modifications of atomic repulsion and molecular 
 attraction, which, as will afterwards be shewn, are only al- 
 tered so that they just counteract the applied mechanical 
 force. Let us, therefore, make the hypothesis that -the 
 common acceleration of all the atoms results from the imme- 
 diate action of molecular forces of an order different from that 
 of the force which has hitherto been called molecular attrac- 
 tion, and due to aetherial waves of greater breadth. The 
 existence of forces, both repulsive and attractive, differing in 
 their order from those by which the atoms of bodies are held 
 in positions of equilibrium, is consistent with the general 
 formula for the translatory action of waves, namely, 
 
 8g t / 2// 2 W*. 
 + 1 \ 3 / 4a" 
 
 2 A 
 
 in which the value of X, while it is so large that H^ 1 nearly, 
 may yet be such that ZT 2 , or 1 + F(-~\ , is either somewhat 
 
 Q 
 
 less, or somewhat greater, than -- . In the former case there 
 
 A 
 
THE THEORY OF ELECTRIC FORCE. 509 
 
 would be a new order of repulsion, and in the other a new 
 order of attraction, the latter being still an order distinct from 
 that of gravity, which may be supposed to correspond to much 
 larger values of X. Further, let us suppose that when a body 
 is made to start from rest by being pushed or pulled, the 
 forces of this order are immediately brought into action, and 
 generally that they are maintained in action whenever a body 
 is accelerated by a mechanical moving force. The mode of 
 action may be conceived to be of the following kind. 
 
 For the sake of distinctness of conception let the body be 
 a cylinder terminated by slightly convex ends, and sym- 
 metrical about a straight axis, and let it be accelerated in 
 the direction of the axis by a cord fastened to the central 
 point of one end. From the theory of the origination of 
 different orders of setherial waves in the interior of any sub- 
 stance (as given in page 490), it will be seen that waves of 
 the breadths proper for producing the above-mentioned attrac- 
 tive and repulsive effects may be supposed to be propagated 
 in the substance at all times, because the composite waves of 
 this order may, as in the case of those of light, have all 
 gradations of the values of X within certain widely separated 
 limits. If no extraneous force be acting, these molecular 
 forces will have no resultant, because the effects of equal 
 waves of attraction, propagated in opposite directions, will 
 everywhere neutralize each other ; and the same will be the 
 case with respect to the waves of repulsion. The application 
 of the external pulling force will destroy this equality in such 
 manner as to cause both the repulsive waves propagated in 
 the direction of the pulling, and the attractive waves pro- 
 pagated in the opposite direction, to be in excess ; and these 
 excesses, as will appear from the following reasoning, give 
 rise to forces of equal amount. It will be assumed, since the 
 waves of the order now under consideration have their origin 
 in molecules that may be regarded as elementary, that com- 
 pound plane-waves, both attractive and repulsive, are gene- 
 rated, not in one plane, but equally in all planes perpendicular 
 
510 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 to the axis of the cylinder. Accordingly the resultant attraction 
 at any point varies as the distance from the preceding end, 
 and the resultant repulsion at the same point varies as the 
 distance from i\\Q following end. If, therefore, a be the length 
 of the axis, and the amount of force at each end be assumed to 
 be the same quantity F, the repulsive force at any distance x 
 
 from the following end will be - - , and the attractive force 
 
 a 
 
 in the same direction at the same point ; so that 
 
 a 
 
 the total force accelerating the atoms in the direction of the 
 pulling, being the sum of these, is equal to F at each point. 
 Thus, since the condition of equal acceleration of all the 
 atoms is fulfilled, the above assumption of the equality of the 
 attractive and repulsive forces is justified. 
 
 If instead of the pulling by a cord, an equal pushing force 
 were applied in the same direction at the other end of the 
 cylinder, precisely the same effects would be produced; and, 
 generally, the same molecular action would be excited at 
 whatever point the force were applied, if it only caused the 
 same acceleration of the cylinder in the same direction. It is 
 clear, therefore, that we are here concerned with the molecular 
 forces on which depends the law, that a force acting on a 
 rigid body in a given direction has the same effect at what- 
 ever point in that direction it be applied ; which was assumed 
 to be one of the fundamental principles of the Statics and 
 Dynamics of rigid bodies. (See the remarks at the bottom of 
 page 155). That law may, therefore, be appealed to as being 
 confirmatory of the reality of the second order of molecular 
 forces. The body was supposed, for the sake of precision, to 
 be cylindrical ; but the reasoning is plainly not dependent on 
 that particular form. 
 
 Resuming now the case of the acceleration of the cylinder 
 by means of a cord, if we suppose a force equal to the common 
 acceleration of the atoms to be impressed on them individu- 
 ally, and on the cord, in the direction contrary to that of the 
 
THE THEORY OF ELECTRIC FORCE. 511 
 
 acceleration, the body will be reduced to rest, and we may 
 regard the total moving force thus impressed as just counter- 
 acting the mechanical action of the cord. But at the same 
 time, since the atoms are severally at rest, the atomic and 
 molecular forces acting on each atom must be in equilibrium. 
 These forces are (1) the force F, the generation of which is 
 above considered; (2) the force of atomic repulsion ; (3) that 
 of molecular attraction of the first order. In the undisturbed 
 state of the body the two latter forces have no resultant, 
 because in its interior the resultants of their separate actions 
 are zero, and in a thin superficial stratum the resultants of 
 their joint action are also zero. But in the case before us of 
 disturbance by an extraneous moving force, the resultant of 
 the conjoint action of the forces (2) and (3) on any atom, 
 must be equal and opposite to the action of the force F upon 
 it. And since, as we have just seen, impressed forces equal 
 and opposite to F have the effect of counteracting the tension 
 of the cord, it follows that the resultants of the forces (2) and 
 (3) counteract the same tension, which, in fact, has alone 
 brought them into action. (See page 508). 
 
 Considering, for instance, the equilibrium of an extreme 
 atom at the preceding end of the cylinder, since, by hypo- 
 thesis, the molecular force of the second order repels it from 
 within to without, the molecular attraction of the first order 
 must exceed the atomic repulsion by the amount of that mole- 
 cular repulsion. This might be the case if either the superficial 
 molecular attraction be greater, or the superficial atomic 
 repulsion be less, than the normal amount in the undisturbed 
 state of the body. As, however, the former is much less 
 altered by disturbance of the superficial atoms than the other 
 it may be presumed that a diminution of the atomic repulsion 
 takes place, and consequently that the superficial density of 
 the atoms is diminished. At the other end the new order of 
 molecular force draws a superficial atom from without to 
 within, and thus acts in aid of the molecular attraction of the 
 first order. Hence to maintain the equilibrium of the atom, 
 
512 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the atomic repulsion must be increased by an increment of 
 superficial density. 
 
 If the motion of the cylinder had been retarded, the incre- 
 ment of density would have taken place at the preceding end, 
 and the decrement at the following end. 
 
 If the acceleration of the cylinder were produced by ano- 
 ther similar cylinder pushing it at its end in the direction of 
 a common axis, the superficial condensation of the second 
 cylinder would, like that of the first, be diminished at the 
 preceding end ; because the excess of the moving force applied 
 to the second cylinder above the reaction at its point of con- 
 tact with the first may be regarded as a single extraneous 
 force urging it in the direction of the acceleration. As the 
 two ends in contact are equally accelerated, the increment 
 of condensation at the end of the preceding cylinder will 
 be equal to the decrement of condensation at the end of the 
 other. 
 
 These preliminaries having been gone through, I now 
 propose the following definition of the electric state of any 
 substance. A fluid or solid body is electrified, when its atoms, 
 after being put by an external moving force into abnormal 
 relative positions, are maintained in those positions by the 
 simultaneous action of atomic repulsion, molecular attraction 
 of the first order, and the molecular forces of the second order 
 that are brought into action by the disturbance*. 
 
 If a solid body be pulled equally in opposite directions by 
 
 * In my first researches on electricity I assumed that the permanency of an 
 electrical state was the result of the action of the atomic and molecular forces on 
 the contiguous air. (Philosophical Magazine for October, 1860, p. 283.) The 
 statements of experimenters led me to suppose that without the presence of the 
 air a body would not be electrified. But I afterwards found that the experimental 
 evidence on this point is not conclusive, and that a diversity of opinion exists as to 
 the influence of the air in electrical phenomena. M. De La Rive states, on the 
 authority of Harris, that " the repulsions and attractions between electrified con- 
 ducting bodies take place in vacuum as in air." (Traite d'Electricite, Tom. i. 
 p. 128.) Although it is certain from experiments that the pressure of the air has 
 an important influence on electrical phenomena, I now think, for the reasons 
 here adduced, that electricity is retained at the surfaces of electrified bodies by 
 molecular forces which act independently of atmospheric pressure. 
 
THE THEORY OF ELECTRIC FORCE. 513 
 
 two cords attached to it, no electric state is induced, because 
 the tensions of the cords, so far as regards the generation of 
 electricity, produce equal and opposite effects, and give rise 
 to no molecular forces of the second order. The result would 
 be the same if the body were pushed equally in opposite 
 directions. The tensions in the one case are simply counter- 
 acted by the excess of molecular attraction of -the first order 
 above the atomic repulsion, consequent upon the general 
 dilatation caused by the stretching, and in the other case the 
 thrusts are counteracted by the excess of atomic repulsion 
 above the molecular attraction, due to a general compression. 
 For like reasons a force pulling or pushing a body detained 
 by a fixed obstacle does not electrify it, so long as there is no 
 motion, or acceleration, of its particles. 
 
 It is known from experiments that although the electric 
 state is originally produced by friction, an electrified body is 
 capable of electrifying by induction other bodies separated 
 from it. I propose, therefore, first to employ the foregoing 
 theoretical inferences in an endeavour to explain the process 
 of electrifying by friction, and afterwards to give the theory of 
 electrical induction. 
 
 When two substances are rubbed together so that the 
 inequalities of their surfaces are brought into collision, the 
 action at the points of contact must produce effects of the 
 same kind as those above attributed to the mutual action 
 between two cylinders one of which accelerates the other by 
 pushing against it. Accordingly the friction puts the super- 
 ficial atoms into new relative positions, in which they are 
 retained by the intervention of the molecular forces of the 
 second order that are simultaneously brought into action by 
 the disturbance of the normal state of the superficial strata. 
 Also, from what is argued above, just in proportion as the 
 superficial stratum of one of the substances is condensed, that 
 of the other is rarefied. This effect is continually being 
 produced so long as the friction is kept up, and consequently 
 the electric state of each substance is continually being aug- 
 
 33 
 
514 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 mented. As friction between two bodies is necessarily 
 reciprocal, it can only be in consequence of some difference as 
 to quality or constitution that the superficial stratum of one is 
 condensed rather than that of the other. The theory would 
 seem to indicate that the substance which is the more yielding 
 of the two, or the more susceptible of changes in the inequali- 
 ties of its surface, is that which has its superficial stratum 
 condensed. It is probable that the amount of electricity gene- 
 rated may depend on a difference of quality in that respect, and 
 that for this reason the quantity is found to be increased by 
 coating with an amalgam one of the two surfaces in contact. 
 
 It has been ascertained by experiment that after rubbing 
 polished glass and resin with the same substance, two kinds 
 of electricity are developed. The theory offers at once an 
 explanation of this fact, inasmuch as, by hypothesis, a body 
 is electrified both when the superficial atoms are condensed, 
 and when they are rarefied, and the two kinds of electricity 
 may reasonably be referred to these two conditions of the 
 superficial stratum. As resin is more yielding than glass, it 
 may be presumed from what is said above, that if the elec- 
 tricity be that which has been called resinous, the superficial 
 atoms are condensed, and the force F is attractive ; and if it 
 be that called vitreous, the superficial atoms are rarefied and 
 the force F is repulsive. The vitreous electricity is also 
 called positive, and the resinous electricity negative. Theore- 
 tically, therefore, electricity is positive or negative according 
 as the superficial stratum is rarefied or condensed, or according 
 as the force F is repulsive or attractive. 
 
 Before proceeding farther it will be proper to advert to a 
 point which, unless it admits of explanation, might be urged 
 as an objection to the theory. In the case of the cylinder 
 accelerated by an attached cord, it was found that both kinds 
 of electricity were excited, and in equal degrees; whereas 
 when electricity is produced by friction each body receives 
 only one kind of electricity, and consequently the analogy 
 between the two cases fails in this particular. The reason 
 
THE THEOItY OF ELECTKIC FOltCE. 515 
 
 for this difference will be sufficiently apparent by noticing 
 that the electricity of the cylinder is excited by an accelera- 
 tion which all its atoms partake of in common, whilst the 
 electricity excited by friction is due to accelerations im- 
 pressed on the superficial inequalities by their mutual col- 
 lisions and pressures, and not communicated equally to all 
 parts of the substances. Although, therefore, there may be 
 both accelerations and retardations of the superficial parts of 
 both, it is conceivable, especially when the result of the 
 friction is increased by an amalgam, that there may be a 
 residual acceleration in one direction of the parts of one of the 
 substances, and an equal residual retardation in the contrary 
 direction of the parts of the other. They would thus receive, 
 one the positive, and the other the negative electricity, in 
 equal amounts, in accordance with experience. 
 
 Again, in another respect the electrical equilibrium of the 
 cylinder, supposed to be accelerated by being pulled or push- 
 ed, differs from that of a body electrified by friction. In the 
 former case the equilibrium is maintained by means of mole- 
 cular forces of the second order which are partly repulsive and 
 partly attractive, because some parts of the cylinder are elec- 
 trified positively and others negatively. But when the electric 
 state is produced by friction the superficial stratum is every- 
 where rarefied, or everywhere condensed, according as the 
 electricity is positive or negative, so that the molecular forces 
 of the second order are wholly repulsive, or wholly attractive. 
 
 Let us take, for example, the simple case of a globe of 
 uniform material equally electrified at all points of its surface, 
 and, first, suppose the electricity to be positive. Then, the 
 superficial stratum being rarefied, the molecular attraction of 
 the first order exceeds the atomic repulsion, and this excess is 
 counteracted by molecular force of the second order. The 
 forces of the second order are in a neutral state so long as the 
 superficial attractions and repulsions are in equilibrium, and 
 the atoms of the stratum are in their normal positions ; but 
 when this equilibrium is by any means disturbed, immediately 
 
 332 
 
516 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 forces of the second order are brought into action, and a new 
 equilibrium is established. In the present case the new forces 
 are wholly repulsive. It will be assumed that they have 
 their origin equally at all parts of the globe, that they ema- 
 nate from cubical molecules of the second order, regarded, 
 however, as of elementary size, and that .according to the 
 general law of accelerative forces due to setherial undulations, 
 they vary inversely as the squares of the distances from the 
 centres of the molecules. Their resultant is consequently a 
 repulsive force varying as the distance from the centre of the 
 globe. The magnitude of this resultant is determined by the 
 condition that its action on an extreme atom must just coun- 
 teract that arising from the assumed excess of molecular at- 
 traction in the superficial stratum, and will, therefore, be 
 greater the more the atoms are displaced from their normal 
 positions. This condition being fulfilled, the equilibrium of 
 the other atoms is effected by an adjustment of the variation 
 of the density of the atoms in the superficial stratum, together 
 with a very small variation of their density in the interior. 
 For it is evident that the interior repulsive forces must have 
 the effect of causing some degree of increment of density from 
 the centre towards the surface. The same reasoning applies, 
 mutatis mutandis, to the electrical equilibrium of a sphere 
 electrified negatively ; in which case the increment of interior 
 density is from the circumference to the centre. 
 
 Next let a spherical shell of the same material as the 
 globe be positively electrified, and, supposing the radii of the 
 exterior and interior surfaces to be r and /, let the radius r 
 be the same as that of the globe. Also, since the molecular 
 forces of the second order may be supposed to act according 
 to the same law in the two cases, let the resultant repulsion 
 on an extreme atom of the globe be ar, and that on an ex- 
 treme atom of the shell be fi(r r). Now if the electric 
 states of the surface of the globe and the exterior surface of 
 the shell, as tried by the usual experimental tests, be the 
 same in kind and degree, it may be presumed that the dis- 
 
THE THEORY OF ELECTRIC FORCE. 517 
 
 placements of the superficial atoms are also the same ; so that 
 by reason of the condition of equilibrium of the extreme atoms 
 we must have ar = /3(r r'). Hence y5 is greater than a in 
 the proportion of r to r r ; or the molecular repulsion is 
 intrinsically greater in the shell than in the globe in that pro- 
 portion. The following reason may be given for this result. 
 
 According to a known theorem, the resultant of the re- 
 pulsions proceeding from the molecules of the shell will be 
 zero at any point within, or upon, the interior surface, and 
 consequently at that surface the superficial atoms are in their 
 normal or undisturbed state. Hence the total variation of 
 density from the interior to the exterior surface due to the 
 molecular repulsions must be the same whatever be the 
 thickness of the shell, and therefore the same as that from the 
 centre to the surface of the globe. Hence the less the thick- 
 ness the more rapid will be the gradation of density ; and 
 since to produce in a given thickness a greater change of 
 density, more force is required, it follows that ft(r r') is 
 greater than a (r r'), or ft greater than a. 
 
 The foregoing theory accounts for the known fact that if 
 a spherical shell of metal, perforated at one part, be electrified, 
 little or no electricity is found on the interior surface when a 
 proof-plane inserted through the orifice is applied to it. The 
 same explanation may be given in the case of an ingenious 
 experiment made by Faraday, in which an electrified hollow 
 cone, having an acute vertical angle, was turned inside out by 
 pulling a thread. The electricity was always on the surface 
 which was exterior, because in this instance, as in that of the 
 hollow sphere, the resultant of the attractions or repulsions on 
 any point within or on the interior surface is zero. From 
 facts thus accounted for by the theory it has been inferred 
 that electricity is confined to the surfaces of bodies*. 
 
 The preceding considerations are only applicable to bodies 
 
 * See Jamin, Cours de Physique, Tom. i. pp. 377 379. The complete expla- 
 nation of the experiment made by applying two hemispherical caps to a sphere 
 will be given under the head of electricity by induction. 
 
518 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 whose surfaces are spherical. In order to treat of electrical 
 equilibrium more generally, it will be necessary to take ac- 
 count of the distinction between bodies, in their relation to 
 electricity, which is expressed by the terms conductors and 
 non-conductors. The facts which these terms indicate may 
 be supposed to depend on the modes of action of the atomic 
 and molecular forces of the bodies, as determined by the 
 magnitudes and arrangement of their component atoms. But 
 in the existing state of theoretical physics it is not possible 
 to explain in what particular manner the conducting or 
 non-conducting property results from such action. The ex- 
 perimental facts admit, however, of being expressed in the 
 language of the present theory as follows. According to the 
 definition in page 512, when electricity is excited by friction, 
 or other means, at any part of the surface of a body, the 
 immediate effect is to put the superficial atoms at that part in 
 disturbed positions. Supposing the body to be a conductor, 
 the disturbed state of the atoms is spread by superficial pro- 
 pagation over the whole of the surface, and while the total 
 quantity of electricity thus distributed is determined by the 
 amount of the local disturbance, the distribution of it is 
 wholly due to the conducting property of the body. In the 
 case of a perfect non-conductor, the disturbed state of the 
 atoms would not be propagated to any sensible distance from 
 the place of excitement ; but if, as is generally true of sub' 
 stances called non-conductors, the power of conducting exists 
 in a small degree, the disturbance either spreads slowly, or to 
 a limited extent. 
 
 In the hypothetical instance of the excitement of elec- 
 tricity both positive and negative in a cylinder by pulling it 
 with a cord (p. 509), if we suppose the cylinder to be a con- 
 ductor, and the cord to be suddenly detached from it, the 
 electric state would instantly cease, because by the conduct- 
 ing property the displacements, both positive and negative, of 
 the superficial atoms would spread over the whole of the sur- 
 face, and being equal in amount would neutralize each other. 
 
THE THEORY OF ELECTRIC FORCE. 519 
 
 Again, if electricity, either positive or negative, were 
 excited at any part of the surface of a perfectly conducting 
 sphere, the electric state would immediately be diffused 
 equally over the whole of the surface, and the displacements 
 of the superficial atoms would quickly become the same at all 
 parts, because otherwise the electrical equilibrium would not 
 be maintained. But if a body has any other form than that 
 of the sphere, there will not be the same reason for uniform- 
 ity of distribution, and we have to enquire what circum- 
 stances may then determine the amounts of electricity to be 
 different at different parts. In fact, inequality of the distri- 
 bution of electricity on the surfaces of conducting bodies is 
 matter of experience, to account for which I propose the fol- 
 lowing theoretical considerations. 
 
 Suppose a conducting body to be electrified positively r , and 
 for the sake of precision let its form be that of a cylinder 
 with hemispherical ends. According to a piinciple already 
 adopted (p. 509), it will be assumed that while the body is in 
 this electrified state, every cubical molecule of the second 
 order is the centre of a repulsive force varying inversely as 
 the square of the distance, and acting on the atoms of all the 
 other molecules. Under these circumstances the extreme 
 superficial atoms will all be urged outwards, and in different 
 degrees at different positions. The resultant repulsive force 
 will be least in the transverse section through the middle 
 point of the cylinder, and will increase towards each end, 
 becoming of maximum value at the points where the hemi- 
 spherical surfaces are cut by the axis of the cylinder. The 
 effect of these repulsions is to dimmish the condensation of 
 the superficial atoms, and thus to cause the superficial mole- 
 cular attraction to exceed the atomic repulsion, in consequence 
 of which excess, in fact, the molecular repulsions are coun- 
 teracted. But it is to be observed that except at particular 
 points the resultant of the molecular repulsions does not cut 
 the surface of the cylinder at right angles, and consequently 
 the counteracting force will not depend exclusively on the 
 
520 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 variation of the condensation of the atoms in the direction 
 perpendicular to the surface. Joined with the force in this 
 direction there must be a force tangential to the surface, and 
 such that the resultant of the two shall just counteract the 
 resulting molecular repulsion. Now the existence of the tan- 
 gential force may be assumed for the following reasons. 
 
 Conceive, at first, the cylinder to be free from electricity, 
 and from one extremity A of its axis let a line be drawn on 
 the surface always in a plane passing through the axis, and 
 terminating at the point P. Let the length of the line AP 
 be 5, and p being a point in advance contiguous to P, let 
 Pp = &?. Supposing, now, the body to be electrified, and the 
 ordinates PM (= z) and pm (= z + 8z) to be equal to the dis- 
 placements of superficial atoms in directions perpendicular to 
 the surface, it will be seen that the molecular repulsive force 
 causes PM to be greater than pm. This inequality of the 
 ordinates will of itself continually tend to produce a propa- 
 gation of the superficial disturbance in the direction from A 
 towards P, and accordingly an atom at the position m will 
 be urged in the direction from P towards p by a force pro- 
 portional to the ratio of PM pm to Pp\ or the force may be 
 
 Cv 
 
 expressed as T^- . It is by the resultant of this tangential 
 
 force and the normal attractive force tending from without to 
 within that the molecular repulsion is counteracted ; and the 
 condition that each superficial atom is held in equilibrium by 
 the combined action of three forces such as these, determines 
 the form of the actual surface. The quantity of electricity at 
 each point is proportional to the ordinate PM. 
 
 Analogous considerations are applicable if the cylinder be 
 supposed to be electrified negatively. In this case the ordi- 
 nates PM and pm are on the opposite side of the line AP 
 and may be considered to be negative ; and, as before, PM is 
 greater than pm. Hence the tendency of the propagation, 
 and of the tangential force, is from P towards A ; and since 
 the normal force acts from within to without, the resultant of 
 
THE THEORY OF ELECTRIC FORCE. 521 
 
 the two forces also tends outwards, and in such manner as 
 just to counteract the resultant of the molecular attractions of 
 the second order, inasmuch as these three forces must be in 
 equilibrium. This condition determines the form of the sur- 
 face as disturbed, and the amount of negative electricity at 
 any point is proportional to the negative ordinate PM. 
 
 It is evident that, according to these principles, there will 
 be an excess of electricity at the two hemispherical ends of 
 the cylinder, and so much the greater as the diameter of the 
 cylinder is less for a given length. The theory thus accounts 
 for the observed accumulation of electricity at sharp points. 
 
 The preceding views may serve to indicate generally the 
 character and mode of operation of the forces that are con- 
 cerned in electrical equilibrium and in the distribution of 
 electricity about the surfaces of conductors. It might, on 
 these principles, be possible to obtain an analytical expression 
 for the quantity of electricity at any point of the surface of an 
 electrified conductor of given form ; but there would be in- 
 volved in the expression certain constants which could only 
 be determined by experiment. Leaving this investigation in 
 the hands of those mathematicians who may feel sufficient 
 confidence in the principles here expounded to be induced to 
 undertake it, I proceed next to the theory of electricity by 
 influence or induction. 
 
 According to the foregoing explanations, electrical equi- 
 librium is produced by the joint action of three kinds of force, 
 atomic repulsion, molecular attraction of the first order, and 
 molecular force, either attractive or repulsive, of the second 
 order. The forces of the first and second kinds, to which the 
 different states of aggregation of bodies, chemical action be- 
 tween their atoms, and the phenomena of capillary attraction, 
 are to be attributed, act sensibly only at insensible distances 
 from their origins. But the forces of the third kind, inasmuch 
 as they are supposed to emanate from molecules intermediate 
 in magnitude to those of the first order of molecular attrac- 
 tion and the molecules of the attraction of gravity, might be 
 
522 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 expected to produce sensible effects at considerable distances 
 from the points of emanation. It is found, in fact, that an 
 electrified body acts upon substances of ail kinds separated 
 from it by considerable intervals. If the bodies exposed to 
 its influence be easily moveable, they are at first attracted 
 towards it, and after contact are repelled; and whether 
 moveable or not, the influence puts them into an electric 
 state, which, as not being immediately caused by friction, 
 is said to be induced. It is necessary to give the theory of 
 induced electricity, before entering upon that of electrical 
 attractions and repulsions. 
 
 Suppose an insulated conductor of the form of a cylinder 
 with hemispherical ends to be brought into the neighbourhood 
 of a like conductor charged positively by being connected with 
 an electrical machine, and let their axes be nearly in the 
 same direction. Experiment shews that under these circum- 
 stances the parts of the first conductor that are towards the 
 charged conductor are electrified negatively, and the remoter 
 parts positively. In order to account for these phenomena it 
 is, first, to be considered that, according to the theory, each 
 of the molecules of the second order that compose the charged 
 conductor is the centre of a repulsive force varying inversely 
 as the square of the distance from the molecule, and that these 
 forces acting on the other conductor displace its atoms from 
 their ordinary positions of equilibrium. Admitting that the 
 waves of this repulsion are readily transmitted through the 
 interior of the conductor, it will follow that the superficial 
 atoms that are situated towards the charged conductor will 
 be urged inwards, and those more remote will be urged out- 
 wards^ the two actions being separated by a neutral line. 
 Thus the nearer end of the cylinder will be electrified nega- 
 tively, and the further end positively, agreeably with expe- 
 rience. It should be noticed that this disturbance of the 
 superficial atoms by a directly impressed force, differs essen- 
 tially from that which was previously supposed to result from 
 the acceleration due to a pulling or pushing force applied at 
 
THE THEORY OF ELECTRIC FORCE. 523 
 
 a certain point. But in the present case, as in the other, 
 equal quantities of the opposite electricities are excited, as 
 I now proceed to shew. 
 
 Selecting a particular molecule of the charged conductor, 
 let us conceive a portion of the repulsion-waves emanating 
 from it to be included in a pyramidal surface of indefinitely 
 small vertical angle, the vertex being at the molecule ; let D 
 and d be the distances from the vertex at which the axis of 
 the pyramid cuts the surface of the other conductor ; and let 
 a and /3 be the angles of inclination of the axis to the surface 
 at the points of intersection. Then the repulsive forces at 
 the distances D and d, resolved perpendicularly to the surface, 
 
 are sin a and ~ sin /3, the former urging a superficial atom 
 
 inwards and the other outwards. Also the elementary sur- 
 faces of the cylinder included within the surface of the pyra- 
 mid are in the proportion of D z cosec a to d z cosec/3. Hence 
 taking the disturbing action perpendicular to the surface to 
 be the measure of the quantity of electricity excited, the two 
 quantities enclosed by the pyramidal surface will be in the 
 
 proportion of -^ sin a x D* cosec a to ^ sin /3 x d 2 cosec /3, 
 
 that is, they will be equal. As the same reasoning may be 
 applied to all such pyramidal surfaces having their vertices 
 at the same molecule, and to all the sets of pyramidal surfaces 
 from the different molecules, it follows that the total quan- 
 tities of electricities of the two kinds are of equal amount. 
 It may not, however, be inferred that the actual quantity of 
 electricity at any point of the surface is just as much as, ac- 
 cording to this reasoning, is excited at that point, because the 
 influence of superficial propagation is to be taken into account, 
 which, while it alters the distribution of the electricities, af- 
 fects the total amount only so far as positive and negative 
 electricities are thereby made to neutralize each other. But as 
 neutralization implies the disappearance of equal quantities 
 of the two kinds, the remainders will still be equal. 
 
524 THE MATHEMATICAL PRINCIPLES OP PHYSICS. 
 
 This, however, is not the complete account of the elec- 
 tricities of the two cylinders. It is evident that as soon as 
 the neutral one is electrified by induction, it reacts on the 
 charged cylinder by means of excited waves of the second 
 order, and thus influences reciprocally the electricity of the 
 latter. As the induced electricity of the neutral cylinder 
 is partly negative and partly positive, part of those waves 
 will act attractively and part repulsively. But because the 
 origins of the former are nearer to the charged cylinder than 
 the origins of the latter, the attractive effect will preponderate, 
 and consequently the positive electricity of that cylinder will 
 be increased at the parts that are nearest the neutral cy- 
 linder. By this reciprocal action the induced electricity of 
 the latter will also be increased. At the same time, as may be 
 inferred from the reasoning in the last paragraph, there will 
 still be equal quantities of positive and negative electricities. 
 
 Hence if the charged conductor be either discharged, or 
 removed to a great distance, the electricity of the other va- 
 nishes, because the equal quantities of opposite electricities 
 neutralize each other by superficial propagation. This theo- 
 retical result is in accordance with experience. 
 
 Let us next consider what would happen if the neutral 
 conductor, after being electrified by induction, were connected 
 by a conducting rod, or chain, with the earth. In the fore- 
 going reasoning the insulated neutral conductor was supposed, 
 for the sake of distinctness, to have a particular form ; but 
 whatever be the form, reasoning of the same kind is applicable. 
 Hence after making communication with the earth, the con- 
 ductor, the chain, and the earth, may be regarded as an in- 
 sulated conductor of unlimited extent. The neutral line will 
 in consequence be far removed from the insulated cylinder, 
 and \h.Q positive electricity will everywhere become extremely 
 small. On this account the quantity of the negative elec- 
 tricity which is neutralized by the tendency of the positive 
 electricity to be propagated superficially is also extremely 
 small. Consequently, at the instant of forming the connection 
 
THE THEORY OF ELECTRIC FORCE. 525 
 
 with the earth, the amount of free negative electricity is in- 
 creased, and at the same time is diffused over the whole of 
 the cylinder. This inference accords with experience. 
 
 Suppose, now, the connection with the earth to be broken 
 by removing the chain, and the conductor to be insulated as 
 at first. It might be supposed that under these circumstances 
 the original state of induced electricity would be restored, and 
 accordingly become partly negative and partly positive. But 
 experiment shews that this is not the case, and that by break- 
 ing the connection with ihe earth the negative electricity of 
 the cylinder is not sensibly altered. No theoretical explana- 
 tion, as far as I am aware, has been given of this significant 
 fact. Before proposing an explanation on the principles of 
 the present theory, it will be necessary to enter upon some 
 additional considerations respecting the molecular forces of 
 the second order. 
 
 The reality of the forces of this order has already been 
 maintained by an argument which may be further illustrated 
 as follows. A body being supposed to be accelerated in a given 
 direction by a cord attached to a certain part of its surface, 
 let this acceleration be impressed on the atoms individually 
 and on the cord in the opposite direction. The body will thus 
 be kept at rest, because the sum of the impressed moving forces 
 will be equal to the moving force of the tension of the cord. 
 But the traction of the cord displaces the atoms on which it 
 immediately acts, causing them to deviate from their original 
 positions, and the disturbance extends by propagation to all 
 the atoms of the body, so that each one is urged towards its 
 original position either by an excess of atomic repulsion above 
 molecular attraction of the first order, or by an excess of the 
 latter force above the other. But since, by what is said above, 
 the whole body and the individual atoms are at rest, these 
 excesses are necessarily counteracted by additional molecular 
 forces not yet taken into account. 
 
 The existence of additional forces that may be proper for 
 producing this effect is compatible with the foregoing general 
 
526 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 theory of molecular forces, according to which as the waves of 
 atomic repulsion produce by composition the waves of the first 
 order of molecular attraction, so the composition of the latter 
 gives, rise to waves of another order, which in their action on 
 spherical atoms are repulsive when their breadths are below a 
 certain value, and become attractive when they exceed this 
 value (p. 508). 
 
 In any case in which a mass is accelerated, not by a force, 
 such as gravity, which acts alike on all the atoms, but by a 
 mechanical pull or thrust, which alters the relative positions of 
 the atoms, the forces of the second order have to fulfil two con- 
 ditions. First, they must be capable of producing accelerations 
 of the atoms individually, equal to the observed acceleration 
 of the mass; and secondly, they must act so as to neutralize, 
 relatively to each atom, the repulsions and attractions of the 
 first order due to the disturbance of the atoms from their ori- 
 ginal positions by the impressed mechanical force. In short, 
 the resultants of these first-order repulsions and attractions 
 are exactly antagonistic to the impressed force, which, there- 
 fore, has the effect of accelerating the body only through the 
 intervention of the molecular forces of the second order, by 
 which the atoms of the body are severally and immediately 
 accelerated. It evidently follows that the forces of the first 
 and second orders acting statically would be in equilibrium *. 
 
 It is an important consequence of these views that when a 
 mass is accelerated by a mechanical pull or push, there will be 
 & gradual increment of density from the parts following to the 
 parts preceding. It having been shewn that in such action 
 
 * This theory gives some account of the interior agency by which a mechanical 
 pull or push, applied at a certain part of a body, causes all its constituents to 
 move equally in the direction of the impulse. Probably an a priori theory of ten- 
 sions and stresses might be founded on like considerations. 1 make this remark 
 because a general theory of physical force, such as that I have proposed, might 
 well be expected to indicate the character of the interior forces concerned in the 
 mutual mechanical actions of masses. This demand I consider to be in some mea- 
 sure met by pointing out the relation of the atomic and molecular forces to surh 
 actions: but it would be beside my present purpose to discuss this subject at 
 greater length. 
 
THE THEORY OF ELECTRIC FORCE. 527 
 
 the molecular forces of the second order accelerate all the 
 atoms of the mass equally, and are at the same time counter- 
 acted by the forces of the first order, the reason for the grada- 
 tion of density is precisely the same in this case as in that of 
 a stratum of the earth, or its atmosphere, acted upon by 
 gravity. In fact, without variation of the density in the in- 
 terior of the mass, there would be no perceptible action of the 
 second order of molecular forces. In consequence of such 
 variation these forces may be supposed to produce effects 
 analogous to those which have been attributed to the repulsive 
 and attractive forces which act in superficial strata of very 
 small thickness. In the second order of waves, as in the 
 first, the values of X are smaller for the repulsion-waves than 
 for the attraction-waves, and it may be that they are very 
 much smaller. Hence applying to the forces of the second 
 order the same kind of reasoning as that which was applied, 
 in the case of superficial disturbance, to those of the first 
 (page 507), we may conclude that the abnormal increment of 
 density towards the preceding boundary of the mass coexists 
 with an excess of repulsion above attraction acting from within 
 to without, and the abnormal decrement of density towards the 
 following boundary, with an excess of attraction above repul- 
 sion acting from without to within. These excesses are the 
 repulsions and attractions which, acting on the preceding and 
 following superficial atoms, counteract respectively the attrac- 
 tions and repulsions due to the forces of the first order. In the 
 interior the sum of the same forces counteracts at each point 
 the resultant of the first order which is due to the gradation of 
 density, and as this sum is equal to the effective acceleration 
 of the atoms, which is the same at all parts of the interior, it 
 follows that the gradation of density is uniform. This infer- 
 ence may be taken to be confirmatory of the expressions 
 
 and ^ - for those forces, adopted in page 510, the con- 
 dition of uniform gradation of density justifying the assump- 
 
528 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 tion there made, that attractive and repulsive plane-waves are 
 generated equally in all planes perpendicular to the axis of the 
 cylinder. 
 
 What is argued above respecting interior variation of 
 density, applies only to bodies which, having been electrified 
 inductively, have equal quantities of the opposite electricities. 
 But if a body be charged with either positive or negative 
 electricity apart from induction, interior variation of density 
 will still be a necessary condition of electrical equilibrium. 
 Taking, for instance, the case of the uniformly electrified 
 globe considered in page 515, it is evident that if the globe be 
 electrified positively, the electrical equilibrium requires that 
 there should be a gradual diminution of density from the cir- 
 cumference to the centre, in order that the molecular repul- 
 sions of the second order may be counteracted by the forces of 
 the first order. So if the globe be electrified negatively, a 
 gradual increase of density from the circumference to the 
 centre is required for counteracting the molecular attraction of 
 the second order. If we made the general hypothesis that 
 when a body is electrified without induction, each molecule of 
 the second order is the centre of waves of the same intensity, 
 acting repulsively or attractively according as the electricity 
 is positive or negative, with forces varying inversely as the 
 square of the distance, the density throughout the interior 
 would be ascertainable by mathematical calculation. And if 
 on the same hypothesis the distribution of electricity about 
 the surface of the body were calculated in the manner indi- 
 cated in pages 519 and 520, it might be possible to test the 
 truth of the hypothesis by comparing the actual with the 
 computed distribution by means of a proof-plane. 
 
 Also on this hypothesis the waves emanating from all th e 
 molecules of the second order have no tendency to produce a 
 motion of translation of the body, because they act as central 
 forces, and their resultant through its centre of gravity is 
 consequently zero. It is accordingly found by experiment 
 that in this respect the electricity which is directly generated 
 
THE THEORY OF ELECTRIC FORCE. 529 
 
 by friction, which for distinction may be called primary elec- 
 tricity, differs essentially from induced electricity. Reasons 
 have been already given for concluding that in any case of 
 a body charged with the latter, the composition of the waves 
 emanating from its molecules of the second order results in 
 plane-waves, both attractive and repulsive, tending in their 
 aggregate effect to accelerate all the atoms equally in a certain 
 direction, and thus to produce a motion of translation of the 
 body. (See pages 510 and 526.) The waves which emanate 
 from the molecules of the second order of the inducing body, 
 and are incident on the other body in different directions, may 
 be assumed to act so as to have a single attractive or repulsive 
 resultant, the direction of which necessarily determines that 
 of the action of the above-mentioned plane-waves. In the 
 representative case of the traction of a body by a cord, this 
 resultant is represented by the tension of the cord, and its 
 direction by that in which the cord pulls. 
 
 It must not, however, be concluded, because these forces 
 of the second order tend in the manner above described to 
 produce motions of translation, that electrical translatory 
 action is to be accounted for in this manner. Although this 
 tendency is a necessary antecedent condition of such action, 
 the motions are immediately produced by forces of a different 
 kind, as will be shewn in the subsequent theory of electrical 
 attractions and repulsions. 
 
 The most general electric state of a conductor is that in 
 which it is charged at the same time both with primary and 
 inductive electricity. Such a state would evidently be pro- 
 duced if two conductors, charged originally with primary 
 electricity, were placed near each other. According to the 
 foregoing theory, the electricity of each would then become 
 partly primary and partly inductive. And since it appears 
 from the reasoning in page 524, that the quantities of opposite 
 electricities generated by induction are equal, if the bodies 
 were afterwards removed far apart, their electricities would 
 become the same as at first, the opposite electricities neutral- 
 
 34 
 
530 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 izing each other by superficial propagation as soon as the 
 molecular attractions, or repulsions, that maintain them cease 
 to have influence. These results agree with known facts. 
 
 We are now prepared to resume the consideration of the 
 case of induced electricity partly discussed in pages 522 525. 
 An explanation having been given of the increment of nega- 
 tive electricity which an insulated conductor, exposed to the 
 influence of the positively electrified conductor of an electric 
 machine, receives upon connecting it with the ground, it is 
 now required to shew why this new electric state is not altered 
 by breaking the connection. The second conductor, the chain, 
 and the earth, being regarded as composing, after the con- 
 nection, an insulated conductor of unlimited extent, it will be 
 seen that the positive electricity will be spread over an im- 
 mense amount of surface, insomuch that its neutralizing action 
 at any given position on the opposite electricity will be inde- 
 finitely diminished. There will, consequently, be no appre- 
 ciable tendency of the positive electricity to modify by super- 
 ficial propagation the negative electricity which pervades the 
 surface of the second conductor, and the molecular repulsion 
 on which such tendency depends will be reduced to zero. 
 The atoms of that conductor will, therefore, take certain po- 
 sitions of equilibrium determined solely by the action of the 
 other forces, and uninfluenced by any emanation of force from 
 the earth. Hence if the connection with the earth be broken, 
 the positions of the atoms and the electric state will remain 
 the same (unless some slight influence be attributable to the 
 electricity of the chain). This appears to be a sufficient 
 theoretical reason for the fact it was proposed to account 
 for. 
 
 It is farther to be remarked that if during the connection 
 with the ground the charged conductor were either discharged, 
 or removed to a great distance, the negative electricity of the 
 ether conductor would disappear by superficial conduction, 
 there being no longer any force to prevent the atoms returning 
 to their natural positions. Or, if we please, we may consider 
 
THE THEORY OF ELECTRIC FORCE. 531 
 
 the disappearance to be produced by the neutralization of 
 opposite electricities, as in the case of an insulated conductor 
 of limited extent. But if the charged conductor be discharged 
 after breaking the connection with the ground, the atoms of 
 the insulated conductor do not return to their ordinary posi- 
 tions of no electricity, for the following reasons. By connec- 
 tion with the ground this conductor became charged with 
 negative electricity only, which, as is shewn above, it retains 
 after the connection ceases. Now this electricity cannot be 
 wholly primary, because of the influence of the conductor of 
 the electric machine ; neither can it be wholly inductive, 
 because in that case there would be equal quantities of the 
 opposite electricities. It must, therefore, be partly primary 
 and partly inductive, which, according to this theory, is the 
 most general state of electrical equilibrium. 
 
 This being admitted we may farther argue that so far as 
 the electricity of the insulated conductor is primary, the equi- 
 librium of its individual atoms is maintained by atomic and 
 molecular forces of the conductor itself; and so far as it is in- 
 ductive, by modifications of the same forces due to the repulsions 
 emanating from the charged conductor, to which, as appears 
 from the reasoning in page 526, the modifications of the 
 first-order forces are antagonistic. If, therefore, the distance 
 between the two conductors were changed, the atoms of the 
 second conductor must take new positions, because the in- 
 fluence of the charged conductor would no longer be the same; 
 and if the latter conductor were discharged, or entirely re- 
 moved, the atoms of the other would take positions of equi- 
 librium determined solely by its own atomic and molecular 
 forces. Its electricity would thus become wholly primary. 
 It is evident that the effacing in this manner of the inductive 
 electricity does not alter the total quantity of -negative elec- 
 tricity, but only its distribution about the surface of the con- 
 ductor. These theoretical results accord with well-known 
 facts of experience. 
 
 The foregoing explanations also complete the theory of 
 
 342 
 
532 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the excitement of primary electricity by friction, inasmuch as 
 they shew why in employing an electric machine for this 
 purpose the charge accumulates by connecting the cushions 
 with the earth by a conducting chain. 
 
 The charged conductor has been supposed throughout the 
 previous discussion to be electrified positively: it is evident 
 that the same reasoning applies, mutatis mutandis, if it were 
 electrified negatively. 
 
 The following are some additional facts which this theory 
 of inductive electricity accounts for: (1) In the instance 
 alluded to in the Note in page 517, the two insulated hemi- 
 spherical caps, when applied to the insulated sphere, may be 
 regarded as forming a spherical shell, which, after being 
 electrified, does not electrify the interior sphere for the reason 
 given in the text to which the Note refers. As soon, how- 
 ever, as the hemispheres are separated, each will electrify the 
 sphere by induction : but these induced electricities, which in 
 the first instance nearly neutralize each other, entirely disappear 
 when the sphere is out of reach of the influence of the caps. 
 
 (2) The theory of the electrical condenser* is contained 
 in the foregoing general theory of primary and induced elec- 
 tricities. Suppose two insulated circular metallic plates A 
 and B of the same diameter to be placed with faces exactly 
 opposite arid separated by a small interval, and let the middle 
 point of the outer face of A be connected with an electric 
 machine, and that of the outer face of B with the ground. 
 Then the conductor of the electrical machine together with 
 the plate A corresponds to the charged conductor of the 
 general theory, and the plate B to the second, or insulated, 
 conductor. Since A is electrified positively by its connection 
 with the machine, its repulsive waves of the second order 
 urge inwards the superficial atoms of the nearer face of the 
 plate j5, and thus put that face into a state of induced nega- 
 tive electricity. The same waves urge outwards the super- 
 
 * For an account of the experimental facts relating to the electrical condenser, 
 see Jamin's Cours de Physique, Tom. i. pp. 427430. ** 
 
THE THEORY OF ELECTRIC FORCE. 533 
 
 ficial atoms of the farther face, and tend to put it into a state 
 of positive electricity. But by reason of the connection with 
 the ground this electricity is spread by conduction over sur- 
 face of indefinite extent, so that at all points of the farther 
 face it becomes of insensible amount and has no tendency to 
 influence by conduction the electricity on the nearer face. 
 The latter (negative) electricity coexists with a gradual in- 
 crease of interior density from the nearer to the farther face, 
 this being a necessary condition of the action of the mole- 
 cular attractive waves of the second order which counteract 
 the induced excess of atomic repulsion above the molecular 
 attraction of the first order. But the same attractive waves 
 of the plate B, acting on the plate A, increase its positive 
 electricity on the nearer or inner face, and diminish that on 
 the farther or outer face. This increment, being on the side 
 towards B, has the effect of increasing .Z?'s negative electricity, 
 and therefore increasing its action on A ; while at the same 
 time the diminution of the positive electricity on A 1 a farther 
 face is supplied by the action of the machine. By this means 
 both the negative electricity of B and the positive electricity 
 of A go on accumulating, and the limit is attained when the 
 positive electricity on the outer face of A is as much as the 
 machine can produce. 
 
 It is to be observed that in the above explanation no 
 account is taken of any effect that may be due to the air 
 between the two plates. This point will be subsequently 
 referred to, when the influence of the air on electrical pheno- 
 mena is under consideration. 
 
 (3) Experiments by M. Matteucci (cited by Jamin, Tom. I., 
 p. 396) shew that non-conductors submitted to the action of 
 an electrified body are electrified inductively in the same 
 mariner as conductors, but with less intensity. On the sup- 
 position that the waves of the second order from an electrified 
 body are transmitted through non-conductors as through con- 
 ductors, the observed induction is accounted for by the theory. 
 The less intensity may be ascribed to the less facility with 
 
534 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 which the atoms of non-conductors take new positions of equi- 
 librium, and the consequent smaller displacements from their 
 natural positions. 
 
 (4) The theory gives the following explanation of the 
 phenomena of the Leyden Jar. It will be necessary, in order 
 to make the explanation intelligible, to commence with point- 
 ing out the distinction between electrifying by induction and 
 by contact. If two spheres, one electrified and the other in 
 its ordinary state, be placed near each other, the latter will 
 become electrified by induction in such manner that the part 
 of its surface which is towards the electrified sphere will be 
 oppositely electrified. As soon, however, as the spheres are 
 brought into actual contact, this is no longer the case. Sup- 
 posing them to be equally good conductors, from the instant 
 of contact they may be regarded as a single body electrified 
 with the electricity of the charged body, because by the pro- 
 perty of conduction this electricity is immediately propagated 
 over the surface of the other. It is found, however, by ex- 
 periment that at the parts of the surfaces immediately sur- 
 rounding the point of contact the electricity is of very small 
 intensity, probably vanishing entirely at the point itself. This 
 fact is at once accounted for theoretically by considering that 
 after the contact the waves of the second order emanating 
 from the two spheres are either wholly attractive or wholly 
 repulsive, and consequently act in opposite directions on 
 atoms situated near the point of contact, so that the resulting 
 action is small, and on this account the electricity at those 
 parts is also small. If the second sphere be now removed 
 from the influence of the other, its electricity will be spread 
 by conduction equally over the whole of its surface, and it 
 will thus acquire a certain amount of primary electricity 
 at the expence of the primary electricity of the charged 
 sphere. 
 
 The facts that electricity is communicated by the contact of 
 a charged conductor with a neutral one, and that the imparted 
 electricity is of the same kind as that of the former, having been 
 
THE THEORY OF ELECTRIC FORCE. 535 
 
 generally accounted for by the above considerations, I propose 
 now to attempt to give a theoretical explanation of the parti- 
 cular process by which the transfer is effected. Suppose the 
 charged body to be positively electrified, so that its second-order 
 waves are repulsive, and its molecular attraction of the first 
 order acting on the superficial atoms exceeds the atomic repul- 
 sion. This attraction will consequently be of sensible amount 
 beyond the extreme atoms, decreasing, however, rapidly with 
 the distance according to its usual law. Now it has been pre- 
 viously shewn (p. 522) that the second-order repulsive waves 
 of the electrified body, acting on the other as a whole and at 
 any distance, only electrify it inductively. But when contact 
 takes place, the above-mentioned excess of first-order mole- 
 cular attraction, extending beyond the confines of the charged 
 body, comes into action, and tends to draw out locally the 
 atoms of the other body, and so to induce positive electricity. 
 But because the bodies are in contact, and the action arid 
 reaction between them must be equal, as much as the atoms 
 of one are drawn out the atoms of the other are pushed in, 
 so that reciprocally an equal quantity of negative electricity is 
 locally induced in the charged body. These local effects exist 
 only for an instant. By reason of the conductive properties 
 of the bodies, the induced positive electricity of one, and the 
 induced negative electricity, or abstraction of positive elec- 
 tricity, of the other, are immediately diffused over the whole 
 of the respective surfaces, the former accompanied by second- 
 order molecular repulsion, and the latter by second-order 
 molecular attraction, acting throughout the interiors. This 
 diffusion of the electricities replaces the bodies in relative 
 electric conditions proper for a repetition of the same process, 
 so that the charged body loses continually positive electricity, 
 and the other continually receives accessions of the same, till, 
 within a very brief interval, the equilibrium of the atoms of 
 both is established. The quantities lost and gained being 
 equal, the total quantity of electricity remains the same, as 
 is found to be the case by experiment. It is plain that 
 
536 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 analogous reasoning might be employed if the charged body 
 were supposed to be negatively electrified. Such appears to 
 be the theory of electrifying a conductor by contact. 
 
 If the second body were a non-conductor, it would still 
 acquire electricity by contact with the charged body, because 
 the electrifying action of the latter would, just as in the 
 former case, locally displace its atoms from their ordinary 
 positions, and the reciprocal local action would be the same. 
 Hence also the imparted electricity will be of the same kind 
 as that of the charged body; but it will not in the same 
 manner be diffused by superficial conduction, or, at least, only 
 to a very limited extent*. It might, therefore, be inferred 
 from the foregoing reasoning that only a very small amount 
 of electricity could thus be communicated to a non-conductor. 
 But here a physical circumstance which has an important 
 bearing on the present enquiry, must be taken into account. 
 It appears that although non-conductors are for the most part 
 devoid of the property of superficial conduction, they exhibit 
 a kind of internal conduction in the normal direction, in virtue 
 of which superficially communicated electricity penetrates 
 gradually some depth below the surface. This law I take 
 as an experimental fact, having already explained (p. 518) 
 that the present theory does not profess to give d priori 
 reasons for the different properties of conductors and non- 
 conductors. (For the experiments from which this property 
 of non-conductors is inferred see Jamin, Tom. I., pages 398 
 and 439.) Now it is conceivable that in consequence of this 
 conduction of electricity inwards the extreme superficial atoms 
 of a non-conductor may be continually brought back to con- 
 ditions fit for receiving additional quantities of electricity from 
 
 * The electrifying of a non-conductor by simple contact with an electrified 
 conductor is well exemplified by an experiment described by M. De La Rive in 
 page 90 of Tom. i. of his Treatise on Electricity. The interior surface of a glass 
 vessel having been touched at various points by the end of a rod connected with 
 the conductor of an electric machine, the presence of electricity was proved by 
 reversing the vessel on a table over a heap of pith-balls, which were thereby set 
 in motion. 
 
THE THEORY OF ELECTRIC FORCE. 537 
 
 the charged body, the process being analogous to that by 
 which a conductor was supposed above to receive accessions 
 to its electricity by means of superficial conduction. But the 
 operation, instead of being instantaneous as in the latter case, 
 is shewn by experiment to proceed by perceptible degrees. 
 In a short time, however, a large amount of electricity might 
 in this way be communicated by a charged conductor to a 
 non-conductor, if the surface of contact be large. 
 
 This inference being admitted, let us now suppose two 
 metal plates A' and B to be under the same electrical circum- 
 stances as the plates of the electrical condenser treated of in 
 section (2), and to be, besides, separated by a non-conducting 
 plate of glass, or resin, of somewhat larger diameter*. Then, 
 the connection of A with the machine and that of B with the 
 ground being broken, the whole of that portion of the surface 
 of the glass-plate with which A is in contact, will, according 
 to the above principles, be electrified positively by J.'s ac- 
 cumulated positive electricity, and the whole of the surface 
 contiguous to B will be electrified negatively by jB's ac- 
 cumulated negative electricity. 
 
 Theory alone cannot determine how much of A's and ,5's 
 electricities are thus transferred to the glass-plate; but by 
 the aid of experiment it may be concluded as follows that 
 they part with the whole of their primary electricities. For 
 assuming this to be the case, they will then retain only in- 
 ductive electricities due to their mutual actions and that of 
 the electricities of the glass. If, therefore, the plate A be 
 removed from the influence of the other two, it will give no 
 indication of electricity because its opposite inductive elec- 
 tricities will neutralize each other by conduction. The case 
 will be the same with respect to B if the glass-plate, as well 
 as A, be entirely separated from it. These inferences are 
 exactly verified by experiment, so that the above assumption 
 respecting the complete transfer of the electricities to the 
 
 * The experiment here referred to is described in pages 436 and 437 of Tom. i. 
 of Jamin's Cours de Physique. 
 
538 THE MATHEMATICAL PEINCIPLES OF PHYSICS. 
 
 glass is justified. The glass-plate, after the separation from 
 A and B, retains the opposite electricities, obviously because 
 they cannot be neutralized by superficial conduction. To 
 complete this explanation it remains to shew how the indi- 
 vidual atoms of the glass-plate are kept in their new positions 
 of equilibrium ; which, I conceive, may be done by the fol- 
 lowing argument. 
 
 Since the face with which A was in contact is positively 
 electrified, its superficial atoms are urged from without to 
 within by an excess of molecular attraction of the first order 
 above the atomic repulsion. (See page 514.) This excess is 
 counteracted by repulsion-waves of the second order, proceed- 
 ing from points of the interior, and acting from within to 
 without. The superficial atoms of the face with which B was 
 in contact, which is electrified negatively, are urged from 
 within to without by an excess of atomic repulsion above the 
 first order of molecular attraction, which excess is counter- 
 acted by attraction-waves of the second order proceeding from 
 points of the interior. These second-order repulsions and 
 attractions conspire in producing a gradual increment of in- 
 terior density from the face electrified negatively to that elec- 
 trified positively, and are in turn counteracted at each point 
 by the excess of interior atomic repulsion in the opposite 
 direction necessarily coexisting with this gradation of density. 
 It may be observed that this reasoning does not shew that 
 there is any necessary relation between the quantities of the 
 electricities on the two faces of the plate ; but if the con- 
 siderations adduced in page 509 may be regarded as proving 
 generally, that plane-waves of the second order, producing 
 equal amounts of attractive and repulsive forces acting in the 
 same direction, are generated simultaneously, it will follow, 
 from the above conditions of the equilibrium of the atoms, 
 that the electricities on the opposite sides of the plate are 
 equal in quantity. This result appears to accord with ex- 
 perience. 
 
 If two points on the opposite faces of the glass-plate be 
 
THE THEORY OF ELECTRIC FORCE. 539 
 
 touched by a discharging rod the electricities are discharged 
 at and very near these points, but not elsewhere, because the 
 equilibrium of the surrounding atoms is not disturbed by 
 tendency to superficial conduction. But if the metal plates 
 be replaced, and the discharging rod be applied to them at 
 any two points, the whole of the sensible electricity, supposing 
 the quantities on the opposite faces of the glass to be equal, is 
 discharged, because by means of the plates each point of one 
 face may be considered to be brought into electrical connection 
 through the rod with a corresponding point of the other. It 
 is found, however, that after an interval small amounts of the 
 same electricities reappear at the surfaces, giving evidence of 
 the reality of that property of gradual internal conduction on 
 which the above explanations essentially depend. It may be 
 supposed that this conduction, which is a gradual transfer of a 
 certain state of the atoms constituting electricity, brings to the 
 surface after the discharge an electrical residuum extending 
 to some distance below the surface. 
 
 The foregoing arguments and results account for well- 
 known properties of the Leyden Jar. 
 
 (5) I now proceed to consider the influence of the air in 
 electrical phenomena. In the first place it is to be observed 
 that the second-order waves of an electrified body act upon 
 the surrounding air to a considerable distance from the surface, 
 condensing it near parts negatively electrified, and rarefying it 
 near parts positively electrified. Consequently if the surfaces 
 of two bodies oppositely electrified be brought near each other, 
 there will be a continuous increase of density of the air from 
 the surface of the body positively electrified to that of the 
 other, produced both by the repulsion-waves of the former and 
 the attraction-waves of the latter. Let us suppose, for the 
 sake of precision, that the surfaces are parallel planes, and 
 that the second-order repulsions and attractions act equally 
 at all points of the space between them. Then the pressure 
 of the air on the negatively electrified surface will be as 
 much greater than that which exists in the case of no electri- 
 
540 THE MATHEMATICAL PEINCIPLES OF PHYSICS. 
 
 city as the pressure on the other surface is less than the same. 
 Also the difference between the two pressures will be propor- 
 tional to the interval between the surfaces. The increased 
 pressure of the air on the negatively electrified surface will 
 increase the condensation of the extreme atoms, and by conse- 
 quence increase the amount of second-order attraction-waves, 
 it having been already argued that there is a fixed relation 
 between the action, attractive or repulsive, of second-order 
 waves, and the degree of the superficial condensation of the 
 atoms. So the diminution of the pressure of the air on the 
 surface of the positively electrified body, by diminishing the 
 condensation of the extreme atoms, increases the excess of 
 first-order molecular attraction, to counteract which increment 
 it is necessary that there should be additional action of the 
 second-order repulsion-waves. These additions to the second- 
 order repulsions and attractions increase the difference be- 
 tween the pressures of the air at the two surfaces. 
 
 But in proportion as the interval between the surfaces is 
 diminished, that difference becomes less, while at the same 
 time the excess of first-order molecular attraction of the posi- 
 tively electrified body, and the excess of atomic repulsion of 
 the other, begin to produce proportionately greater effects. 
 Both these forces tend to increase the pressure of the air on 
 the positively electrified surface, and to diminish that on the 
 negatively electrified surface, and, therefore, to diminish both 
 the attraction-waves and the repulsion-waves of the second 
 order. These are just the opposite effects to those mentioned 
 above. When, therefore, in consequence of the diminution of 
 the distance between the surfaces the latter action prevails 
 over the other, a discharge takes place for the same reason, 
 and in the same manner, as when electricity is communicated 
 by contact (see p. 535) : only being effected by the interven- 
 tion of the air, it occurs while the bodies are still separated by 
 a small interval. 
 
 From the mode in which atomic repulsion and molecular 
 attraction of the first order operate to produce a discharge, it 
 
THE THEORY OF ELECTRIC FORCE. 541 
 
 may be inferred that if the opposite surfaces were curved and 
 convex towards each other, the axis of the discharge would 
 be the straight line of least distance between them, because 
 in that direction the sum of the radii of the spheres of 
 activity of these forces has a maximum ratio to the course 
 traversed. If, however, the resistance to the discharge were 
 not the same in all directions, that circumstance might modify 
 the course it would take. The fact that an electric discharge 
 is sudden may be accounted for by the rapidity of change of 
 value of the two above-mentioned forces with change of dis- 
 tance from the surfaces. 
 
 Simultaneously with the sudden diminution of the second- 
 order repulsive and attractive forces, the density of the air is 
 suddenly increased towards the positively electrified body, 
 and diminished towards the other. These sudden changes of 
 the density of the air may be the cause of the crepitations 
 which accompany discharges. 
 
 The atoms of the air are made to take the new positions 
 by the action of aetherial waves which impress on them, toge- 
 ther with the motions of translation, very rapid vibrations. 
 The disturbance of the aether produced by these vibrations 
 may be supposed to be the cause of the electric spark. 
 
 It is known by experiment that air is a non-conductor. 
 We may, therefore, suppose it to have the property of interior 
 conduction discussed in page 536, and consider the electric 
 discharge to be unaffected by lateral or superficial conduction. 
 On this account an electric spark does not spread laterally, but 
 is generally confined to a very narrow course. Similar effects 
 might be produced by the intervention of any other non-con- 
 ductor. For instance, it is found that if a plate of glass be 
 placed between two points oppositely electrified, on causing 
 the points to approach each other, a discharge takes place in 
 such manner that a hole is made through the glass. This 
 effect may be attributed to vibrations of the same kind as those 
 which are supposed above to agitate the air so as to cause the 
 electric spark, it being assumed that they may be of such 
 
542 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 extent and rapidity as to be capable of breaking the con- 
 tinuity of the solid matter of the glass. 
 
 The foregoing theory of the communication of electricity 
 by contact may account for the loss of electricity which con- 
 tinually takes place when an electrified body is surrounded 
 by air, even when the air is dry and non-conducting. Air in 
 this state will, like any other non-conductor, receive electricity 
 by contact with the electrified body at the expence of the 
 electricity of the latter, in the manner explained in page 536 ; 
 and if, besides, as we have already supposed, it possesses the 
 property of interior conduction, a transfer of electricity of the 
 same kind as that previously attributed to solid nori-conduc- 
 tors will account for the observed continuous abstraction of 
 electricity in air. Experiments seem to shew that the rapidity 
 of this conduction and the distance to which it extends are 
 much greater in a large mass of air than in a solid, and also 
 greater in air as the air is rarer. 
 
 This communication of electricity to the air by internal 
 conduction will, as in all cases of change of atomic equili- 
 brium, be accompanied by rapid vibrations of the atoms. It is 
 found, in fact, that when the conduction is intensified by being 
 made to take place from a sharp point connected with the 
 conductor of an electric machine, a conical luminosity, called 
 the brush discharge, is seen in the dark, giving evidence of 
 the agitation of the aether by the vibrations of the aerial atoms. 
 The appearance of the luminosity is different according as the 
 electricity is positive or negative, being broader and longer at 
 a point charged positively than at one charged negatively, if 
 the electricity be of low intensity. The difference in this, as 
 in other respects, between the phenomena of the two electri- 
 cities, might, according to the theory, be attributable to the 
 circumstances that in positive electricity the second-order 
 waves are repulsive, and in negative electricity attractive, and 
 that the values of X are less for the former class of waves than 
 for the other (p. 509). Besides, as will afterwards be shewn, 
 an electric current coexists with each of these forces, flowing 
 
THE THEORY OF ELECTRIC FORCE. 543 
 
 from or towards the point according as it is positively or 
 negatively electrified. 
 
 In the experiment of the electric egg (Jamin, p. 403),- the 
 electric luminosity is made to extend between two points con- 
 siderably distant from each other, by simply rarefying the air 
 in which the discharge passes from one point to the other. It 
 would seem that in rarefied air the excursions of the atoms 
 are large and easily excited. The luminosities due to the 
 vibrations of the atoms of different gases exhibit different 
 colours, as is shewn by means of Geisster's tubes. In abso- 
 lute vacuum, according to the theory, no luminosity would be 
 seen, as, also certain experiments, which will be referred to 
 under the head of galvanism, appear to indicate. It is possible 
 that the repulsive force of the second-order waves, or the ac- 
 companying current mentioned in the preceding paragraph, 
 may detach particles from the surface of the positively electri- 
 fied point. In that case the electric spark might be modi- 
 fied by the addition of these particles to those of the air, and 
 thus the different colours it presents, with the apparent de- 
 pendence of the colours on the substance of the points, might 
 be accounted for. (Jamin, pp. 401 and 402.) 
 
 Air mixed with vapour of water is found to be a conductor 
 of electricity. It seems that the admixture of the atoms of 
 water communicates to the air the property of superficial con- 
 duction at the expence of that of internal conduction, the rea- 
 son probably being that these atoms, since they are mechani- 
 cally in suspension, but not chemicall}*- united with the atoms 
 of the air, are more susceptible than the latter of change of 
 positions of equilibrium. The rapid loss of electricity which 
 occurs when an electric machine is worked in moist air, may 
 be attributed to conduction of the electricity to the ground 
 along the superficies of the air in contact with the solid parts, 
 whether conductors or not, of the machine. This loss is dis- 
 tinct from that which may be due to actual condensation 
 of moisture on the insulating supports. (Jamin, pp. 366 
 and 375.) 
 
544 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 A discharge of electricity is accompanied by heat (Jamin, 
 p. 402). The heat may be attributed to such vibrations of 
 atoms as those to which the luminosity of the brush discharge 
 was considered to be due (p. 542), inasmuch as they may be 
 supposed to generate setherial vibrations proper for producing 
 heat as well as light. 
 
 The phenomenon of the electric wind (Jamin, p. 405) is 
 reserved for subsequent consideration because it will come 
 under the head of electric attractions and repulsions, to the 
 theory of which I now proceed. 
 
 It might at first sight appear that the theory which ac- 
 counted for induced electricity would also explain the attrac- 
 tions and repulsions by which bodies in a state of induced 
 electricity are found to act upon each other so as to produce 
 motions of translation. But these motions cannot be due to 
 the second-order attractions and repulsions, because, if this 
 were the case, two bodies would repel each other when 
 positively electrified and attract each other when negatively 
 electrified ; whereas the fact is, in both cases they repel each 
 other. Hence the dynamical action of the aether by means of 
 waves fails to give an explanation of the phenomena. We 
 are, therefore, compelled by the mathematical theory of Hydro- 
 dynamics to make the supposition that it acts by currents; 
 and this, I think, will be found to be the true explanation. 
 The general theory of physical forces proposed in this work 
 would be altogether at fault if it did not account for electrical 
 attractions and repulsions without having recourse to any 
 other principles than those already enunciated. I propose to 
 shew by the following argument that these principles are 
 sufficient for the purpose. 
 
 It has been concluded (p. 510) that the induced electricity 
 of any body is always accompanied by a uniform increase of 
 its interior density from the parts negatively electrified to the 
 parts positively electrified, and that the interior density is the 
 same at all points of any plane perpendicular to the direction 
 in which the second-order molecular forces of the inducing 
 
THE THEORY" OF ELECTRIC FORCE. 545 
 
 body ten d to move the atoms of the other. These laws, being 
 general, are applicable to the reciprocal action of the induced 
 electricity on the inducing body. Therefore, since the induced 
 electricities of the two bodies are of opposite kinds at their 
 nearest parts (p. 522), it follows that their interior densities 
 increase in the same direction, the increase being in the direc- 
 tion from the originally electrified body to the other if the 
 original electricity be positive, and in the contrary direction 
 if it be negative. This being understood, it will next be shewn 
 that there are other forces, distinct from the second-order at- 
 tractions and repulsions, which also tend to move the bodies. 
 
 It appears from a previous argument (bottom of page 499) 
 that a uniform stream of the aether might enter into and pass 
 through an aggregation of atoms constituting either an aeri- 
 form, or liquid, or solid substance, without sensible alteration, 
 at any point, of the mean stream, that is, without variation of 
 the quantity of fluid which passes in a given interval a given 
 portion of any plane transverse to the direction of the stream. 
 But if the space occupied by the matter of the atoms contained 
 in a given small space have to that space a different ratio at 
 different points of the plane, the mean velocity of the fluid 
 will also be different at different points. For on this suppo- 
 sition the same quantity of the aether flows in the same time 
 through virtual channels of different transverse sections. 
 
 Let, for instance, the average velocity of the stream at a 
 certain point A be F, and at another point B be V, and let 
 the points be so situated that the straight line drawn from A 
 to B is in the direction in which the density of the body 
 increases. Then if the average density of the sether be p at 
 A and p at B, by the law just stated we shall have Vp = V'p. 
 Suppose V Q to be the velocity of the aether where the above 
 mentioned ratio is D , and let the ratio be D at A and D' at 
 B. Now since the variations of the velocity of the aether and 
 of the density of the body are supposed to be extremely small, 
 if k be an unknown constant factor, we shall have 
 
 D,)} and F = F,{1 +k(lf - D,)}. . 
 
 85 
 
546 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 By substituting these values of V and V in the equation 
 Vp = V'p', it will be found that 
 
 Hence supposing that p = p -f A/D, D r = D + AZ>, and that 
 the distances of A and B from a fixed plane at right angles 
 to the direction of the straight line joining them are z and 
 z -f Ag, we shall have 
 
 Ap k A# . AD 
 
 r 1 = - T-TF; - TTT -j = # T~' ver y nearly, 
 p Az 1 + A; (Z> - 2? ) Az As 
 
 because, "by hypothesis, k (D Z> ) is an extremely small 
 fraction. 
 
 But -- jr- may be considered to be a mean accelerative 
 pA0 
 
 force urging the fluid in the positive direction of z, that is, the 
 direction in which the density of the body increases. Hence 
 
 since that force is shewn above to be equal to ha* -r , which, 
 
 on the supposition that the density increases uniformly, is a 
 positive constant, it follows that throughout the space in 
 which this variation of the density of the body operates, the 
 fluid is urged at each point in the direction of the increase of 
 density by a constant force due to the variation of its own 
 density. Consequently a secondary steady stream is generated 
 in that direction, which remains steady as long as the original 
 stream is uniform. 
 
 If we conceive to be impressed on the asther and the atoms 
 of the body motions equal and opposite to that of the original 
 stream, the aether will be reduced to rest, and the body will 
 move through it with a uniform velocity. We have already 
 concluded (Note, p. 500) that under these circumstances there 
 is no permanent transfer of any of the fluid in the direction of 
 the motion of the body, as much flowing backwards as its 
 atoms displace. Hence it appears, since the impressed motion 
 produces no change of relative motion, that the same 
 
THE THEORY OF ELECTRIC FORCE. 547 
 
 secondary steady stream is generated as before, and in the 
 same direction relative to the moving body. 
 
 A necessary condition of this effect is that the motion of 
 the body be uniform. But clearly the result would be the 
 same if instead of moving in a rectilinear course it revolved 
 uniformly in a circle of large radius compared to its linear 
 dimensions. Also if the motion, although variable, be com- 
 posed of uniform motions, as, for instance, when it consists of 
 simultaneous uniform motions in a straight line and a circle, 
 the steady secondary motion will still be produced, because, 
 as is shewn in pages 242 and 243, steady motions due to 
 independent causes may coexist*. Hence in the above instance 
 the resulting secondary stream is the sum of the streams 
 which the rectilinear and the curvilinear motions would gene- 
 rate separately, the previous reasoning having shewn that a 
 secondary stream is generated by forces which are independent 
 of the angle made by the direction of their action with that of 
 the original stream, being determined both as to amount and 
 direction solely by the gradation of density of the moving 
 body. 
 
 The foregoing results are immediately applicable to the 
 case of an electrified body, -supposing the variation of internal 
 density to be that which, according to the theory, always 
 accompanies induced electricity, and the secondary streams to 
 be produced by the stream which relatively passes through the 
 earth in consequence of its motion about the sun. To the effect 
 of this stream may be added, as the foregoing reasoning 
 shews, the comparatively small effect of that due to the 
 rotation of the earth about its axis, as well as the effect of the 
 relative stream passing through the earth in consequence of 
 the motion of the solar system in space, which, for a very long 
 
 * This law is proved generally by shewing, as may readily be done, that in any 
 
 du dv dw 
 case of the steady motion of a compressible fluid -v- + -3- + -7- = 0, if terms of 
 
 the third and higher orders be omitted. In the proof cited above, the reasoning 
 5 restricted to the case in which ittix + vdy + wdz is an exact differential. 
 
 352 
 
548 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 period, may be regarded as uniform and rectilinear (see p. 505). 
 Accordingly, each of these three motions being supposed to 
 generate a secondary stream by the agency of the gradation 
 of density in the inductively electrified body, the resulting 
 stream is the sum of the separate streams, and is sensibly 
 constant, because the sum of the three motions is very nearly 
 constant, it being evident that the secondary streams will 
 cceteris paribus be proportional to the primary streams. This 
 result agrees with the observed general constancy of electrical 
 translatory action under the same conditions at different epochs. 
 The explanation I propose to give of electrical attractions and 
 repulsions depends essentially on this generation of secondary 
 steady streams, the preceding theory of which is a strict 
 deduction from antecedent principles without involving any 
 new hypothesis, and the reality of which will be subsequently 
 confirmed by its being applied in accounting for phenomena 
 of Galvanic Force and Magnetic Force. It remains to shew 
 in what particular manner the attractions and repulsions result 
 from the action of the secondary streams. 
 
 For the sake of distinctness suppose the two electrified 
 bodies to be spheres, and one to be originally electrified either 
 positively or negatively, and the other to be electrified by its 
 influence. Then the electricity of the former is partly primary 
 and partly inductive, and that of the latter wholly inductive. 
 The primary electricity of the first sphere will, as we have 
 seen (p. 516), be accompanied by a gradual variation of the 
 density from the centre to the surface equally in all directions 
 from the centre. This gradation of density gives rise to no 
 streams, because opposite pyramidal portions of the sphere 
 having their vertices at its centre will produce equal and 
 opposite effects. But the induced electricity of each sphere 
 will be accompanied by a gradual variation of density from 
 plane to plane perpendicular to the straight line joining their 
 centres, because the resultants of their second-order molecular 
 forces act in this line. (See pp. 509 511). Also since the 
 adjacent parts of the two spheres are oppositely electrified the 
 
THE THEORY OF ELECTRIC FORCE. 549 
 
 increase of density will be in the same direction for one as for 
 the other. These variations of density will generate in the 
 interiors of the spheres secondary streams, whose mean 
 direction is opposite to the direction of the increase of density. 
 Now without attempting to calculate the velocity at any 
 point resulting from the secondary streams, we may assert 
 (1) that the total motion will be symmetrical about the line 
 joining the centres of the spheres ; (2) that the velocity due 
 to the action of each sphere will be less on any given radius 
 prolonged the greater the distance from the centre, and on any 
 spherical surface concentric with the sphere the points of its 
 intersection with the axis of the motion will be points of 
 maximum velocity ; (3) the velocities due to the actions of the 
 two spheres coexist, since each of the two motions is steady. 
 Also with respect to each motion it may be assumed that on 
 every line of motion there is a distant point at which the 
 velocity may be considered to be zero, and the density to be 
 that of the fluid in its undisturbed state. Hence by the 
 reasoning in pages 241 243, an equation such as (26) in 
 page 241 applies to each motion, and an equation such as (27) 
 in page 243 applies to the motion resulting from the compo- 
 sition of the two motions : that is, if p represent the resultant 
 density and V the resultant velocity, we have 
 
 ' " *^ -Z2 / -y*\ 
 p = p*e ** = Po (l - jjjjjj very nearly, 
 
 p being the undisturbed density. This equation indicates 
 that the density, and therefore the pressure, is greater as the 
 composite velocity is less. 
 
 It is next to be observed that for each sphere the lines of 
 motion of the secondary streams will be symmetrical with 
 respect to a plane through its centre perpendicular to the axis 
 of the motion, and the velocities will be equal on the opposite 
 sides of the plane at corresponding points of the lines. For 
 while the variation of the interior density of the sphere acts 
 as a constant force maintaining the steady motion, the fluid at 
 
550 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 any distance from its centre on the side from which the motion 
 proceeds is accelerated by variations of its own density due 
 solely to its inertia, and the fluid on the other side is retarded 
 by variations of its density equally due to its inertia. These 
 two forces, being referable to the same cause, and alike related 
 to the disturbing action, will be equal in amount at equal 
 distances from the centre of the sphere, and will consequently 
 produce the above-mentioned symmetrical motion. It hence 
 follows, from the equation (26), that the density also, so far as 
 it depends on this motion, will be the same at corresponding 
 points on the opposite sides of the above-mentioned plane, 
 increasing equally in both directions from that plane. 
 
 But when the coexistence of the secondary streams of the 
 two spheres is taken into account, if on the axis of the motion 
 any two points be chosen equally distant from the centre of 
 one of the spheres, the velocity at the point which is the 
 nearer of the two to the other sphere will be more added to by 
 the stream of the latter than the velocity at the more remote 
 point, because the velocity of the stream is less the greater the 
 distance from the centre of its originating sphere. Thus if the 
 velocities at the two points due to the sphere between them 
 have the same value V l , and those due to the other sphere be F 2 
 at the nearer point and V z r at the more remote, the composite 
 velocity V l -f F 2 at the former point will be greater than the 
 composite velocity V l + V^ at the other, because F 2 is greater 
 than F 2 '; also if the densities corresponding to these composite 
 velocities be p and p', we shall have 
 
 , /, (p.+r.n , / (v t +v 
 
 P-P'V "a? J' f'f^ 1 -& 
 
 and consequently 
 
 Since F 2 exceeds F 2 ', the right-hand side of the last equation 
 is negative, so that p is less than p', or the density is greater 
 at the remoter point than at the nearer. It may be remarked 
 
THE THEORY OF ELECTRIC FORCE. 551 
 
 that for a given distance between the points p p is a greater 
 negative quantity as the excess V z F 2 ' and the sum F 2 -f F 3 ' 
 are greater, and also as F z is greater, which quantity depends 
 immediately on the amount of induced electricity of the second 
 sphere. The general inference from the foregoing result is 
 that the above-mentioned symmetrical increase of density on 
 the opposite sides of the transverse planes through the centres 
 of the spheres, is altered so that the increase becomes less 
 rapid in spaces included within the planes than in the cor- 
 responding spaces outside. 
 
 This point being ascertained, we can now determine the 
 dynamical action of the steady composite waves on the 
 spheres. It is evident that their action on the individual 
 atoms is the same as that which is treated of under Example 
 viii. in page 313, and we may accordingly at once adopt the 
 conclusion arrived at in page 315, namely, that the accelera- 
 tive action on an atom has a constant ratio to the accelerative 
 force of the fluid where the atom is situated. Hence it fol- 
 lows that the atoms in the two nearer halves of the spheres, in 
 which the increase of the density of the sether is less rapid 
 than in the other two halves respectively, will be urged from 
 each other in less degree than the atoms of the latter halves 
 are urged towards each other. Upon the whole, therefore, 
 the spheres will be attracted towards each other. 
 
 As the reasoning has all along been equally applicable 
 whether the electricity of the originally electrified sphere 
 were positive or negative, we may conclude that in both cases 
 the spheres attract each other. A fortiori there would be 
 attraction if the two spheres were originally electrified with 
 opposite electricities, because the addition of primary elec- 
 tricity to the induced electricity of the second sphere would 
 have the effect of increasing the inductive electricities of 
 both. 
 
 Let us now suppose that two spheres, both positively 
 electrified, are brought within each other's influence. Then 
 their repulsive second-order waves will cause the electricities 
 
552 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 to be minima at the nearest points of their surfaces, and 
 maxima at the most distant Since the tendency of these 
 repulsions is to push the spheres farther apart, the interior 
 variation of density from plane to plane perpendicular to the 
 axis through the centres of the spheres, will in each sphere be 
 an increase of density in the direction from the middle point 
 between them. The secondary streams due to these varia- 
 tions of density will consequently flow towards that point, 
 and, therefore, in opposite directions. Also, as in the previous 
 case, apart from the effect of their composition, these streams 
 will be symmetrical as to velocity and density with respect to 
 planes through the centres of the respective spheres perpen- 
 dicular to the axis of the motion. This symmetry will be 
 disturbed by the composition of the contrary motions in such 
 manner that the increments of the velocity of the aether, and 
 the consequent decrements of its density, in the directions 
 towards those planes, will become more rapid in the adjacent 
 hemispheres than in those more separated. The spheres will, 
 therefore, on the whole be urged asunder, or the electric action 
 will be repulsive. 
 
 If the two spheres be negatively electrified, the second- 
 order attractions will tend to draw them towards each other, 
 their interior densities will increase in the directions towards 
 the middle point between them, and therefore, as before, the 
 secondary streams will flow in opposite directions. Hence by 
 just the same reasoning as that above the electric action will 
 be repulsive. 
 
 The theory has thus shewn, in conformity with experience, 
 that when two spheres, one electrified either positively or 
 negatively, and the other either in a neutral state, or oppo- 
 sitely electrified, are brought within each other's influence, 
 they mutually attract; and that two spheres, both electrified 
 either positively or negatively, mutually repel. If, however, 
 the electricity of one of these spheres be small, its primary 
 electricity may happen to be less effective than the electricity 
 induced by the other, in which case there would be attraction. 
 
THE THEORY OF ELECTRIC FORCE. 553 
 
 Although the preceding reasoning is restricted to the 
 mutual electric action of two spheres, it may serve to indicate 
 the principles on which electric attractions and repulsions 
 between bodies of other forms might be investigated. 
 
 For instance, in the gold-leaf electroscope (Jamin, p. 410), 
 the strips of gold-leaf diverge whether electrified positively 
 or negatively, because their second-order waves, repulsive or 
 attractive, produce gradations of interior density increasing in 
 opposite directions, and, as consequences of these gradations, 
 oppositely flowing secondary streams. The velocity in each 
 stream is a maximum at the middle points of the originating 
 strip, and has equal values at points equally distant from the 
 two faces. Under these circumstances the gradations of den- 
 sity due to the composition of the velocities may be shewn, 
 as in the case of the two spheres, to exceed in the space 
 between the strips those in the spaces outside; and conse- 
 quently, since the former gradations tend to separate the 
 strips and the latter to bring them together, upon the whole 
 they will be made to diverge, and that too whether the elec- 
 tricity be positive or negative. It might seem to be an ob- 
 jection to this explanation that the gold-leaf is so thin that 
 the interior variation of its density would be insufficient to 
 generate the secondary streams. But relative to this point it 
 is to be considered that the opposite superficial disturbances 
 of the atoms at the opposite faces of a plate by the second- 
 order waves, and by consequence the total variation of in- 
 ternal density from face to face, are independent of the thick- 
 ness of the plate ; so that the interior gradation of density is 
 the more rapid, and therefore the more effective, the thinner 
 the plate, but acts during a proportionally less time on a 
 given particle of the sether. For this reason it may be pre^ 
 Burned that an extremely thin plate might generate perceptible 
 secondary streams. 
 
 When a small body in a neutral state has been electrified 
 inductively and attracted by a large electrified body, on 
 coming into actual contact with the latter it is observed to 
 
554 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 be quickly repelled. This fact is to be accounted for by the 
 communication of electricity from the larger to the smaller 
 body in the manner explained in page 535, in consequence 
 of which they are charged with the same kind of electricity 
 and become mutually repellent. 
 
 The electric wind observed to flow from a point connected 
 with the conductor of an electric machine admits of the fol- 
 lowing explanation*. The conductor communicates to the 
 air in contact with it a portion of its electricity, according to 
 the law explained in page 536, and the quantity received 
 from any part of the surface is in proportion to the electricity 
 jat that part. Hence as the pointed form tends to intensify 
 the electricity, the air will be electrified in greater degree at 
 and near the point than at other parts of the surface of the 
 conductor. Now as the air is a non-conductor this excess of 
 electricity does not spread by superficial conduction, and will 
 only be subject to some degree of internal conduction. Con- 
 sequently two bodies in contact, the air and the solid point, 
 have the same electricity, and, therefore, by the intervention 
 of their secondary streams they become mutually repellent. 
 Thus the air is driven off, and as it is immediately replaced 
 by the flow of surrounding air, which in turn is electrified, 
 a continuous stream is produced. Also, as the effect is the 
 same whether the conductor be electrified positively or nega- 
 tively, it follows that if a point be brought near the conductor 
 so as to be electrified by induction, the electric wind should 
 be produced in this case just as in the other. This result 
 accords with experience, it being found that in both cases the 
 air flows from the point. 
 
 It remains to account for the law of electric attractions 
 and repulsions as determined by experiment. With refer- 
 ence to this enquiry, it is first to be remarked that the 
 second-order waves of the theory tend to produce motions 
 of translation of a mass, and that they are attractive or re- 
 
 * See Jamin, Cours de Physique, Tom, i. p. 405, and Ganot's Physics, trans- 
 lated by Atkinson, Art. 651. 
 
THE THEORY OF GALVANIC FORCE. 555 
 
 pulsive according as tlie body from which they emanate is 
 electrified negatively or positively. But since, as we have 
 seen, electrical attractions and repulsions do not conform to 
 this rule, it must be concluded that the translatory force 
 of the second-order waves is very small compared with that 
 of the streams which they generate by the intervention of the 
 internal gradation of density of the atoms. It would appear 
 that with respect to the motion of translation of the mass 
 this mediate action is much more effective than the immediate 
 action of the waves on the atoms. Since this inference de- 
 pends both on the degree of elasticity of the aether and on 
 the ratio of the space occupied by the matter of the atoms 
 to the intervening space, it could not be deduced solely by 
 d priori investigation. 
 
 It follows from this consideration that electrical attrac- 
 tions and repulsions are very nearly in proportion to the 
 amount of gradation of density produced by the action of the 
 second-order waves. Supposing, therefore, the waves from a 
 given sphere to be incident on a small body at rest, since 
 their disturbances of its superficial atoms would vary in- 
 versely as the square of the distance from the centre of the 
 sphere, the gradation of density generated in the body would 
 vary with its mean distance very nearly according to the 
 same law. Hence the attraction or repulsion of small bodies 
 at rest by an electrified sphere varies inversely as the square 
 of the distance from its centre. The experiments by which 
 this law was ascertained by Coulomb were made under the 
 circumstances supposed in the theory. (Jamin, p. 354.) 
 
 The Theory of G-alvanic Force. 
 
 In the theory of Electric Force the phenomena treated 
 of were only such as are produced originally by friction. 
 On that account they are properly called electrical, and are 
 distinctly separated from the class of phenomena the theory 
 of which I now enter upon, which, as having been first 
 noticed by Galvani and Volta, will be termed for distinction 
 
556 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 galvanic or voltaic. In both classes the superficial atoms 
 of substances, and, as a consequence, their interior atoms, 
 are put into abnormal positions, differing by very small 
 quantities from those which, as being occupied anterior to 
 any disturbance, may be called their natural positions. This 
 displacement of the atoms results in galvanic phenomena 
 solely from the contact of two dissimilar substances. It 
 appears, however, from experiment that the contact, or sol- 
 dering together, of two solids has no such effect in any 
 perceptible degree, and that for the production of galvanic 
 phenomena one of the substances must be a fluid. The case 
 which ordinarily occurs is that of acidulated fluid in contact 
 with a metal, the admixture of the acid having the effect 
 of either originating, or increasing, the galvanic action. 
 
 It will be assumed theoretically that the forces by which 
 a solid and a fluid in contact disturb each other's superficial 
 atoms are those which I have named atomic repulsion and 
 molecular attraction of the first order. The phenomena of 
 capillary attraction may be regarded as giving evidence that 
 such forces are brought into action by the mere contact of a 
 fluid with a solid. The kind and degree of the mutual action 
 between the two substances will evidently depend much on 
 the laws of chemical affinity, whether or not it result in 
 actual analysis. I have already stated (p. 485) that I do 
 not profess to give in this work a theory of chemical analysis 
 and synthesis, although I believe that for propounding such 
 a theory it would be required to take account only of the 
 magnitudes and arrangement of atoms, and of the action 
 of the atomic and molecular forces the laws of which have 
 been the subject of previous mathematical investigation. As 
 far as regards the bearing of chemical affinity and analysis 
 on galvanic phenomena, the facts of experience will be cited 
 without attempting to give a priori reasons for them. 
 
 It is proper to state here that in the following theory of 
 galvanic action I assume, for the reasons stated in the theory 
 of electrical action, that any disturbances of superficial atoms 
 
THE THEORY OF GALVANIC FORCE. 557 
 
 from their natural positions are accompanied by related dis- 
 turbances of the interior atoms, producing regular gradations 
 of the density of the substance, and that simultaneously with 
 the disturbances the second-order molecular forces proper for 
 maintaining the atoms in their disturbed positions are called 
 into action. 
 
 Before proceeding with the theoretical considerations, it 
 will be necessary to determine by reference to experiments 
 what is the seat, and what are the conditions, of galvanic 
 action. For this purpose I cite, first, the following experi- 
 ment described by M. De La Eive in his Treatise on Elec- 
 tricity (Tom. II., p. 785). In two separated and well in* 
 sulated vessels, containing the same acidulated liquid, the 
 extremities of a zinc and copper couple are plunged, the two 
 metal plates being soldered together at their upper parts. 
 The liquid of the vessel in which the zinc plate is inserted 
 is connected with the ground by a platinum wire, and by 
 means of another platinum wire furnished with an insulating 
 handle, the zinc, the copper, and the liquid in the other 
 vessel, are brought successively into communication with a 
 Volta's condensing electroscope. . By this means it is shewn 
 that while the liquid in the first vessel is rendered neutral by 
 the connection with the ground, the zinc, the copper, and 
 the liquid in the other vessel are negatively electrified. The 
 platinum wire forming connection with the ground is then 
 transferred to the liquid in which the copper plate is inserted, 
 and the consequence is that this liquid, the copper, and the 
 zinc become neutral, and that the other liquid is positively 
 electrified. Evidently, therefore, the seat of the develop- 
 ment of the electricity is not at the junction of the zinc and 
 copper, but at the points of contact of the acidulated liquid 
 and the zinc. 
 
 From this experiment it may also be inferred that by the 
 mutual action between the liquid and the zinc, the liquid 
 acquires positive electricity, and the zinc negative electricity. 
 This is so important an inference that to confirm it I shall 
 
558 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 adduce another experiment*. A zinc capsule containing 
 dilute sulphuric acid is placed on a disk of moistened paper 
 in contact with the upper plate of a Volta's condensing elec- 
 troscope. The liquid is connected with the ground by dip- 
 ping in it the extremity of a platinum wire, and the lower 
 plate of the condenser is also connected with the ground by 
 being touched with the moistened finger. On breaking the 
 connections and removing the upper plate by its insulating 
 handle, the gold leaves are found to be positively electrified, 
 and consequently indicate that the zinc is negatively electri- 
 fied. In order to account for this phenomenon the following 
 considerations are proposed. 
 
 By reason of the chemical affinity between the sulphuric 
 acid and the zinc, it may be presumed that atomic and mo- 
 lecular action tending to produce, if not actually producing, 
 a chemical analysis of the fluid, takes place at its points of 
 contact with the zinc. In consequence of such action the 
 atoms in a thin superficial stratum, whether of the zinc or 
 the fluid, do not continue in their natural positions, the 
 change in each substance being such that the gradation 
 of density in the thin stratum becomes at all points either 
 more rapid, or less rapid, in a given small interval than it 
 was before the disturbance of the natural state. Without an 
 exact knowledge of the forces concerned in chemical affinity 
 and of their laws of action, it is not possible to determine by 
 theory alone the precise character of the changes that take 
 place in the state of the superficial atoms of the two sub- 
 stances. I shall, therefore, now make the provisional hypo- 
 thesis that in cases of such action as that between the zinc 
 capsule and the dilute sulphuric acid, the liquid and the parts 
 of the metal in contact with it are positively electrified, and 
 that the opposite side of the zinc is negatively electrified. 
 
 The truth of this hypothesis will subsequently be tested 
 by the applications that will be made of it in explaining 
 
 * This experiment is described in Atkinson's Edition of Ganot's Physics, 
 Art. 081. 
 
THE THEORY OF GALVANIC FORCE. 559 
 
 phenomena. But an a priori reason for adopting it may be 
 drawn from the considerations entered into in pages 507-510, 
 where it is shewn that the traction of a body by a cord 
 applied at any point of its antecedent part may give rise to a 
 positive electric state on that side, and a negative electric 
 state on the opposite side, the former accompanied by second- 
 order repulsive waves, and the other by second-order attractive 
 waves, both acting in the direction of the impressed force. 
 Also the mutual action between the cord and the body may 
 be taken to represent the immediate action between the acid 
 and the zinc due to chemical affinity, inasmuch as the simul- 
 taneous stretching of the cord and of the parts of the body 
 to which it is applied, corresponds to that diminution of 
 density of the contiguous superficial parts of both the sub- 
 stances, which theoretically co-exists with the positive electri- 
 cities ascribed by hypothesis to both. It is true that in the 
 case of galvanic action the substances are not moved bodily 
 as in that adduced for illustration; but if we suppose that 
 by such action an analysis of the fluid, and an actual transport 
 of constituent atoms, take place, there would then be an 
 effect corresponding to the movement of the body as a whole, 
 and in some degree equivalent to it. According to this view 
 it might be argued that proper galvanic action is always 
 accompanied by chemical analysis, because otherwise the 
 liquid and metal in contact could not both be positively 
 electrified. 
 
 From the application of the foregoing hypothesis to the 
 second of the two experiments adduced above, it follows that 
 the liquid and the contiguous side of the zinc capsule are 
 positively electrified, and the opposite side negatively elec- 
 trified. This being admitted, it may be assumed that the 
 positive electricities both of the liquid and the zinc will be 
 abstracted by the platinum wire which communicates with 
 the ground, while the remaining negative electricity of the 
 zinc is imparted to the upper plate of the condenser, the 
 conduction being facilitated by the intervening moistened 
 
560 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 paper. This electricity acts through a covering of varnisli 
 inductively on the lower plate, the induced positive electricity 
 of which is retained, while the negative electricity is ab- 
 stracted by the connection of the under part of the plate with 
 the ground. Thus on breaking the connection there remains 
 only positive electricity, causing the gold-leaves to separate 
 just as if the original electricity had been frictional. It 
 should, however, be observed that the electricities we are 
 here concerned with are much feebler than those usually pro- 
 duced by friction, and particularly it should be noticed that 
 the retention of the negative electricity of the capsule after 
 the liquid is connected with the ground is effected by the 
 insulation of the condenser and the galvanic action between 
 the liquid and the zinc, which still goes on after that connec- 
 tion has been established. 
 
 The theoretical explanation of the first experiment above 
 cited may now be given as follows. Since the zinc plate 
 forms part of a galvanic couple, the action of the liquid on it 
 is influenced by the copper plate to which it is joined. The 
 mode of this influence appears to be such as follows. Since, 
 by hypothesis, the galvanic action between the liquid and 
 the zinc on one side of the latter electrifies oppositely and 
 equally the opposite sides, and a like effect is produced by 
 the galvanic action on the other side, it follows that the 
 electric state of the zinc is rendered neutral. There remains 
 the positive electricity of the liquid, accompanied, as usual, 
 by second-order repulsive waves. These waves are propa- 
 gated from the liquid towards the zinc on both sides in equal 
 degree. Those propagated in the direction from the copper 
 to the zinc have no effect on the copper because they originate 
 in liquid separated from it. But those propagated in the 
 opposite direction, being incident on the copper plate, elec- 
 trify it on both sides according to the laws of electrical induc- 
 tion ; that is, the side towards the zinc is electrified negatively 
 and the other positively. The copper in this state reacts upon 
 the zinc, electrifying positively the adjacent side and nega- 
 
THE THEORY OF GALVANIC FORCE. 561 
 
 tively the farther side. Consequently the positive electricity 
 of the gold-leaves of the electroscope when the liquid in con- 
 tact with the zinc is connected with the ground, is accounted 
 for in this experiment precisely as in the other. The positive 
 electricities of the liquid and zinc being abstracted by the 
 connection, the negative electricity of the latter is diffused by 
 the laws of conduction over the zinc, the copper, and the 
 liquid in contact with the copper, and is retained by means 
 of the continuous galvanic action between the liquid and the 
 zinc and the insulation of the second vessel. This electricity 
 in the three substances is detected by the condensing elec- 
 troscope. 
 
 After transferring the platinum wire to the liquid in con- 
 tact with the copper, the negative electricity of each of these 
 substances is abstracted by conduction; and the copper be- 
 coming neutral has no longer any influence on the zinc, which, 
 therefore, from what is argued above, itself also becomes 
 neutral. This explains why under these circumstances the 
 electroscope indicates that no other electricity remains than 
 the positive electricity of the liquid in contact with the zinc. 
 It is to be observed that this electricity is retained by the 
 continuous galvanic action between the acid and the zinc, 
 and the insulation of the vessel in which the operation pro- 
 ceeds. 
 
 In the foregoing explanation it has been assumed that 
 there is no galvanic action between the acid and the copper. 
 If there should be such action, or if for the copper a metal be 
 substituted on which the acid acts more energetically than 
 on zinc, the sensible electricity would result from the differ- 
 ence of the two actions. 
 
 The foregoing explanations of these two fundamental ex- 
 periments may be regarded as giving prim & facie evidence 
 of the truth of the hypothesis I have adopted relative to the 
 superficial conditions of the contact of a solid and liquid 
 between which there is chemical affinity, and of the conse- 
 quent distinction between frictional electricity and galvanic 
 
 36 
 
562 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 electricity*. The hypothesis will receive confirmation by 
 the application I now proceed to make of it in giving a theory 
 of the galvanic battery, and of the currents it generates. 
 
 It will suffice for indicating all that is essential in the 
 theory to take the case of a single couple consisting of a plate 
 of zinc and a plate of copper partially immersed in dilute 
 sulphuric acid, with faces opposite and parallel. To the 
 upper part of each plate a metallic wire is attached, arid the 
 galvanic circuit is closed or opened according as the two 
 wires are brought into contact, or are separated t. This 
 arrangement differs from that of the experiment described in 
 page 557, in so far as the plates are immersed in fluid con- 
 tained in a single vessel, and insulation is unnecessary. But 
 on the same hypothesis respecting galvanic action, the account 
 given in page 560 of the mutual action between the liquid 
 and the zinc, and of the effect of the influence of the copper, 
 applies equally to both experiments. Consequently the zinc 
 is electrified positively on the side facing the copper, and 
 negatively on the opposite side, and the copper is electrified 
 negatively on the side towards the zinc, and positively on 
 the other side. Hence since the interior gradation of density 
 is always from negative electricity, and towards positive elec- 
 tricity, and the secondary streams always flow towards the 
 denser parts, it follows that the galvanic action tends to pro- 
 duce a current in the direction from the zinc to the copper. 
 
 The original positive electricities of the liquid on the two 
 sides of the zinc tend to produce equal currents in opposite 
 directions and therefore neutralize each other, and the liquid 
 in contact with the copper is by hypothesis originally in a 
 neutral state. But the inductive action of the zinc electrifies 
 
 * In conformity with the practice of experimentalists, and because the electro- 
 scope shews that galvanic action and friction may produce identical effects, I desig- 
 nate the effect in either case by the same word " electricity," although it properly 
 applies to that produced by friction. The term "galvanic" will be used as 
 opposed to "frictional" when it may be necessary to distinguish between the 
 two modes of generating the electric state. 
 
 f See the figure in Art. 682 of Atkinson's Edition of Ganot's Physics. 
 
THE THEORY OF GALVANIC FORCE. 563 
 
 positively the liquid in contact with the copper on the inner 
 side, and negatively that on the outer side ; and the inductive 
 action of the copper electrifies positively the liquid in contact 
 with the zinc on the outer side, and negatively that on the 
 inner side. Hence the effective electricity of the liquid tends 
 also to produce a current from the zinc towards the copper. 
 
 But by experiment it is found that no current flows unless 
 the two metallic wires are put in contact, and that as soon 
 as the current commences the galvanic action is increased. 
 We must next endeavour to account theoretically for these 
 two phenomena. 
 
 Assuming that the galvanic currents of experiment are 
 identical with the theoretical setherial currents, it follows, 
 according to experimental indications, that a good conductor 
 allows an astherial current to permeate freely through its 
 interior, and that its bounding surface contains the stream 
 within limits just as a closed channel of any shape contains 
 a stream of water. It would seem too from experiment that 
 the aether flows through some substances more readily than 
 through others, and that, according to the general law of 
 streams, it takes the course in which it can flow most freely. 
 I shall not attempt to account for the difference of conducting 
 powers of bodies relative to galvanic currents, this being a 
 property of the same class as, and apparently connected with, 
 the conducting and non-conducting qualities of substances 
 relative to factional electricity. But it is possible, I think, 
 to explain by considerations such as those I am about to 
 adduce, in what manner conducting bodies are in effect chan- 
 nels of galvanic currents. 
 
 With this object in view I propose, first, to submit to 
 mathematical treatment the case of a steady stream disposed 
 symmetrically about a rectilinear axis, and to investigate the 
 laws of such motion, this hydrodynamical problem not having 
 been discussed in the earlier part of the work. I begin by 
 asserting that the motion cannot be wholly parallel to the 
 axis and at the same time be different at different distances 
 
 362 
 
564 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 from it, because in that case, according to the laws of steady 
 motion, the pressure would generally be less where the velo- 
 city is greater, and consequently there would be variation 
 of pressure in directions transverse to the axis, which would 
 destroy the parallelism of the lines of motion. Let us, there- 
 fore, assume that at any distance r from the axis the motion 
 is compounded of the velocity F(r) parallel to the axis, and 
 the velocity /(r) in circles the planes of which are transverse 
 to the same. Then supposing the axis of x to be coincident 
 with the axis of the motion, we have 
 
 It may readily be shewn that these values of u, v, and w 
 satisfy the equation 
 
 du dv dw _ 
 
 dx dy dz " 
 
 Also by substituting the same values in the equation 
 
 dp du du du du 
 
 j + ^r + j~ u + ^~ v + -j-w = 0, 
 
 dx dt dx dy dz 
 
 and the two analogous equations 
 
 . , 
 dy dz 
 
 since the motion is supposed to be steady, the resulting equa- 
 tions will be found to be 
 
 Hence 
 
 dr 
 
 It thus appears that the proposed kind of motion satisfies the 
 second and third general hydrodynamical equations inde- 
 pendently of the particular forms of the functions F and /. 
 
THE THEORY OP GALVANIC FORCE. 565 
 
 But according to the principles of Hydrodynamics I have 
 heretofore maintained, it is, besides, a necessary condition 
 that the lines of motion should always and everywhere be 
 normals to surfaces of continuous curvature ; which principle 
 is analytically expressed by saying that udx + vdy + wdz 
 must either be integrable of itself or by a factor. The sub- 
 stitution of the foregoing values of w, v, and w shews that the 
 equation is not integrable of itself, so that we must conclude 
 that the supposed motion is not possible unless the equation 
 udx -f vdy -f wdz = be integrable by a factor. The condition 
 of such integrability is, as is well known, that the values of 
 u, v, and w satisfy the equation 
 
 dv dw\ fdw du\ fdu dv\ - 
 ^---j-)+v(-j -- -7-+w-7---r-=0. 
 dz dyj \dx dz) \dy dx) 
 
 The result of substituting these values is the equation 
 
 I - f(r) F'(r] -F(r) /(r) + = 0. 
 
 Ifence by integration, 
 
 /(*-)_ a 
 F(r) r> 
 
 a being an arbitrary constant. This result establishes a re- 
 lation between the functions f(r) and F(r) which must be 
 satisfied by the supposed kind of motion. 
 
 By taking account of this relation the equation 
 
 udx + vdy + wdz 
 gives 
 
 dz = - rpp) (y dx - *%) = p ( xd y ~ v dx )- 
 
 Hence by integration, 
 
 z a tan" 1 - -f 5, 
 x 
 
 which is the general equation of the surfaces of displacement. 
 By changing the sign of the constant a the direction 'of the 
 circular motion about the axis is changed. 
 
566 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 The above relation between f(r) and F(r) would have 
 been equally obtained if the axis had been supposed recti- 
 linear only through a very small length, and may conse- 
 quently be considered to be applicable to the case of a curvi- 
 linear axis. 
 
 The preceding reasoning has proved the possibility of the 
 flowing of a steady stream symmetrical with respect to a 
 rectilinear or curvilinear axis, and has indicated that such 
 motion must be compounded of uniform rectilinear motion 
 parallel to the axis and uniform circular motion about the 
 same. The resulting motion is therefore spiral, as appears 
 also from the foregoing equation of the surfaces of displace- 
 ment. The mathematical investigation has conducted to a 
 definite expression for the ratio of the components f(r) and 
 F(r), and as this result was obtained independently of any 
 particular mode of disturbance, it may be considered to denote 
 a law of the mutual action of the parts of the fluid. If the 
 problem were solved with the requisite degree of generality, 
 the forms and values of these functions would be deduci- 
 ble from given conditions of the motion. But since in the 
 existing state of Hydrodynamics such general solution does 
 not appear to be practicable, I shall only attempt to deduce 
 from the preceding argument some definite conclusions rela- 
 tive to the case in which the axis of the motion is the axis 
 of a metallic wire known from experiment to be capable of 
 transmitting galvanic currents. 
 
 The setherial stream being supposed to be steady, and 
 such that the same quantity of fluid passes in the same time 
 every section transverse to its mean course, it is evident that 
 any contraction of the channel will increase the velocity of 
 flowing. Hence when a stream flows within and contiguous 
 to a metallic wire, the atoms of the wire, so far as they con- 
 tract the channel by the occupation of space, increase the 
 velocity of the stream. The augmented velocity will be the 
 same at all parts of the interior of the wire where the density 
 of the atoms is the same ; and consequently at those parts 
 
THE~THEORY OF GALVANIC FORCE. 567 
 
 the pressure of the fluid will be uniform, or (dp) = 0. Hence 
 it appears from the foregoing value of this differential that 
 / (r) = 0, or that there is no circular motion. It must not, 
 however, be inferred from the relation between f(r) and F (r) 
 that the latter function also vanishes, inasmuch as that rela- 
 tion depends on udx + vdy 4- wdz being only integrable by 
 a factor, whereas for the motion now under consideration this 
 differential is integrable of itself. Or, we may say that 
 f(r) = because the arbitrary constant a vanishes when the 
 velocity parallel to the axis does not vary with the distance 
 from the axis. 
 
 But when the variation of the density of the atoms, which 
 necessarily exists within a thin superficial stratum of the wire, 
 is taken into account, it may be shewn as follows that this 
 physical circumstance may be accompanied by circular motion 
 of the fluid. For the decrement of density towards the super- 
 ficies, by causing the effective channel of the stream to in- 
 crease from point to point in that direction, produces a dimi- 
 nution of the velocity of the stream, and, in consequence of 
 the steady motion, an increase of the density of the fluid, also 
 towards the superficies. Thus there is generated an accele- 
 rative force urging the fluid towards the axis ; and as, by 
 hypothesis, no motion takes place in that direction, this force 
 must be counteracted, if not in any other way, by centrifugal 
 force due to circular motion. 
 
 Let us, therefore, now assume that the accelerative force 
 due to variation of pressure in the direction of the radius r 
 is exclusively counteracted by centrifugal force. Then the 
 equation 
 
 (dp) = {f(r)Y 
 dr r 
 
 is applicable at all distances from the axis, because it is a 
 necessary consequence of that assumption, the right-hand side 
 being the centrifugal force. But this equation requires to be 
 differently treated according as it applies to parts of the fluid 
 within the wire, or to the parts exterior. With respect to 
 
568 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the fluid within the wire bounded by a cylindrical surface 
 the radius of which is equal to the .distance from the axis at 
 which the superficial gradation of density commences, it has 
 already been argued that F(r) is constant, and (dp)=0; so 
 that f(r) = 0. From that distance to the extreme boundary 
 of the wire, the variations of the longitudinal velocity and 
 of the pressure of the fluid have been supposed to depend 
 only on the variation of the density of the atoms of the wire 
 in the thin superficial stratum. Consequently if the latter 
 variation were given, the quantities F (r) and p should admit 
 of being calculated for all points of the stratum. And p being 
 known, /(r) might be calculated for all points. 
 
 These calculations being assumed to be effected, let F(r^ 
 p^ and /(rj be the values thus obtained for the extreme 
 boundary of the wire, r l being its radius. It is next to be 
 considered that the relation obtained in page 565 between 
 f(r) and F(r) antecedently to any supposed conditions of 
 disturbance, applies only so far as the mutual action of the 
 parts of the fluid is concerned. Hence the equation express- 
 ing that relation is not applicable to the fluid within the wire, 
 the motion and pressure of which have already been differ- 
 ently determined; but is to be applied solely to the fluid 
 at, and external to, its extreme boundary. Accordingly we 
 have 
 
 which equation determines the arbitrary constant a, / (rj and 
 F(r^ being supposed to be already known; and for any 
 point outside the wire at the distance r from the axis, 
 
 Supposing, for the moment, the form and value of F(r), 
 or w, to be known, the last equation gives f(r), the constant 
 a having been determined. Hence u and v become known, 
 
 as also -~- , so that p can be found by integration. Thus 
 
THE THEORY OP GALVANIC FORCE. 569 
 
 the circumstances of the motion would on this supposition be 
 wholly ascertained. 
 
 With respect to the determination of x the value of F(r) 
 for any value of r, as depending on the particular values /(rj 
 and F(r^ we have here a problem analogous to that of 
 limited lateral divergence of vibrations discussed in pages 
 288 292. In the case of vibrations it was shewn that un- 
 limited lateral divergence is prevented by the composition of 
 vibrations which are partly longitudinal and partly trans- 
 versal. In the present case of a stream symmetrically dis- 
 posed about an axis, it appears that the like effect results 
 from the composition of circular motion about the axis with 
 the longitudinal motion parallel to it. In the one case, as in 
 the other, the longitudinal motion varies laterally according 
 to a law depending only on the mutual action of the parts of 
 the fluid, (that for the stream being expressed by F(r)), but 
 in neither case can the law be ascertained except by methods 
 of solving hydrodynamical problems which in respect to com- 
 prehensiveness are in advance of those adopted in this work. 
 Although, however, the foregoing solution of the present 
 problem is on this account incomplete, we may yet legiti- 
 mately draw from it the important conclusion that cetherial 
 streams along cylindrical wires are accompanied by transverse 
 circular motions, 
 
 We may now proceed to consider why the stream is 
 stopped if the galvanic circuit is not closed. Eesuming the 
 case of the zinc and copper couple immersed in sulphuric 
 acid, it appears from what has been argued, that the galvanic 
 action between the liquid and zinc gives rise to a stream of 
 aether, which after flowing through the liquid between the 
 plates and through the copper plate, is concentrated in the 
 wire attached to the copper, this being the channel in which 
 it can most readily proceed. When the current, by propa- 
 gation of the original impulse, has reached the extremity of 
 the wire, it is stopped, or sent backwards, for the following- 
 reasons. In the tirst place, it ceases to have an axis for the 
 
570 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 circular motion, which, by the previous argument, is neces- 
 sary for maintaining the steadiness and limited transverse 
 extent of the stream. If then it actually flowed beyond the 
 extremity of the wire, being no longer guided and kept within 
 limits, it would tend to diverge in all directions from that 
 position as from a centre. The motion may, in fact, be com- 
 pared to that which would be produced by impulses from the 
 preceding half of a small sphere moving in the fluid. Now 
 in the case of the sphere it has been proved (p. 264) that the 
 movement does not on the whole transfer any portion of the 
 fluid, just as much as the sphere displaces being sent back- 
 wards by a reflex action due to the inertia of the fluid mass. 
 The same reflex action, on account of the divergence of the 
 lines of motion, will take place in the case of motion before 
 us ; and as the reflected stream can flow without divergence 
 only along the wire as its channel, it is restricted to this 
 course, and consequently stops the original stream first at 
 the end of the wire, then through its whole length to the 
 copper plate, and finally at the zinc plate. At the same time 
 the galvanic action between the liquid and the zinc tends to 
 draw the aatherial fluid along the wire attached to the zinc, 
 and the effect of the traction at the extremity of the wire on 
 the adjacent aether is comparable with that resulting from the 
 motion in an elastic fluid of a small sphere, the following half 
 of which, while it draws the fluid after it, produces on the whole 
 no transfer of fluid in the direction of its motion. Accordingly 
 by the reaction due to the inertia of the aether and the con- 
 vergency of the lines of motion at the terminal, the stream is 
 neutralized along this wire also. Simultaneously with the 
 stoppage of the streams the circular motion also ceases. Thus 
 no dynamical effects result from the action between the liquid 
 and the zinc, the tendency to produce such effects being stati- 
 cally opposed by the reaction of the inert mass of aether*. 
 
 * The Theory of Galvanic Force which I proposed in the Philosophical Maga- 
 zine for December, 1860, agrees in all essential respects with that now given, 
 except as to the reasons assigned for the return current. I there say incorrectly, 
 " the current is driven back by the insulation of the battery and the non-conduc- 
 
THE THEORY OF GALVANIC FORCE. 571 
 
 KB. The wire attached to the copper plate will be 
 called the positive rheophore, and its extremity (P) the positive 
 pole; and the wire attached to the zinc plate the negative 
 rheophore, and its extremity (N ) the negative pole. Whatever 
 two metals constitute the couple, the positive pole is the 
 extremity of the wire out of which there may be reason to 
 conclude that a galvanic current tends to flow. 
 
 By the argument just concluded, so long as the circuit is 
 open, there is neither efflux of fluid from P, nor influx at N. 
 The considerations that have led to these results relative to 
 the points P and JV, may be extended to the angular points 
 and edges of all rheophores, (i. e. conductors of galvanic cur- 
 rents) and, in fact, to any points of their surfaces, because in 
 the absence of an axis of steady motion the fluid could not 
 issue at any point except as a divergent stream, nor enter at 
 any point except as a convergent stream, and therefore by 
 reason of the reaction from inertia it neither issues nor enters. 
 It is plain, however, that there is an exception to this law in 
 the case of the stream flowing from the zinc plate and into 
 the copper plate (p. 562). By the very process of generation 
 this stream is caused to issue from the zinc and to enter into 
 the copper: but it is a necessary condition of the flowing, that 
 the intermediate liquid be a rheophore of some degree of con- 
 ductivity*. After entrance into the copper the courses of the 
 lines of motion are influenced by the convergence of the stream 
 towards the positive rheophore. 
 
 For the above reasons the plates, the liquid, and the wires 
 may be regarded as irregular channels, of different degrees of 
 conductivity, confining the galvanic current as an irregular 
 
 tiveness of the air." At that time I had not recognised, in the case of the motion 
 of a small sphere in a fluid, the reflex effect arising from the inertia of the fluid. 
 See the remarks and the Note on this point in page 492. 
 
 * The degree of roughness of the surface of the plate may also have effect. It 
 is stated in Ganot's Physics (Art. 683) that a current flows from cast zinc to rolled 
 zinc. The action of the former might be more energetic than that of the other 
 because, on account of the greater roughness of its surface, the amount of effective 
 surface might be greater. But supposing an equality in this respect, the stream 
 incident on the smoother surface would be less obstructed than the other. 
 
572 THE MATHEMATICAL PEINCIPLES OF PHYSICS. 
 
 solid tube confines a stream of water. This would be true 
 also if the external rheophores, instead of being wires through- 
 out, were made to consist in part of interposed conducting 
 bodies of various forms. For according to the foregoing 
 argument, aetherial steady streams are capable of flowing 
 through such bodies as channels without suffering dispersion 
 or diminution. These theoretical results admit of being 
 applied in the explanation of various galvanic phenomena 
 and laws, as I next proceed to shew. 
 
 Let us, first, suppose the circuit to be closed by bringing 
 the ends of the two rheophores into contact. It is not neces- 
 sary for this purpose to put the wires end to end so that the 
 axis of one shall be continuous with that of the other; it 
 suffices if their surfaces be made to touch each other in any 
 manner. The reason for this is, that at the instant of contact 
 the negative rheophore becomes a channel for the stream which 
 is ready to issue from the positive rheophore, and as soon as 
 the channel is furnished, the resistance from the inertia of 
 the surrounding fluid to efflux from the positive rheophore at 
 any point, and to influx into the negative rheophore at any 
 point, is neutralized at the points of contact. At these points, 
 therefore, the stream passes out of the positive into the nega- 
 tive rheophore, and the fluid through the whole circuit is set 
 in motion. The motion is steady because the stream flows, 
 as we have seen, in a constant channel, and is maintained by 
 a constant action between the zinc and the liquid, operating 
 so as to analyse the latter by a process of which some account 
 will be given at a subsequent part of the theory. 
 
 From these considerations we may now draw the import- 
 ant inference that when the aetherial current is established, 
 however irregular and heterogeneous the conducting channel 
 may be, the same quantity of aether passes each section of it 
 in the same time. If the quantity of aether which passes a 
 section in a unit of time be assumed to be what experimental- 
 ists call the intensity of the galvanic current, we shall thus 
 have a theoretical explanation of the experimental law, that 
 
THE THEORY OF GALVANIC FORCE. 573 
 
 " the intensity is the same in all parts of one and the same 
 circuit*." 
 
 This comparison of the theory with experiment particu- 
 larly points to the hydrodynamical character of galvanic 
 currents. It appears that the intensity of the current is the 
 same at all sections of the circuit for the hydrodynamical 
 reason that the same quantity of incompressible fluid must 
 at each instant pass every section of a confined rigid channel, 
 independently of any transverse circular motion the fluid may 
 have about the axis of the channel. But the calculation by 
 theory alone of the intensity of a galvanic current under given 
 circumstances is a difficult problem, to the solution of which 
 physical science is not yet adequate. I propose, however, to 
 bring forward here certain considerations, which may, at least, 
 serve to shew what kind of research may be required for 
 effecting a complete solution. 
 
 Let us take the simple case of a current flowing in and 
 along a homogeneous cylindrical wire from the copper plate 
 to the zinc plate, and endeavour to ascertain on what circum- 
 stances the intensity depends. We have already seen that 
 the current has its origin in the galvanic action between the 
 liquid and the zinc, and that after traversing the liquid and 
 entering the copper, it passes out through the wire, and thus 
 returns to the zinc. But it is here to be remarked that the 
 current is not capable by its own impulse of either issuing 
 from the zinc or entering the copper, because, from the cir- 
 cumstance that the conductivity of the liquid is less than that 
 of either metal, a resistance from the inertia of the aether has 
 to be overcome the same in kind as if no liquid were interposed, 
 but in degree proportional only to the differences of the con- 
 ductivities. Hence if the galvanic action were to cease, the 
 current would not continue to circulate as water does in a close 
 re-entering channel after receiving a momentary impulse, but 
 would instantly be stopped. This being the case, continuous 
 
 * See Jamin, Cours de Physique, Tom. m., p. 102, and Ganot's Physics, 
 Art. 701. 
 
574 THE MATHEMATICAL PEINCIPLES OF PHYSICS. 
 
 galvanic impulses are necessary for maintaining the current. 
 Evidently also the momentum, estimated along the axis of 
 the stream, of the whole of the fluid that is at each instant 
 in motion, will be in some direct proportion to the galvanic 
 motive force. Supposing, now, the velocity parallel to the 
 axis to be a function of the distance from it, let I t be the 
 mean quantity of fluid which passes through a unit of area 
 transverse to the axes in a unit of time, and ra 2 be an area 
 such that I i x m z is equal to the whole quantity of fluid which 
 at any instant passes a transverse section. This product is 
 what was before defined to be the intensity of the current, and 
 is, therefore, the same for a given couple at all points of the 
 circuit. Hence, / being this intensity, and X the length of the 
 circuit, the product /X is the above mentioned momentum ; so 
 that if G be the specific galvanic motive force, and A be the 
 area of the zinc plate> we have. 
 
 ~ A I\ T Gc.A 
 GA = , or J= ^ , 
 Cj X 
 
 the factor c, representing the specific conductivity of the wire 
 by whatever conditions it may be determined. 
 
 In these formulae X is to be taken only through a portion 
 of the circuit for which c t is the same, and, therefore, in the 
 present instance does not include the interval between the 
 plates. Since we found that / is constant under different 
 forms and conditions of different parts of the channel, if 
 c i> c \i c i'> ^ c> > be * ne specific conductivities of portions whose 
 lengths are X, X', X", &c., and GA be the total galvanic force, 
 we shall have the general formula 
 I 
 
 It is now to be taken into account that so far as regards 
 the current along the wire, the lines of motion are spiral 
 (see p. 566), and that the conductivity of the wire, as ex- 
 pressed by the quantity c t , depends as well on this circum- 
 stance in the motion of the fluid, as on the specific quality of 
 the wire as a galvanic conductor. The more the lines of 
 
THE THEORY OF GALVANIC FORCE. 575 
 
 motion are spiral, the less will be the effective conductivity, 
 because /, which may be regarded as the projection of the 
 mean spiral stream on the axis, will be in the same proportion 
 less for a given value of GA. It is, further, to be observed 
 that the motion is wholly generated by the galvanic action, 
 and that the forces acting in planes perpendicular to the axis 
 only divert it into spiral courses; but the amount of this 
 diversion, or the ratio of the mean spiral motion to the mean 
 motion parallel to the axis, cannot, as I have already inti- 
 mated, be calculated theoretically for want of the appropriate 
 physical data and mathematical reasoning. Yet we may, I 
 think, infer from the argument -maintained in page 568, that 
 this ratio depends only on the ratio of the circular motion 
 f(r^) to the longitudinal motion F (r^ at a distance r^ from 
 the axis equal to the radius of the wire. In that case the 
 required ratio will be a function of r t , and supposing c to 
 represent the specific conductivity of the wire apart from any 
 effect of the motion of the fluid, we may put c(j> (r^) for c r 
 Consequently IX = GrcA<f> (r^). In this formula the part of 
 the circuit between the plates is left out of account. 
 
 Having thus proceeded in the theory as far as appears at 
 present to be practicable, I shall now adopt, as established by 
 experiment, Ohm's Law respecting the intensity of a galvanic 
 current. According to this law*, if co be the transverse section 
 of the wire, /A, = GrcAay. Hence, by comparison with the the- 
 oretical formula, < (r t ) = co. It is thus found that the conduc- 
 tivity, so far as it depends on the spiral motion, varies as the 
 square of the radius of the wire. Or rather, con sidering resistance 
 to be the inverse of conductivity, the resistance depending on 
 the spiral motion varies inversely as the square of that radius. 
 
 It is known from experiment that a fine wire along which 
 a galvanic current is transmitted emits heat and light, if the 
 current is not of low intensity. The present theory gives the 
 following explanation of this phenomenon. We found (p. 
 567) that the variation of density subsisting in a thin super- 
 
 * See Jamin, Tom. in., pp. 103 105, and Ganot's Physics, Art. 701. 
 
576 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 ficial stratum of every solid substance, produces in the case of 
 a wire traversed by an aetherial current a variation of the 
 density of the aether within the stratum, the density increas- 
 ing towards the exterior surface. The effect of this variation 
 on the aether itself is counteracted by the centrifugal force due 
 to the circular motion. But the same gradation of density 
 must have a motive effect on the superficial atoms of the wire, 
 urging them inwards. This disturbance necessarily excites 
 atomic and molecular forces by the action of which the atoms 
 in a thin superficial stratum are put into a state of continual 
 vibration. The rapidity or intensity of the vibrations will 
 plainly increase, in a wire of given transverse section, with 
 any increase in the intensity of the galvanic current. It is 
 assumed in this theory that these vibrations by disturbing the 
 aether generate the observed heat and light. 
 
 According to this explanation the developement of heat is 
 due to the same forces as those which produce the resistance 
 consequent upon the spiral motion. Hence the one effect will 
 be proportional to the other, and from the law found above 
 for the resistance, we may conclude that the heat developed 
 varies inversely as the square of the radius of the wire, the 
 intensity of the current being given. The heat thus gene- 
 rated at the superficies, after being diffused through the sub- 
 stance of the wire, will produce an increment of temperature 
 varying as the quantity of heat directly and as the transverse 
 section of the wire inversely, and therefore varying inversely 
 as the fourth power of the radius. This law may be extended 
 to a wire the different parts of which have different transverse 
 sections, the intensity of the current being supposed to be 
 the same at all parts. For according to the laws of the atomic 
 and molecular forces concerned in the equilibrium of heat, 
 the temperature at any point will depend on the action of 
 aetherial waves emanating from atoms whose distances from 
 it do not exceed a very small interval*. 
 
 * The law of increase of temperature at different parts of the wire, as ascer- 
 tained by experiment, is stated in Ganot's Physics, Art- 703, in terms exactly 
 
THE THEORY OF GALVANIC FORCE. 577 
 
 It is ascertained by experiment that the increment of tem- 
 perature is the same at all parts of a wire of any length, if its 
 radius and the intensity of the current be the same through- 
 out*. The theory evidently gives the same law. 
 
 We have found that the heat developed in any wire by 
 a current of given intensity is proportional to the wire's 
 resistance, and therefore varies inversely as the square of its 
 radius. The resistance and the size of the wire remaining 
 the same, what would be the effect of changing the intensity? 
 The experiments of Joule have proved that in that case the 
 amount of developed heat varies as the square of the intens- 
 ity t. Perhaps this law may be accounted for theoretically 
 as follows. It^has been argued (p. 576) that the accelerative 
 effect on the aether itself of the gradation of its density which 
 disturbs the superficial atoms of the wire and generates the 
 heat, is counteracted by the centrifugal force due to the cir- 
 cular motion at the distance r l very nearly. This force is 
 
 Now it was found (p. 568) that /(r t ) = - F (r^ ; 
 
 so that the centrifugal force is equal to 3 {F(r^} z . But since 
 
 r i 
 within the wire the velocity parallel to the axis is F(r^) very 
 
 nearly, and outside the wire we have 
 
 it follows that F(r^) varies nearly as the mean velocity parallel 
 to the axis, and, therefore, as the intensity /. Hence the heat- 
 producing force, being equal and opposite to the centrifugal 
 force, varies as / 2 , the radius r, of the wire being given. 
 
 From the equation I\ GfcAco we may infer that for a 
 given couple and given radius of the wire, the intensity of 
 
 agreeing with this theory. According to a formula employed by Jamin, (Tom. lit., 
 pp. 181 and" 182), the increment of temperature varies inversely as the cube of the 
 radius. 
 
 Ganot's Physics, Art. 703. 
 
 f Jamin, Tom. in., p. 172, and Ganot, Art. 703. 
 
 37 
 
578 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the current is less in proportion as the length of the wire is 
 greater. This law agrees with observation*. 
 
 Hitherto we have only treated of the course of the current 
 through the conducting wire joining the copper and the zinc. 
 If we now neglect this part of the circuit and attend only 
 to that between the plates, very simple considerations will 
 suffice. For here there is no circular motion, or it is so small 
 that it may be left out of account; and the intensity esti- 
 mated along the mean direction of the current, that is, the 
 direction perpendicular to the surfaces of the plates, is still 
 the quantity /. Hence if \ be the distance of the copper 
 from the zinc, the momentum of the aether between them is 
 I\, and if this were the whole of the momentum, it would be 
 equal to the total galvanic motive force ; so that I\ = GA. 
 This equation shews, since Cr is absolutely constant, that if \ 
 remain the same, I is increased proportionally to any increase 
 of the area of the zinc plate. The same inference may be 
 drawn from the general formula in page 574, the quantity in 
 the brackets being supposed to be constant t. 
 
 I come now to an important part of the theory of galvanic 
 force, requiring for its discussion some considerations that 
 have hitherto not been brought forward. From experiments 
 that will be presently cited it appears that on approaching 
 the terminals of a galvanic battery an electric spark passes, 
 indicating the presence of electricity properly so called. Now 
 it seems evident that the terminals do not derive their elec- 
 tricity by conduction from those of the plates with which 
 they are respectively connected, because the wires do not 
 require to be insulated, and when they are touched by a con- 
 ducting substance the electricities are not carried off. How 
 then, it may be enquired, are these electricities generated and 
 retained? To answer this question, I remark, first, that it 
 has been shewn, in the instance of a couple (p. 574), by rea- 
 soning which may be extended to a battery, that the galvanic 
 
 * Jamin, Tom. m., p. 182, 4. 
 
 f Jamin, Tom. HI., p. 182, 5; Ganot, Art. 701, v. 
 
THE THEORY OF GALVANIC FORCE. 579 
 
 motive force is at each instant equivalent to a momentum ex- 
 pressed by /X, / being the intensity of the current, and X its 
 length, reckoned from the positive pole through the battery 
 to the negative pole. When the current is open, / = by 
 reason of the reaction of the aether at the terminals, as already 
 explained. On making the terminals approach this reaction 
 is partially neutralized, and the current begins to flow. At 
 the same time by the impulse which causes it to flow, the 
 fluid surrounding the positive terminal is impressed in such 
 manner that a decrement of density and accompanying acce- 
 lerative force are produced in directions diverging more or less 
 from the the terminal. The same impulse acting in the same 
 direction by dragging the fluid in the negative rheophore, 
 produces at its terminal a decrement of density and consequent 
 accelerative force in the fluid surrounding it, in directions 
 more or less converging towards the terminal. In fact, it is 
 the action of these two accelerative forces at the ends of the 
 mass. of fluid between the terminals, that overcomes its inertia, 
 and sustains the current. 
 
 It will be seen that under these circumstances the accele- 
 rative force of the aether at and just within the positive, ter- 
 minal will push out its extreme atoms, and thus generate an 
 excess of molecular attraction of the first order above the 
 atomic repulsion, which excess, in case of equilibrium, just 
 counteracts the disturbing force. By this process positive 
 electricity is induced. So at the negative terminal the acce- 
 lerative force of the aether pushes in the extreme atoms, till it 
 is counteracted by an excess of atomic repulsion above mole- 
 cular attraction of the first order, by which means negative, 
 electricity is induced. It is particularly to be noticed that the 
 equilibrium of the atoms does not in these cases require the 
 action of second-order molecular forces, on which account 
 these electricities are not susceptible of being conveyed away 
 by conduction, and in that respect are distinguished from 
 frictional electricities. 
 
 From these theoretical considerations it may be inferred 
 
 372 
 
580 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 that after the generation (by chemical affinity between a metal 
 and a liquid) of the state of tension proper for producing a 
 galvanic current, on the approach of the terminals the reaction 
 of the aether abates, and the current commences flowing before 
 actual contact ; and at the same time the positive terminal is 
 electrified positively and the negative terminal negatively. 
 These electricities tend to neutralize each other by the inter- 
 vention of the air, as in the case of frictional electricity (p. 
 539), and a discharge takes place when the accelerative forces, 
 which, as explained above, maintain both the electric state of 
 the terminals and the flow of aether between them, are dimi- 
 nished to a certain point in consequence of the diminution, 
 by the approach of the terminals, of the quantity of aether to 
 be moved. 
 
 Davy states in his "Elements of Chemical Philosophy" 
 (Vol. i., p. 152), that with the same battery an electric dis- 
 charge took place in air of ordinary density between charcoal 
 points distant from each other about one-thirtieth of an inch, 
 and in rarefied air which supported one-fourth of an inch 
 of mercury, the sparks passed through nearly half an inch : 
 also that in the former case the constant or galvanic discharge 
 passed in the heated air through a space equal at least to four 
 inches, and in the latter through a space of six or seven 
 inches. These facts prove experimentally, not only that air 
 resists the passage both of electricity and of the galvanic 
 current in greater degree as its density is greater, but seem 
 to shew also that the galvanic discharge may go on while the 
 battery is in a state of electric tension. This inference is in 
 accordance with the present hydrodynamical theory, which 
 distinctly separates the electric from the galvanic discharge, 
 inasmuch as it indicates that an electric discharge is the 
 dynamical effect of aetherial vibrations which produce a sudden 
 return of superficial atoms in disturbed positions to their 
 natural state, whilst a galvanic discharge is the continuous 
 flow of a stream of aether due to a state of constraint of 
 interior atoms. For the maintenance of the current the elec- 
 
THE THEORY OF GALVANIC FORCE. 581 
 
 trical conditions of the superficial and interior atoms of the 
 plates are necessary; so that we must suppose such condi- 
 tions to be kept up in the required degree, after closing the 
 circuit, by the continuous galvanic action between the acid 
 and the active metal plate. 
 
 Discharges of great length may be passed from the ex- 
 tremity of one rheophore to that of the other within glass tubes 
 of small bore very nearly exhausted, such as Geissler's glass 
 tubes referred to in page 543*. These discharges are dif- 
 ferent from the electric egg spoken of in page 543, which is 
 strictly an electric phenomenon. Experiments by Mr Gassiot 
 (Phil. Trans. Vol. 149, page 156) have proved, in accordance 
 with what the hydrodynamical theory indicates, that a gal- 
 vanic current cannot flow in a perfect vacuum. It seems, 
 however, that while dense air is a non-conductor, air, or any 
 gas, in a very rarefied state admits of the flow of considerable 
 currents. In Geissler's apparatus the length of the discharge 
 is promoted by the form and quality of the tube, which are 
 favourable to the maintenance of the spiral movement, with- 
 out which the current between the terminals could not subsist. 
 The coloured light in these tubes may be caused by the vibra- 
 tions of the gaseous substance enclosed within them, the 
 atoms of which in its state of extreme rarity may be made 
 to perform large excursions by the dynamic action of the 
 current, and reciprocally may produce agitations of the aether 
 rendered sensible in the form of light. The stratification of 
 the light may be due to the positions of maximum and mini- 
 mum vibration which always result from exciting vibrations 
 of elastic gases in cylindrical tubes. The colour may very 
 probably be ascribed to the particular quality of the gas. The 
 glow at the terminals appears to have a different origin from 
 that of the stratification, being possibly due to the agitations 
 of the aether caused by the sudden changes of the stream on 
 issuing from, and entering into, the terminals. The difference 
 of the luminous appearances at the terminals may very well 
 
 * See Ganot's Physics, Art. 771. 
 
582 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 be accounted for by the difference between the circumstances 
 of a stream when it diverges from, and when it converges 
 towards, a narrow channel. The appearances favour the 
 hypothesis that the stream is from the positive towards the 
 negative terminal. 
 
 The phenomena of the voltaic arc* may be explained on 
 the principles of this theory as follows. The small electric 
 discharges perceived when the terminals of a battery are 
 brought very nearly into contact, indicate that during the 
 galvanic action the terminals are in a state of electric tension. 
 Hence, according to the theory explained in page 579, when 
 the circuit is a little open, and as soon as the galvanic cur- 
 rent has commenced, accelerative forces due to gradations of 
 density of the aether at and near the terminals, act repulsively 
 from the positive terminal, and attractively towards the nega- 
 tive terminal. If, therefore, the points are placed near and 
 opposite to each other, these forces urge outwards the extreme 
 atoms of the positive terminal, and inwards those of the other. 
 At the same time the atoms of the terminals are maintained 
 in vibration by the displacements thus caused and the anta- 
 gonistic action of the forces of aggregation, and consequently 
 heat is produced. Also heat might result from successive 
 small discharges through the air, causing partial returns of 
 the superficial atoms of each terminal to their natural state, 
 and alternating with resumptions of the normal state of elec- 
 tric tension. Since the tendency of the generated heat is 
 always to diminish the cohesiveness of the extreme particles 
 by opposing the attraction of aggregation, it may be presumed 
 that the above-mentioned accelerative forces are capable by 
 their joint action of continually detaching some of the extreme 
 atoms of the positive terminal, and transferring them to the 
 other. It is found, in fact, by observation that the substance 
 of that terminal continually wastes by a process of volatiliza- 
 tion. (A like effect has already been adverted to in the ex- 
 
 * See Jamin, Tom. in., pp. 188192; and Ganot, Art. 704 
 
THE THEORY OF GALVANIC FORCE. 583 
 
 planation given in page 543 of the electric egg.} These con- 
 siderations will also account for the circumstance that the 
 positive terminal is more heated than the other, it being rea- 
 sonable to conclude that atoms urged outwards to the point 
 if undergoing separation from the mass will be more agitated, 
 and thus generate more heat, than atoms compressed by equal 
 forces inwards. The light accompanying the heat accounts 
 for the luminous appearance of the terminals. 
 
 As soon as the volatilized matter is detached, the galvanic 
 current begins to flow more freely, because that matter, con- 
 stituting a conducting medium, acts as a rheophore* '. The 
 resistance being thus diminished, the galvanic action is in- 
 creased, and the gradations of density of the aether at the 
 terminals become on the whole larger, and extend to greater 
 distances. Hence, if the interval between the terminals be 
 increased, although the resistance from the inertia of the 
 intermediate aether becomes larger at the same time, the cur- 
 rent will go on flowing till the distance reaches a limit im- 
 posed by the amount of reaction of the aether. Experience 
 shews that this is the case, the voltaic arc admitting of large 
 extension under the supposed circumstances. 
 
 The detachment of matter at the positive pole, and its 
 transfer towards the negative pole, receive confirmation from 
 the fact (Jamin, p. 189), that when the positive pole is directly 
 over the other, the limiting length of the arc is greater than 
 in the contrary case. Evidently the explanation which the 
 theory gives of this fact is, that when the positive pole is 
 above, gravity assists the action of the disruptive and trans- 
 latory forces, and opposes it when that pole is below. This 
 explanation is decisive as to the direction of the galvanic cur- 
 rent being from the positive to the negative pole. 
 
 The brightness of the arc may be supposed to be caused 
 by the agitation of the aether resulting from the rapid transit 
 of disengaged atoms, and, therefore, to be in some direct pro- 
 
 * The term rheophore. or electrode, will be used for distinction according as a 
 substance is regarded as conducting a galvanic current, or frictional electricity. 
 
 I 
 
 
584 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 portion to the velocity of their motion and the number which 
 pass any transverse section in a given time. Now both these 
 quantities will, cceteris paribus, be greater the greater the 
 intensity of the current ; and as, in the case of a single couple, 
 the intensity is increased by increasing the size of the plates, 
 the brightness of the arc will also be thereby augmented. 
 Experiments cited by Jamin (p. 192) agree with this result, 
 and at the same time shew that the brightness increases but 
 very little with the number of the couples. The reason for 
 this most probably is that while the surface of galvanic action 
 is enlarged by adding to the number of couples, the resistance 
 to be overcome in the passage of the current through the liquid 
 and the plates is augmented in nearly equal degree. 
 
 But it appears from experiment that the length of the arc 
 is greater the greater the number of couples (Jamin, p. 188). 
 This fact may be accounted for by considering that when the 
 arc has its greatest length, the motive forces of the aether at 
 the terminals just suffice to overcome the resistance from the 
 inertia of the intermediate aether, and at the same time are 
 equivalent to the galvanic motive forces which maintain the 
 current through the battery. (This will be expressly shewn 
 subsequently.) But the latter forces are greater, for a given 
 intensity, the greater the resistance to be overcome, and, there- 
 fore, increase with the number of couples. Hence the former 
 forces augment under the same circumstances, and can thus 
 sustain a longer voltaic arc. 
 
 From experiments adduced by Jamin (p. 190) it appears 
 that matter is transferred from the negative to the positive 
 pole as well as from the positive to the negative. This fact 
 obviously seems to indicate that volatilization takes place at 
 loth terminals, in consequence of the great heat they are sub- 
 ject to from the aetherial vibrations excited by their own 
 atoms, and from those due to the transported matter of the 
 voltaic arc. Admitting this to be the case we may, farther, 
 assume that the volatilized matter generated at a terminal, 
 and remaining for an instant contiguous to it, will partake of 
 
THE THEORY OF GALVANIC FORCE. 585 
 
 its electricity (see pages 535 and 536). Consequently, not 
 only will there be repulsion between each terminal and the 
 contiguous volatilized matter (see p. 554), but each will also 
 attract the oppositely electrified matter volatilized at the other 
 terminal. Thus there will be transfer of matter loth ways, 
 and, it may be, in equal quantities, as far as this action is 
 concerned. But the before-considered effect of the flow of 
 the galvanic current from the positive to the negative terminal 
 will tend to make the transfer preponderate in that direction. 
 This argument would remain the same if it should happen 
 that the detached atoms immediately combine with the oxygen 
 of the air. 
 
 The quantity of transported matter will plainly depend 
 also on the degree of volatility of the terminals, and if they 
 consist of different substances, and the particles of the nega- 
 tive one be more susceptible of disruption than those of the 
 other, it may happen that the transfer of matter from the 
 negative terminal will be equal to, or even greater than, that 
 from the positive one. Again, from what has been already 
 argued, the brilliancy of the arc will be in some direct pro- 
 portion to the total quantity of matter transported in the 
 opposite directions, and will, therefore, be greater when the 
 less tenacious of two substances is at the positive, or more 
 energetic pole, than in the contrary case. Thus the arc is 
 found to be more brilliant when the current is from a silver 
 to a charcoal terminal than when it passes in the contrary 
 direction, apparently because silver is more subject to atomic 
 disintegration than charcoal. (See Jamin, Tom. III., p. 190.) 
 
 The form of the arc seems to depend on the translatory 
 action of the galvanic current, as distinct from that due to the 
 electrical state of the terminals. Since the galvanic current 
 is a steady stream diverging from the positive pole, and con- 
 verging towards the negative pole, its velocity will be least 
 and the condensation of the aether greatest, at positions about 
 mid-way between the poles. Hence each of the opposite 
 streams of volatilized matter will be retarded by the eether 
 
586 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 and made divergent through the first half of the course, and 
 be accelerated and made convergent through the other. The 
 boundary of the stream will thus be a curved surface the 
 transverse section of which is greatest about the mid-position. 
 It is also to be considered that the course of the stream may 
 be influenced by the resistance opposed to it by the air, and 
 that, if the line joining the poles be horizontal, the rising of 
 air rarefied by the generated heat might cause the courses of 
 least resistance to be such that the axis of the stream would 
 be curved upwards. This, in fact, is found to be the case. 
 
 From the foregoing theory of the voltaic arc, we may now 
 proceed to the theoretical consideration of the analysis of 
 liquids by galvanic currents. As, however, I have not given 
 a mathematical theory of chemical action, I am not prepared 
 to treat of this subject completely, inasmuch as the operation 
 by which such analysis is effected depends generally on a 
 chemical relation between the liquid and the plates of the 
 battery. Yet in some important particulars the present theory 
 of galvanic force may be brought into comparison with expe- 
 rimental facts of this class. I take, first, the experiment by 
 which Carlisle and Nicholson decomposed water*. 
 
 In this instance, although chemical affinity is concerned 
 in the generation of the galvanic current, the analysis of water 
 at two platinum terminals, which chemically are neutral to- 
 wards it, appears to result solely from their electrical state, as 
 accounted for in page 579. It was there argued that for main- 
 taining the current there must be a sudden decrement of density 
 of the gether outwards at the positive terminal, and a sudden 
 decrement inwards at the negative terminal. These variations 
 of density are distinct from those resulting from the con- 
 traction, at the terminals, of the channel for the galvanic 
 current, the latter being decrements of density towards the 
 terminal in both cases, and, it may be presumed, in equal 
 degree; and as producing on that account equal and opposite 
 
 * Jamin, Tom. in., p. 46; Ganot, Art. 709. 
 
THE THEORY OF GALVANIC FORCE. 587 
 
 effects, they may be left out of account in respect to the 
 phenomena under consideration. But the variations of density 
 necessary for the maintenance of the current will have the 
 effect at the positive terminal of pushing the extreme atoms 
 outwards, and at the negative terminal of pushing the ex- 
 treme atoms inwards. To counteract these disturbing forces 
 there will be an excess of first-order molecular attraction 
 above atomic repulsion at the positive terminal, and an excess 
 of atomic repulsion above first-order molecular attraction at 
 the negative terminal. It has already been explained that in 
 the production of this state of electrical tension, the second- 
 order molecular forces are not concerned as in frictional 
 electricity. 
 
 Now it may be laid down as a principle in this theory 
 that the chemical analysis, or synthesis, of substances can 
 only be effected by forces of the order of those by which the 
 constituent atoms in their natural state are held in positions 
 of equilibrium, the forces, namely, which I have called first- 
 order molecular attraction and atomic repulsion. This being 
 admitted, we may suppose that the excess of first-order attrac- 
 tion acting beyond the limits of the positive terminal on the 
 constituent atoms of the adjacent water, draws the atoms of 
 oxygen in greater degree than the atoms of hydrogen, reasons 
 having been given (pages 484 and 485) for concluding that 
 the attraction of this order accelerates small spheres in dif- 
 ferent degrees according to their magnitudes*. If the differ- 
 ence of the attractions of the atoms of oxygen and the atoms 
 of hydrogen exceed the attraction of aggregation, an analysis 
 of the liquid will take place, such as is exhibited in the 
 
 * If the reasoning in page 485 be good, the atom of oxygen is larger than the 
 atom of hydrogen. Whe'her as being the larger it would be the more attracted, 
 the mathematical theory of the first-order molecular attraction is not sufficiently 
 complete to determine. The above supposition, that the oxygen is attracted at 
 the positive terminal in greater degree than the hydrogen, is supported by the fact 
 that when the circuit of the usual zinc and copper couple is open, bubbles of 
 hydrogen are seen to adhere to the zinc plate. This plate, which acts as an 
 analyzer like the positive terminal, seems thus to have abstracted from the bubbles 
 their oxygen. 
 
588 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 experiment by the rising of bubbles of oxygen from the 
 positive terminal. At the same time the excess of atomic 
 repulsion extending beyond the limits of the negative terminal 
 may be supposed to repel in different degrees the atoms of the 
 contiguous water, and, as in the case of the attractive force, to 
 act more energetically on the oxygen than on the hydrogen. 
 In this way it is possible to account for the observed fact, that 
 hydrogen is set free at the negative terminal and rises up in 
 bubbles. 
 
 We have, farther, to take into consideration that as by a 
 chemical law oxygen and hydrogen are combined in definite 
 proportions to form water, the analyzed portion must contain 
 the constituents in the same definite proportions; as is found 
 to be the case by experiment. The physical views I advocate 
 admit of accounting for this law of decomposition by the 
 hypothesis of Grottlms*, according to which the hydrogen 
 equivalent from which the oxygen at the positive terminal is 
 liberated unites with the adjacent oxygen equivalent, the 
 hydrogen combined with which passes over to the next 
 oxygen equivalent, and so on, till the oxygen is reached from 
 which the hydrogen was set free at the negative terminal. 
 By this process the analysis necessarily takes place in the 
 definite proportions of the composition. It would seem that 
 the same conditions of the aether determine both the course of 
 the stream from the positive to the negative terminal, and 
 that taken by the successive decompositions and recomposi- 
 tions of the atoms. 
 
 It may be concluded from the above theory that the 
 quantity of liquid analyzed in a given time is in a certain 
 proportion to the intensity of the current, or to the quantity of 
 aether which in a unit of time passes out of the positive 
 terminal, or into the negative terminal. Hence if there were 
 
 * Jamin, Tom. in., p. 78; Ganot, Art. 713. The same hypothesis applies to 
 the remarkable experiment (Jamin, p. 81; Ganot, Art. 712) by which Davy 
 demonstrated the passage of constituent atoms in opposite directions along the 
 same rheophore. 
 
THE THEORY OF GALVANIC FORCE. 589 
 
 two or more sets of terminals in the same current, although 
 the intensity would not be the same as for a single set, since, 
 from what has been shewn (p. 573), it would be the same 
 throughout the current, the results of the analysis of a given 
 liquid would be the same at all the sets. The amount of de- 
 composition of the liquid at the plates is to be included in 
 this law, because, although the total moving force of the current 
 depends on the chemical relations of the liquid to the plates, 
 the amount of analyzed liquid depends on the quantity of 
 aether which flows out of a positive plate, or flows into a 
 negative plate, which quantity is the same measure of intensity 
 as that which flows at the terminals. This is the law referred 
 to parenthetically in page 584. It is one of the laws of 
 galvanic analysis discovered by Faraday. (Ganot, Art. 714.) 
 
 Experimentalists have inferred the identity of "common 
 electricity" and "voltaic electricity" from the fact that 
 chemical decomposition may be effected by the former. Ex- 
 periments recorded by Faraday in his Experimental Researches 
 in Electricity (Vol. I. Arts. 309 331) seem to establish the 
 fact. But whatever frictional electricity and the galvanic 
 current may have in common, it is certain that in some 
 respects they are widely different. For instance, the amount 
 of chemical decomposition that can be effected by a current of 
 frictional electricity is so extremely small compared to that 
 produced by even a feeble galvanic current, that there must 
 be some great diversity in the two modes of operation. 
 I shall now endeavour to indicate, by means of the foregoing 
 theories of electric force and galvanic force, the circumstances 
 on which this difference depends. 
 
 In the first place it is to be remarked that the electrodes in 
 Faraday's experiment require to be strictly insulated, and in 
 that respect differ from a galvanic rheophore*. The electric 
 machine does not send a stream of aether along the electrode, 
 but generates a certain state of its superficial atoms, which is 
 
 * See the Figure in Jamin, Tom in., p. 58. 
 
590 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 transmitted along the electrode by conduction, and on this 
 account insulation is necessary. The electricity continuously 
 generated by the machine, being conducted by the first 
 platinum wire to the liquid drop, or moistened paper, after 
 traversing the same by conduction, enters the other platinum 
 wire, and is conveyed to the earth. Now it may be presumed 
 that by this operation some degree of electric polarity is pro- 
 duced at the end of the first platinum wire, both on account of 
 its pointed form, and the difference between its electric con- 
 ductivity and that of the liquid. This end accordingly be- 
 comes a positive electric pole, differing from a galvanic pole 
 in the respect that the electric state is maintained by second- 
 order attractive waves, accompanied by internal gradation of 
 density of the wire. Hence, by the laws of frictional elec- 
 tricity (pages 522 and 544), the end of the other wire is 
 electrified negatively by induction, and a stream of aether is 
 caused to flow in the direction from the positive to the 
 negative pole. This current, it seems, can be detected by a 
 galvanometer, but is found to be very feeble. Since, however, 
 a current exists, although it does not originate in chemical 
 affinity as in the case of a galvanic current, it may yet be 
 expected to produce, in proportion to its intensity, chemical 
 decomposition at the two poles, just as this effect takes place 
 at the ends of the platinum wires in the experiment for 
 decomposing water. And such is found to be the case in per- 
 ceptible degree. This theory, I think, explains why only a 
 very small amount of decomposition is detected, and in what 
 respect this action differs essentially from galvanic action. 
 
 The following experiment by Faraday is also readily ex- 
 plained on the principles of this theory*. From one end of a 
 delicate balance a piece of copper- wire of the form of U was 
 suspended and equipoised, and its ends were deeply immersed 
 in mercury contained in two separate .cups. On passing a 
 galvanic current from the mercury in one cup through the 
 
 * Ganot, end of Art. 724. 
 
THE THEORY OF GALVANIC FORCE. 591 
 
 wire into the mercury in the other, the wire rose very per- 
 ceptibly, and sank again when the current ceased. Now the 
 increments of the density of the aether at the extremities of the 
 wire, so far as they are caused by any difference between 
 the conductivities of the wire and the mercury, are, according 
 to the theory (p. 579), equal and in opposite directions relative 
 to the direction of gravity. Hence the accelerative forces due 
 to these variations of density will produce no translation of 
 the wire. But since, apart from this accelerative action re- 
 quired to overcome resistance, the galvanic current is an 
 instance of steady motion, and the pressure is consequently 
 less where the velocity is greater, there will be a decrease of 
 pressure towards the end at which the aather enters, as well as 
 towards that from which it issues, because in the former 
 position its velocity is increased by entering into a narrow 
 channel, and in the other it is diminished by issuing from the 
 same. Thus at both ends there is an accelerative force acting 
 upwards, and capable, it seems, of raising the wire. The 
 space through which it can be lifted is limited by the increase 
 of the weight to be raised as the wire emerges from the 
 mercury. 
 
 A galvanic current is maintained when the rheophores 
 connected with the poles of the battery, instead of being 
 brought into contact, are made to terminate in the ground. 
 The reason appears t$ be, that the earth may be regarded as 
 a conductor of unlimited extent, in which the current may 
 diverge, or converge, in innumerable channels none of which 
 end abruptly. There is no stoppage, or revulsion of the 
 current, from the inertia of the aether contiguous to the ter- 
 minals, because the fluid is actually conducted along the 
 diverging or converging courses by the materials of the earth. 
 Thus the motion of the current through the battery is the 
 same as that of a re-entering one of the same intensity. 
 Supposing the total galvanic motive force to be be represented 
 as before by I\ what, it may be asked, will be the value of 
 X in this case? To answer this question 1 remark that if a 
 
592 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 large number of points of the rheophores, or of points in 
 connection with them, be in contact with materials of the 
 earth, there will be equal facility of ingress and egress of the 
 getherial fluid, in the manner above stated, at all the points. 
 Hence the stream, entering and emerging in curved courses 
 at an unlimited number of points, will cease to be of sensible 
 magnitude at small distances from the terminals estimated in 
 its mean direction. On this account X will but little exceed 
 the length of the current between the terminals. 
 
 I proceed now to the theory of the mutual action between 
 galvanic currents. Experiment shews that two rectilinear 
 and parallel rheophores, placed near each other, and traversed 
 by galvanic currents, are mutually attracted if the currents 
 pass in the same direction, and mutually repelled if they pass 
 in opposite directions*. These facts admit of being explained 
 as follows by the hydrodynamical theory of currents contained 
 in pages 564 569. According to this theory, the galvanic 
 current along a cylindrical metallic rheophore is composed of 
 steady circular motion about its axis and a steady current 
 parallel to the same. Each, motion is a function of the dis- 
 tance from the axis, and decreases with the increase of dis- 
 tance, and, consequently, the resulting spiral motion is a 
 function of the same quantity, and is less as the distance is 
 greater. But, by Hydrodynamics, in steady motion the 
 pressure is a function of the velocity^ and is less as the 
 velocity is greater. (See page 243.) Hence the pressure is a 
 function of the distance from the axis, and is greater as the 
 distance is greater. These inferences apply to a single rheo- 
 phore. Let us now suppose two straight and parallel rheo- 
 phores, traversed by currents in the same direction, to be 
 brought near each other. Then again by Hydrodynamics 
 
 * The details of the large class of experiments that arose out of Ampere's 
 discovery of the mutual action of galvanic currents, and relate for the most part 
 to their mechanical action under different circumstances, are given in Jamin's 
 Leron 70 (Tom. HI., pp. 195222), and in Ganot's Physics, Arts. 723739. I 
 shall attempt to explain theoretically only the more elementary facts and laws. 
 
THE THEORY OF GALVANIC FORCE. 593 
 
 (p. 243), the two sets of motion coexist without interference, 
 and the pressure is less where the resultant velocity is greater. 
 Hence since by the composition the velocity becomes greater 
 at any point in the space between the axes of the wires than 
 at a point equally distant from either axis in the spaces out- 
 side, it follows that there is a decrement of pressure in and 
 near each wire towards the intermediate space, and therefore 
 urging the individual atoms in that direction. Thus the 
 wires are apparently attracted towards each other. If the 
 currents were in opposite directions, the resulting velocity 
 would be less, and the pressure greater, at points between the 
 axes than at corresponding distances outside ; so that the de- 
 crements of pressure within the wires will be in the directions 
 from the intermediate space, and the wires will consequently 
 be repelled from each other. 
 
 It should, however, be observed that the circular motions 
 by themselves would produce just the contrary effects, because 
 when the currents are in the same direction, these motions in 
 the space between the axes of the wires are opposed to each 
 other, and conspire when the currents are in opposite direc- 
 tions. It must, therefore, be concluded, since the foregoing 
 results agree with experience, that the circular motion has 
 less effect than the motion parallel to the axis. Also the 
 
 relation /(r) = - F(r) shews, that whatever be the law of de- 
 crease of the latter motion with distance from the axis, the 
 circular motion decreases in a higher ratio. 
 
 If a rnoveable rheophore and one that is fixed be placed 
 with their axes in the same straight line, and the ends be 
 separated by a small interval, on passing a galvanic cur- 
 rent in either direction, the moveable rheophore is repelled 
 from the other*. The explanation of this fact is, that by the 
 break of continuity of the rheophore, the action by which, 
 as stated in page 567, the superficial stratum sustains the cir- 
 cular motion is interrupted, and in such manner that the 
 
 * Jamin, Tom. ni., p. 200- 
 
 38 
 
594 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 motion In the intervening space expands laterally and di- 
 minishes in magnitude. Hence the pressure has a maximum 
 value between the ends of the rheophores, in consequence of 
 which the decrements of pressure at and near the end of the 
 moveable one put it in motion and repel it from the other. 
 
 When a sinuous form is given to a galvanic rheophore, 
 certain remarkable consequences are found to result, which 
 I shall now attempt to explain theoretically. For the sake of 
 simplicity it will be supposed, at first, that the rheophore has 
 the form of a regular helix with a rectilinear axis. It has 
 been argued (p. 566) that an setherial current may flow along 
 a curved cylindrical rheophore, the motion being partly in 
 lines parallel to the axis, and partly such motion combined 
 with circular transverse motion. But it is evident that the 
 motion cannot be in all respects the same as that along a 
 rectilinear rheophore, because the curvature of the course 
 gives rise to centrifugal force, the effect of which it is neces- 
 sary to take into account. In the instance of the helix, there 
 must be centrifugal force relative to its axis, apart from that 
 due to the circular motion about the axis of the wire, because 
 the motion of a given aetherial particle compelled to move 
 along the curve is resolvable at each point into motion parallel 
 to the axis and motion about it in a transverse circle. Now 
 it may be presumed that the circular motion immediately im- 
 pressed by the helix affects the fluid generally in such manner, 
 that the total motion may be regarded as a case of steady 
 motion relative to a fixed rectilinear axis, namely, the axis of 
 the helix. Consequently the general formula rf(r) = aF(r) 
 is applicable to this case, the compulsory motion along the 
 helix being one of the conditions which determine the arbi- 
 trary quantities. If r 1 be the distance of the helix from the 
 axis, and a be the constant angle which its course makes with 
 the direction of the axis, we have, since /(rj and F(r^ are in 
 this case resolved parts of the velocity along the helix, 
 
 l 
 
 
 tan a = - ; and .'. a = r l tan a. 
 
THE THEORY OF GALVANIC FORCE. 595 
 
 The constant a being thus found, if F(r) were known for all 
 values of r greater than r l9 the whole of the motion exterior 
 to the helix might be calculated by the process indicated in 
 page 568. Within the helix, that is, at distances from the 
 axis less than r^ there is no cause producing either circular 
 motion or variation of pressure ; so that, as in the analogous 
 case considered in pages 566 568, the velocity is wholly 
 parallel to the axis, and equal to F(r^}. 
 
 The velocity of the current in the rheophore is Ffr^ sec a, 
 
 or/(rj cosec a. Hence if &=-, the velocity is f(rj, and 
 
 F(r 1 ) = Q; so that the motion along the rheophore is wholly 
 in circles of radius r r The velocity at any distance r from 
 the axis greater than ^ depends on the form and value of the 
 unknown function F(r\ The case in which a is nearly equal 
 
 to - is that of a Solenoid* which has the turns of the helix 
 
 very close to each other. 
 
 If a = 0, the velocity along the rheophore is Ffa), and 
 f( r i) = 0; or there is no transverse velocity due to the spiral 
 form. In this case the value of F(r^ is the velocity along 
 the axis of a rectilinear rheophore. There is reason to sup- 
 pose that for all values of a not very near to - the value 
 
 of F(r^} is the same, inasmuch as on this supposition the 
 following fact can be accounted fort. A galvanic current 
 sent along a helix is made to return by a rectilinear continua- 
 tion of the same coincident in position with its axis, and this 
 duplex rheophore, arranged so as to be moveable about an 
 axis, is brought to be near and parallel to a fixed rheophore. 
 On passing a current through the latter, no attraction or 
 repulsion of the first is produced such as is treated of in 
 pages 592 and 593. The reason, on the above supposition, evi- 
 dently is, that as the same value of F(r^) applies to the axis 
 
 * Jamin, Tom. HI., p. 224; Ganot, Art 735. 
 f Jamin, p. 200; Ganot, Art. 725. 
 
 382 
 
596 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 of the helix and to the straight wire coincident with it, arid 
 the current passes along them in opposite directions, they 
 neutralize each other with respect to the fixed rheophore. 
 
 It should, however, be observed, that the function F(r t ) 
 represents not only the velocity along the axis, but, as was 
 before explained, that along any parallel less distant from 
 it than the helix ; and also that outside the helix the law of 
 decrement of the velocity is expressed by the general function 
 F(r) as for a rectilinear rheophore. For these reasons the 
 neutralization of a helix by a straight rheophore will be the 
 more exact the less the distance of the helix from the axis ; 
 as is found to be the case by experiment. 
 
 If the sinuous rheophore be of any irregular form, and 
 no part be distant from the axis, or be abruptly inclined to 
 it, the neutralization by a rectilinear rheophore will still take 
 place. For each small portion may be regarded as a part of 
 a helix, and as for each portion FfrJ will have the same 
 value as for a rectilinear rheophore, it will have this value for 
 the whole. Jamin states (p. 200), as a result of experience, 
 " that we may always replace a rectilinear current by any 
 sinuous one having the same mean direction, provided the 
 latter deviates very little from the other." 
 
 The mutual action between two of Amperes solenoids, so 
 far as it depends on the motion of the aether parallel to the 
 axis of the wire, is readily shewn to result from the law 
 obtained in page 593, namely, that two parallel currents at- 
 tract or repel according as they are in the same or opposite 
 directions*. But according to the hydrodynamical theory, 
 these motions are accompanied by circular transverse motions, 
 the effect of which is also to be taken into consideration. The 
 necessity for so doing will be expressly shewn when the 
 theory of the mutual action between galvanic currents and 
 magnets is under discussion. At present I propose to investi- 
 
 * Explanations of the action of a straight current on a solenoid, and of that 
 between two solenoids, are given on this principle in Jamin, Tom. in., Le^on 71. 
 See also Ganot, Articles 736 and 7/39. 
 
THE THEORY OF GALVANIC FORCE. 597 
 
 gate the influence of the transverse motion only with respect 
 to the mutual action of solenoids. 
 
 The turns of the solenoid being very close to each other, 
 planes transverse to them will all pass, quam proxime, through 
 its axis. The consequence will be that since, as is shewn 
 in page 242, the circular motions coexist, the parts of them 
 resolved parallel to the axis coalesce within the solenoid, and 
 in greatest degree along the axis, and partially neutralize 
 each other outside, while the parts resolved perpendicular to 
 the axis everywhere neutralize each other. The result is, 
 therefore, a stream parallel to the axis, and of greatest in- 
 tensity where it is coincident in direction with the axis. But 
 coexisting with the circular motions there will be the motions 
 parallel to the axis of the wire at each of the turns. Now 
 these neutralize each other within the solenoid in such manner 
 that their resultant is zero at the axis, and small at small 
 distances from it ; while outside the solenoid they conspire. 
 Thus the total motion agrees with that obtained for a cylin- 
 drical rheophore by different considerations in pages 567 569, 
 and the more nearly as the transverse section of the solenoid 
 is less. It may also be observed that in this case, as in that 
 just referred to, the values of f(r) may vary discontinuously, 
 or even per saltum, if only the variations of p do not vary per 
 saltum. 
 
 A stream, accompanied by diminution of pressure, being 
 thus excited along the axis of the solenoid, there will be con- 
 vergent lines of motion at the end where the stream enters, 
 and divergent lines at the end where it issues. Consequently 
 if two solenoids be placed with their axes in the same direc- 
 tion and the ends a little apart, there will (by hydrodynamics) 
 be repulsion between them if the streams issue from both, or 
 enter both, and attraction if the stream of one enters, and that 
 of the other issues. These effects may be such as to conspire 
 with those of the mutual action between the solenoids which 
 results, as stated above, from the influence of the currents 
 parallel to the axes of the respective wires. 
 
598 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 But in order that the effects of the longitudinal and trans- 
 verse motions may be of the same kind, a condition is neces- 
 sary which demands particular attention. It is necessary that 
 the spiral motion about a cylindrical rheophore should always 
 le dextrorsum, that is, looking in the direction of a horizontal 
 current, that the motion above the axis be from left to right. 
 It might, in fact, be assumed on abstract grounds that these 
 circular motions, if they exist, are determined as to direction 
 by certain physical conditions. Also it may be supposed that 
 the direction is given to them ab initio, that is, by the chemi- 
 cal action at the seat of the galvanic motive force. Here, 
 then, the theoretical evidence for the necessity of actual 
 chemical analysis for the production of galvanic currents 
 seems to be complete; for without chemical analysis, or trans- 
 fer of atoms, it is presumed that the circular motion is not 
 generated, or determined as to direction, and without circular 
 motion there is no permanent current. 
 
 Presuming, therefore, that the forces which determine the 
 circular motion to be dextrorsum rather than sinistrorsum are 
 those concerned in chemical action, I shall not here attempt 
 to give a mathematical theory of them. It may, however, be 
 said that the circular motion about the rheophore can be con- 
 ceived to be the resultant of a vast number of very small 
 circular motions generated at the points of chemical action, 
 Their composition might be supposed to produce this result in 
 the manner exhibited by a diagram in Art. 740 of Ganot's 
 Physics, which refers to the composition of molecular currents 
 in Ampere's theory of magnetism. It is, however, to be 
 understood that the elementary currents of aether applied thus 
 in the present theory are quite distinct from the molecular 
 currents of Ampere, inasmuch as they only account for circu- 
 lar motions about a rheophore, whereas Ampere, after having 
 employed his currents to account for the motion about a 
 solenoid, transferred the conception of them to the pheno- 
 mena of magnetism. In the theory of magnetism I am about 
 to propose there is nothing analogous to molecular currents. 
 
THE THEORY OF GALVANIC FORCE. 599 
 
 The direction of the circular motion about a rheophore 
 being assumed to be dextrorsum, it will follow that the cur- 
 rent generated in the manner above described along the axis 
 of a solenoid the turns of which are also dextrorsum, will 
 have the same direction as the original galvanic current, that 
 is, it will enter and issue at the same ends as the latter. But 
 if the turns of the solenoid be sinistrorsum, the direction of 
 the current along the axis will evidently be changed and the 
 two currents will enter and issue at opposite ends. Hence if 
 a dextrorsum and a sinistrorsum solenoid be placed end to- 
 wards end, there will be attraction or repulsion between them 
 according as the directions of their galvanic currents are 
 opposite or identical. But the case is the same with respect 
 to the attraction or repulsion due to the currents along the 
 wires. So far, therefore, as two solenoids are concerned, 
 there is no circumstance to distinguish one effect from the 
 other, or to determine the ratio of one to the other. The 
 theory that will subsequently be given of the mutual action 
 between a solenoid and a magnet will demonstrate that the 
 effect depends largely on the current along the axis of the 
 solenoid. 
 
 The rate of propagation of the beginning and end of an 
 astherial stream within a wire is obviously very different from 
 the rate of flowing of the stream in the wire, the latter being 
 the velocity of the aatherial particles, and, therefore, very small 
 compared to the velocity of propagation. The sending of 
 messages by an Electric Telegraph is effected by the generation 
 and propagation of streams of limited lengths. At the ex- 
 tremities of a given stream the variation of the condensation 
 of the aether may be comparable with the variation in a soli- 
 tary wave of condensation or rarefaction for which the value 
 of \ is very much larger than for waves of light. Hence the 
 retardation of the rate of propagation by the medium may be 
 so small that a stream of limited length may be propagated 
 in a metallic wire at a rate very little inferior to that of propa- 
 gation in vacuum. (See in pages 501 and 502.) 
 
600 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 The account given in the preceding paragraph of the cha- 
 racter of currents of limited length is preliminary to the fol- 
 lowing theory of the induced currents discovered by Faraday*. 
 It seems that in all cases of the generation of such currents, 
 either the beginning or the end of a current sent along a rheo- 
 phore acts so as to induce a momentary current in a neigh- 
 bouring rheophore. Hitherto we have regarded the currents 
 as instances of steady motion, which it is allowable to do, so 
 long as the reasoning is confined to established currents. But 
 it is evident that at the beginning of a current an setherial 
 particle acquires by degrees its eventual permanent velocity, 
 and at the end of the current loses by the same degrees the 
 same velocity. Hence at the beginning and end of each 
 limited current there is a portion in which the variations of 
 velocity are directly proportional to the variations of condens- 
 ation, in accordance with the law expressed by the equation 
 V= a S. The same equation applies to the intermediate 
 parts of the current, the condensation S being positive if the 
 current be produced by pushing the fluid forward, and nega- 
 tive if produced by drawing it backward. The portions in 
 which Fand S are variable are propagated at the rate proper 
 to the rheophore, which, for the reason assigned above, pro- 
 bably differs very little from the rate of propagation of light in 
 vacuum. Hence the time during which a given particle passes 
 from a state of rest to that of uniform velocity, or from uni- 
 form velocity to a state of rest, is extremely brief, and at the 
 positions where the transitions occur the fluid may be con- 
 sidered to receive sudden impulses. These are, in fact, the im- 
 pulses which originally set the current in motion and limited 
 its extent, only modified in a certain manner and degree by 
 the circumstances of their transmission to the positions in 
 question. 
 
 Now, according to a hydrodynarnical law which has al- 
 ready been several times applied, these impulses produce 
 reaction of the surrounding fluid by reason of its "inertia. The 
 
 * Jamin, Tom. in., pp. 273 275; Ganot, Article 752. 
 
THE THEORY OF GALVANIC FORCE. 601 
 
 kind of reaction in this case may be appropriately illustrated 
 by the before-cited instance of impulses given to a fluid by a 
 small sphere moving in it. It appears that whether the 
 sphere be moving uniformly or variably, there is a return 
 current in constant proportion to its forward movement ; and 
 although in this instance the amount of backward flow is 
 doubled by the form of the sphere, it is yet wholly the result 
 of resistance from the fluid's inertia. In the case of a limited 
 galvanic current the reactions at the ends neutralize each 
 other, if no other channel than the rheophore is furnished for 
 the streams which the reactions tend to produce. But during 
 the brief interval the impulse lasts, there appears to be nothing 
 to prevent a partial reflection of the stream into the opposite 
 direction, if only a contiguous rheophore be provided in which 
 the reflected portion can permanently flow. This, in fact, is 
 done in every case in which an induced current has been 
 detected. It is evident that the reflected current may be 
 supposed to partake in every respect of the character of the 
 original one. 
 
 At the following end of the inducing stream, a sudden 
 impulse, transmitted from the original disturbance, puts a 
 stop to the motion of a given particle, and therefore acts in 
 the direction contrary to that of the stream. This impulse, 
 just as in the case of that at the preceding end, produces by 
 means of the reaction of the fluid a momentary stream in the 
 opposite direction, which is now the direction of the original 
 current. It has hitherto been supposed that the inducing 
 stream is a condensed one, that is, that S is a positive quan- 
 tity. The same argument applies, mutatis mutandis, and the 
 inductive results are the same, if the stream be a rarefied one. 
 The foregoing theoretical inferences agree with well known 
 laws of induced currents. 
 
 It is an evident Corollary from the above theory, that if 
 the intensity of an existing current were suddenly increased, 
 or suddenly diminished, a current would be induced in a con- 
 tiguous rheophore in precisely the same manner as when 
 
602 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 there is no previous current, and would be proportional to the 
 increase or diminution of the intensity. This is found to be 
 the fact. (Ganot, Art. 753.) 
 
 It is also found that an induction current is generated on 
 diminishing or increasing the interval between a rheophore 
 along which a current is transmitted, and a parallel one which 
 is neutral, that is, one which does not form part of a galvanic 
 circuit, and consequently has no circular motion about its axis. 
 (Jamin, p. 273.) The neutral rheophore, however, acts as a 
 channel for the stream, and has the effect of increasing the con- 
 ductivity. By applying here Ohm's Law, as expressed by the 
 formula I\ = GrcAc* (p. 575), we may infer that as the neutral 
 rheophore does not add by spiral motion to the resistance, it 
 has simply the effect of increasing intensity by increasing the 
 virtual transverse section a) of the conducting wire. Since the 
 velocity of the current diminishes with the distance from the 
 conducting rheophore, the effect of the neutral one will be 
 greater the less the distance between them, because it will be 
 traversed by a stream of greater velocity. Hence by sud- 
 denly varying that distance the intensity / may be varied at 
 pleasure, and induction currents be by that means produced. 
 
 This theory may also serve to explain why on inducing a 
 current by the contiguity of two coils*, the effect is increased 
 by increasing the number of turns of the secondary coil, and 
 insulating them ; for these arrangements augment the quan- 
 tity of reflected current which the secondary coil is capable 
 of appropriating and transmitting. Although a galvanic 
 stream is not conductible from a rheophore by contact, as 
 frictional electricity is conducted from an electrode, there is 
 reason from experiment to say that some amount of con- 
 duction takes place by mere contact of the rheophore with a 
 galvanic conductor ; on which account the enveloping of the 
 wire with a non-conducting substance, as silk or shellac, tends 
 to maintain the intensity. It is also evident from the theory 
 that the quantity of induced current will be increased by any 
 
 * Jamin, Tom. in., pp. 275277; Ganot, Art. 752. 
 
THE THEORY OF GALVANIC FORCE. 603 
 
 arrangement, such as thatinRuhmkorfFs apparatus, by which 
 primary currents of limited length may be produced in rapid 
 succession. 
 
 I have now, I think, sufficiently exemplified the principles 
 and processes involved in the explanations which this theory 
 gives of the facts of galvanism. It would be possible to ex- 
 tend its applications much farther ; but enough, perhaps, has 
 been advanced to justify the conclusion, that galvanic pheno- 
 mena are governed by hydrodynamical laws. Before proceed- 
 ing to the theory of Magnetism, I propose to say a few words 
 on thermo-electric currents. 
 
 Since, according to the views I have been advocating, a 
 current of the aether is produced wherever there is a regular 
 gradation of density in the interior of any substance, it might 
 be expected that a lamina of metal heated unequally at the 
 two ends would generate a current ; and experiment shews 
 that such is the fact. If a closed circuit of any metal be 
 heated at any point, currents, according to this theory, will 
 be generated in consequence of the different degrees of expan- 
 sion of the metal at different parts, and as they will be 
 caused to flow in opposite directions, if they be equal they 
 will neutralize each other. But if by any mechanical means 
 they are made unequal, as by contortions of the metallic 
 circuit, the result is that a current is detected by the galvano- 
 meter. 
 
 Since these currents act on the galvanometer, they will, 
 according to the theory, be accompanied, like galvanic cur- 
 rents, by transverse circular motions. It must, therefore, be 
 admitted that there are other means besides chemical action 
 by which the circular motions can be produced. Various 
 methods of generating thermo-electric currents in a single 
 substance are given by Jamin (Tom. in. pp. 3940), which 
 seem all to point to the conclusion that besides the condition 
 of gradation of density induced by heating, there must be 
 some circumstance determining the circular motion and its 
 direction. The connection shewn by M. Matteucci to exist 
 
604 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 between the phenomena and crystallization allows of the sup- 
 position that the circular motion may be due in every case to 
 a distortion of the crystalline arrangement of the atoms. 
 Perhaps even chemical action operates in this way. 
 
 When the circuit is formed by two metals soldered to- 
 gether at their extremities, and heat is applied at one of the 
 positions of junction*, inequality of the opposite currents may 
 be supposed to arise from difference of the capacity of the two 
 metals for generating the proper currents, owing to difference 
 in their atomic constitutions. It is found, in fact, that under 
 these circumstances a differential current is generated. 
 
 Such currents, however, are distinct from galvanic cur- 
 rents,, because chemical action is not concerned in producing 
 them ; and they are also distinct from the currents of fric- 
 tional electricity, because the interior gradation of density is 
 maintained by heat apart from any electrical condition of the 
 superficial atoms. The existence of these currents is pecu- 
 liarly corroborative of a theory which ascribes the immediate 
 generation of currents by any substance solely to variation of 
 its interior density. 
 
 THE THEORY OF MAGNETIC FORCE. 
 
 In the preceding Theories of Electricity, Galvanism, and 
 Thermo-electricity, the generation of secondary setherial cur- 
 rents from the primary ones (see in pages 545 548), has been 
 ascribed to a gradation of interior density produced by exter- 
 nal agency. In electricity the agent is friction; in galvan- 
 "ism, the mutual molecular actions of dissimilar substances 
 in contact ; and in thermo-electricity, the application of heat. 
 In the Theory of Magnetism which I now enter upon, the 
 fundamental hypothesis is, that in certain substances a gra- 
 dation of interior density exists independently of any such 
 
 * Seebeck's Experiment, respecting- which see Jamin, Tom. HI., p 41, and 
 Ganot, Art. 777. 
 
THE THEORY OF MAGNETIC FORCE. 605 
 
 agency ; that it exists, for instance, in iron found in nature 
 in a magnetized state, and without being accompanied by an 
 electric state of the superficial atoms. It is assumed, in short, 
 in this theory that the gradation of density is maintained 
 solely by interior atomic repulsions and first-order molecular 
 attractions, the particular atomic constitution of the substance 
 allowing to a limited extent these forces to be in equilibrium 
 in its interior consistently with the gradual variation of dens- 
 ity. Also it is supposed that the direction of the gradation 
 of density depends on the form of the magnetized body. By 
 experience it is known that the magnetized state may be in- 
 duced in steel by mechanical means, with different degrees of 
 permanence ; and that it may be momentarily induced in 
 soft iron. All the subsequent reasoning will rest on these 
 fundamental hypotheses and facts, the antecedent reasons for 
 which, since they relate to the intimate constitution and qua- 
 lities of magnetic bodies, will not come under consideration. 
 
 Taking, first, the case of a bar of magnetized steel of the 
 form of a rectangular parallelepiped, let us assume that there 
 exists permanently a uniform decrease of its atomic density from 
 the end^l to the end B. By the argument in pages 545 548, 
 the aether within the bar will in consequence be urged by a 
 uniform accelerative force acting throughout its length from 
 B towards A. To the accelerative force in that half of the 
 bar which terminates at B may be ascribed the agency which 
 overcomes the inertia, or reluctance to motion, of the aether with- 
 in and without the bar on the side of B, and to the accelerative 
 force in the other half that which overcomes the resistance 
 opposed to the flow of the current by the inertia of the aether 
 on the side of A. Thus the motion will be maintained so as to 
 be symmetrical with respect to a neutral position JV, mid-way 
 between A and B, the lines of motion converging towards 
 parts about B, and diverging in like manner from parts about 
 A. The density of the aether will be least, and its velocity 
 greatest, at N, and the density will increase, and the velocity 
 decrease, in both directions from this point by the same gra- 
 
606 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 dations. Hence the spherical atoms, by reason of the excess 
 of pressure on the halves of their surfaces turned from N, will 
 be urged on both sides towards that position, and the total 
 moving forces in the opposite directions will be equal. Thus 
 the accelerative forces of the sether will have no tendency to 
 produce a motion of translation of the bar. Also as there 
 can be no permanent transfer of the eetherial fluid in the 
 mean direction of the motion through the bar, it must be 
 concluded that the fluid returns by a slow stream, distant 
 from the bar on all sides, in such manner that the motion is 
 altogether circulating. 
 
 But experiment shews that motion ensues if a magnetized 
 bar be placed in iron filings, these being immediately attracted 
 towards it. It is found also that the filings arrange them- 
 selves about the bar in tufts, chiefly at the two ends, but in 
 equal degree on each side of the middle part, where there 
 are none. (This effect is exhibited by a diagram in Ganot's 
 Physics, Art. 582.) The kind of action by which iron filings 
 are thus attracted will be a subject of future enquiry; at pre- 
 sent I am only concerned with remarking that their arrange- 
 ment about the bar corresponds closely with the directions of 
 the lines of motion of the aether which resulted from the fore- 
 going theory. This analogy affords a presumption, which 
 will be fully confirmed by subsequent investigation, that mag- 
 netic force is identical with the dynamical action of steady 
 secondary streams of the cether, generated, as already stated, 
 by a regular gradation of density of the atomic constituents of 
 the magnetized body. I proceed, next, to draw some imme- 
 diate inferences from these premisses. 
 
 (1). The theory accounts at once for the well-known fact 
 that the magnetism of a steel bar is equal and opposite on the 
 opposite sides of a middle neutral position. 
 
 (2). It also follows, conformably with experience, that 
 if a magnetized bar be divided into two or more parts by 
 being cut transversely, each part becomes a magnet, it being 
 assumed that the gradation of density in each from end to end 
 
THE THEORY OF MAGNETIC FORCE. 607 
 
 is the same, and in the same direction, as when it formed a 
 part of the whole bar. 
 
 (3). Since reason was given above for concluding that a 
 magnetic stream generated in a rectangular steel bar is re- 
 entering, its intensity may be estimated in a manner analogous 
 to that adopted with respect to the current in a galvanic cir- 
 cuit. (See pages 573 575.) There is the same hydrodyna- 
 mical reason in the one case as in the other for the intensity 
 of the stream being constant throughout the course. Hence if 
 I be the intensity of the magnetic current, defined just as that 
 of a galvanic current, \ the length of the circuit, Ma, constant 
 proper to the substance of the magnetized bar, but in a given 
 bar always proportional to the accelerative force due to the 
 change of density in a given space, I the length of the bar, 
 and m* its transverse section," we shall have on this principle 
 
 I\ = Mlm*, and .'. 7= . 
 
 A. 
 
 It follows from this result that, cseteris paribus, the intensity 
 is proportional to the transverse section of the bar. The value 
 of M in a given bar, being proportional to the rate of change 
 of its density in a given length, is the exponent of the degree 
 of magnetization. When this rate is the greatest that the 
 atomic constitution of the substance admits of, the bar is mag- 
 netized to saturation. Supposing two bars of the same sub- 
 stance, but of different lengths, to be magnetized in the same 
 degree, both, for instance, to saturation, the intensities of 
 the currents will be nearly the same, because if I be increased 
 beyond a certain small length, X must be increased in very 
 nearly the same proportion. It may, I think, be affirmed that 
 these results accord with experience. 
 
 (4). The mutual attractions and repulsions of magnetized 
 bars are accounted for by the same hydrodynamical laws of 
 steady motions as those already applied in explanation of 
 electric and galvanic attractions and repulsions. (See pages 
 551 and 592.) Let A, A' be the positive poles of two mag- 
 netized bars, i.e., those from which the currents flow, and 
 
608 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 B, B the negative poles, or those into which they enter. Then 
 if either A, A' or B, B' be turned towards each other, by the 
 coexistence of the opposing streams the equality and sym- 
 metrical distribution of the densities on the opposite sides of 
 the neutral positions are in such manner destroyed, that the 
 forces urging the bars from each other become more effective 
 than those urging them in the contrary directions. Ap- 
 parently, therefore, the bars are mutually repulsive. If either 
 A be turned towards B, or B towards A', the streams be- 
 tween the adjacent ends have the same mean direction ; in 
 consequence of which the bars are urged from each other by 
 forces that are less effective than those which urge them 
 towards each other, and they appear, therefore, to be mutually 
 attractive. These results agree with the experimental law, 
 that like poles repel, and unlike poles attract*. 
 
 From the foregoing theory of magnetic currents com- 
 bined with the previous theory of galvanic currents, we 
 may now proceed to give an account of the mutual action be- 
 tween a galvanic rheophore and a magnetic needle (Oersted's 
 Experiment)!. Conceive a straight horizontal rheophore to 
 be fixed in the plane of the meridian, and a current to flow 
 through it northward; and let a horizontal magnetized needle, 
 moveable about a vertical axis, be placed directly underneath, 
 and parallel to, the rheophore at a small distance from it, with 
 its positive pole also northward. It will be assumed that the 
 influence of the earth's magnetism on the needle is counter- 
 acted by appropriate means. Since the motion and lines of 
 motion of the magnetic current are symmetrical about the 
 axis of the magnet, and the motion of the galvanic current 
 resolved parallel to the axis of the rheophore is the same at 
 all equal distances from the axis, it follows that so far as 
 regards the dynamical effect of the pressures due to these 
 motions, there is no force tending to move the needle out of 
 the plane of the meridian. 
 
 * Jamin, Tom. i., p. 479; Ganot, Art. 583. 
 f- Jamin, Tom. in., p. 4; Ganot, Art. 698. 
 
THE THEORY OF MAGNETIC FORCE. 609 
 
 But by the theory of galvanism, the stream along the 
 rheophore is accompanied by circular motion about its axis 
 in the direction from above towards the right hand of a person 
 looking along the axis northward (see page 598). Thus the 
 circular motion produces a stream which crosses the magnet 
 from east to west. With respect to the north portion of the 
 needle, this stream opposes on the east side, and flows with 
 on the west side, the parts of the issuing magnetic streams 
 resolved horizontally and at right angles to the axis of the 
 needle. There is consequently an increase of density and 
 pressure on the east side, and a diminution of the same on 
 the west side, and the needle is consequently urged towards 
 the west. About the south portion of the needle, where the 
 entering streams converge in curved courses, the parts re- 
 solved horizontally at right angles to the axis of the needle 
 flow with the circular streams on the east side, and oppose 
 them on the west side. The greater density of the aether is 
 therefore on the west side, and the south end of the needle is 
 consequently urged towards the east. Thus since the mo- 
 menta of the forces acting equally on the two halves of the 
 needle tend to produce rotation in the same direction, the 
 result is a deviation of the north end towards the west. 
 
 The positive pole of the needle still pointing northward, 
 if either the direction of the galvanic current be reversed, or 
 the needle be placed above the current instead of below, it 
 may similarly be shewn that the deviation of the north end 
 would be towards the east. 
 
 These inferences from the theory agree with the well- 
 known facts of the experiment. It is, however, to be said 
 that they strictly apply only to the initial motion of the 
 needle. As soon as an angular separation takes place be- 
 tween the axes of the rheophore and magnet, it fcright be ex- 
 pected that the mutual action of the parts of the currents 
 parallel to the axes would, according to a known law of 
 " angular currents" (Ganot, Art. 724), check the separation, 
 and prevent its exceeding a certain limit. But there are 
 
 39 
 
610 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 reasons for concluding that this action is very small. In the 
 first place, the lines of motion in the magnetic current 
 are wholly different from the spiral lines of the galvanic 
 current, inasmuch as there is no circular motion in the former, 
 and all its lines of motion, if the magnet were symmetrical 
 about an axis, would be in planes passing through the axis. 
 The courses of these lines are determined generally by the 
 law of circulation, adverted to in page 605, which is a neces- 
 sary consequence of the resistance of an unlimited mass of 
 fluid to the permanent transfer of any portion across a fixed 
 plane. The particular forms of the curves will depend partly 
 on the transvere dimensions of the magnet, but chiefly on .its 
 length. According to experiments by Coulomb, cited by M. 
 De La Eive (Tom. I., p. 184), the distribution of magnetism 
 in rods, or bars, of different lengths, is exactly the same in 
 all through four inches from each extremity, but through the 
 intermediate interval, of greater or less length, there is no 
 perceptible magnetism, and iron filings are not attracted. (See 
 p. 606.) Since in order to maintain the curvature of the lines 
 of motion there must be decrements of density of the fluid in 
 the directions towards the centres of curvature, it is a conse- 
 quence of the re-entering courses (see the figure in Jamin, 
 Tom. I., p. 477), that the density contiguous to the above- 
 mentioned intermediate part is less than the mean density. 
 It seems, therefore, allowable to conclude that where there is 
 no perceptible magnetic attraction, an equilibrium is estab- 
 lished between the density of the aether adjacent to the mag- 
 net and of that within it, and that the forms of the magnetic 
 curves are partly determined by the fulfilment of this con- 
 dition. Also since consistently with hydrodynamical principles 
 the different filaments of a rectilinear steady stream may 
 differ in any manner as to velocity if the pressure be every- 
 where the same, it is possible that the magnetic stream, where 
 its attraction is imperceptible, may not extend beyond the 
 limits of the magnet. In this respect it would differ alto- 
 gether from a galvanic stream along a rheophore. 
 
THE THEORY OF MAGNETIC FORCE. 611 
 
 Again, it is to be considered that the courses of the lines of 
 motion towards the ends of the magnet (in the above experi- 
 ment) are favourable to coincidence with the motion about the 
 rheophore on the farther side, and to opposition on the other, 
 after the angular separation of the axes. This circumstance, 
 taken in connection with the foregoing considerations respect- 
 ing the character of the magnetic current, may suffice to ac- 
 count for the observed fact that the separation increases up to 
 90, independently of the intensity of the current. It is plain 
 that when this angle is reached all the lines of motion are 
 symmetrically disposed relatively to the two axes, and there 
 can be no further tendency to motion. 
 
 The reciprocal action of a magnet on a moveable rheophore 
 may be considered with reference to the same hydrodyna- 
 mical laws, that is, by taking account of the variations, from 
 point to point of space, of the density of the aether resulting 
 from the coexistence of steady motions. It is evident that if 
 in Oersted's experiment the magnet be supposed fixed, and 
 the rheophore moveable about a vertical axis, the distribution 
 of density will be the same for the same angular separation 
 of their axes as in the contrary case. Hence, from what is 
 argued above, the galvanic current will tend to repel the 
 magnet, if the angle of separation be of any value less than 
 90. But it does not appear legitimate to infer conversely 
 that under the same condition the fixed magnet will repel the 
 moveable rheophore. For that repulsion comes into operation 
 only when the magnetic stream so alters the symmetrical dis- . 
 tribution of density about the rheophore as to cause an excess 
 of density on the side turned towards the magnet ; but it can- 
 not be affirmed, without taking account of the intensity of the 
 stream and the courses of the lines of motion, that such would 
 be the case for every angle of separation less than 90. To 
 ascertain this point theoretically, it would be necessary to 
 bring both the galvanic and the magnetic streams under exact 
 mathematical treatment, which I am not prepared to do. But 
 although the present theory is on this account incomplete, it 
 
 392 
 
612 THE MATHEMATICAL PRINCIPLES OP PHYSICS. 
 
 is yet not inconsistent with the experimental fact, that " a 
 moveable rheophore immediately begins to turn when a strongly 
 magnetized bar is held in any direction below it, and stops 
 after some oscillations in a plane perpendicular to the axis of 
 the magnet." (Ganot, Art. 731.) 
 
 It will be proper to point out here the difference between 
 the action of a fixed galvanic rheophore on a magnet, the 
 theory of which is given above, and its action on another 
 rheophore holding the place of the magnet, and moveable in 
 like manner about a vertical axis. In the mutual action of 
 the two rheophores it may be supposed, from what is said in 
 page 593, that the coincidence or opposition of the currents 
 parallel to their axes is much more effective in producing 
 variation of density than the coincidence or opposition of the 
 transverse circular currents. The moveable rheophore being 
 supposed to be parallel to the other, and a little below it, and 
 the currents to be in the same direction, by the consequent 
 opposition of the circular currents in the space between the 
 rheophores, a maximum of density will be produced there ; 
 but as the density will be equally distributed on the opposite 
 sides of a vertical plane through the axes, no horizontal 
 motion will result. If, however, the moveable rheophore be 
 separated from the other by a small angle, the tendency of 
 the pressure due to the circular motions will be to increase 
 the separation. But the attraction due to the motions parallel 
 to the axes will prevail over this force, and the rheophore will 
 consequently be made to perform small oscillations about the 
 original position parallel to the fixed rheophore, which, there- 
 fore, is a position of stable equilibrium. These are just the 
 opposite effects to those produced when the rheophore acts on 
 a magnet, in which case, as we have seen, the variations of 
 pressure are due mainly to the transverse motions, repulsion 
 takes place when the axes are parallel, and stable equilibrium 
 when the axes are at right angles to each other*. 
 
 * The theoretical proof of the "laws of angular currents" (Ganot, Art. 724), 
 which was omitted under the head of Galvanism, might readily be supplied from 
 the considerations contained in this paragraph. 
 
THE THEORY OF MAGNETIC FORCE. 613 
 
 On proceeding now to give a theory of terrestrial mag- 
 netism, I make the preliminary remark, that no other hypo- 
 theses are admissible than those on which the preceding 
 theories of the physical forces have been based. It will ac- 
 cordingly be assumed that the earth is composed of spherical 
 inert atoms of different constant magnitudes, and in different 
 states of aggregation, and that the gether pervading its in- 
 terior has the same mean density as in the external spaces. 
 With respect to the motion impressed on the aether by the 
 moving mass of the earth, the very same argument may be 
 employed as that which is applied in pages 545 548 to any 
 aggregation of atoms constituting an aeriform, liquid, or solid 
 substance. The effect of interior gradation of density is to be 
 calculated in exactly the same manner. In short, the earth 
 may be regarded as a vast magnet, which generates secondary 
 streams just as any other magnet does, that is, in consequence 
 both of internal gradation of density, and the primary streams 
 that relatively pass through it. These streams, as is explain- 
 ed in page 547, are due to the earth's motions in its orbit and 
 about its axis, and the motion it has in space in common with 
 all the members of the solar system. 
 
 It is, however, to be taken into consideration that if the 
 form of the earth were perfectly spherical, and the density 
 were a function of the distance from the centre, the gradation 
 of density would produce no secondary streams, because the 
 accelerative forces would be just equal and opposite in op- 
 posite directions from the centre. In consequence of the law 
 of fluid reaction which has already been several times applied, 
 the sether would, under these circumstances, relatively per- 
 meate the interior of the earth, without being in any sensible 
 degree transported in the direction of the earth's motion. By 
 reason of the same law the motion of the earth about its axis 
 would produce no rotatory motion of the asther. 
 
 But the earth has, in reality, the form of an oblate sphe- 
 roid, and the normals to the surfaces of equal density do not 
 generally pass through its centre. Also the form of the super- 
 
614 THE MATHEMATICAL PRINCIPLES OF PHYSICS. > 
 
 ficies is very uneven, and the materials of the crust are very 
 irregularly distributed as land and water. Under these con- 
 ditions it may be presumed that secondary streams will be 
 generated by the residual effects of gradations of density. In 
 fact, according to this hydrodynamical theory of magnetism, 
 the directions and intensities of these streams are actually 
 ascertained by the experimental determinations of magnetic 
 declination, dip, and intensity. 
 
 It may be supposed that terrestrial magnetic streams are 
 constantly the same as to direction and intensity at the same 
 positions relative to the earth, if its structural circumstances 
 and its motions in space and about its axis remain unchanged, 
 for the same reason that the magnetism of any magnetized 
 body has fixed relations to its form and structure independ- 
 ently of any motion we may choose to give to it. This is 
 known to be true generally. There are, however, periodic and 
 secular changes in the indications by magnets of the direction 
 and intensity of the earth's magnetism, of which some theo- 
 retical explanations will be proposed subsequently. 
 
 I proceed, in the next place, to make certain applications 
 of the theory, which will serve to determine in what direction 
 the cetherial currents pass through the dipping needle. For 
 this purpose the experiment illustrated by a figure in Art. 732 
 of Ganot's Physics will be first employed. This experiment 
 proves that a vertical galvanic current, moveable about a 
 vertical axis, places itself in a position of stable equilibrium 
 . on the east side of the axis in a plane perpendicular to the 
 magnetic meridian, if the current be ascending, and on the 
 west side if it be descending. To account for these facts let us 
 suppose that the terrestrial magnetic current (supposed to 
 be wholly rectilinear) enters the north or lower end of the 
 needle, and issues at the south end. Then if it be resolved 
 vertically and horizontally, the vertical part will ascend, and 
 the horizontal part will flow in the magnetic meridian south- 
 ward. The former will, therefore, meet a descending galvanic 
 current ; but as it will not alter the symmetrical distribution 
 
THE THEORY OF MAGNETIC FORCE. 615 
 
 of density about its axis, it has no tendency to produce mo- 
 tion about the axis of rotation. The circular motion about 
 the rheophore being dextrorsum, and the current descending, 
 the part of the terrestrial current flowing southward meets it 
 on the side which is to the left hand of a person looking 
 northward, and concurs with it on the opposite side. It fol- 
 lows that the rheophore cannot be in equilibrium unless it is 
 in the plane perpendicular to the magnetic meridian, and to 
 the right of the axis of rotation, that is, on the east side. If 
 the current be ascending, the rheophore may in like manner 
 be shewn to be in equilibrium only in the same plane on the 
 west side of the axis. As these results agree with observa- 
 tion, they confirm the hypothesis that the magnetic stream 
 enters the north end of the needle, passing, consequently, out 
 of the earth. Similarly it may be argued that on the south 
 side of the magnetic equator the terrestrial magnetic current 
 enters into the earth. 
 
 The same inference respecting the direction of the earth's 
 current is deducible from another experiment, which exhibits 
 the action of terrestrial magnetism on a horizontal galvanic 
 current. In this instance the galvanic current, after ascend- 
 ing to the horizontal rheophore by a vertical column enclos- 
 ing its axis of rotation, and proceeding to equal distances 
 along the opposite arms, descends by two vertical rheophores. 
 (Ganot, Art. 733.) From the theory of the preceding ex- 
 periment it appears that the descending currents, being equally 
 distant from the axis of rotation, tend to produce equal rota- 
 tory motions in opposite directions, and thus neutralize each 
 other. Also the horizontal component of the terrestrial cur- 
 rent, by acting on the circular motions of the rheophore, 
 tends only to make it rotate in a vertical plane, and is con- 
 sequently ineffective as regards horizontal rotation. Thus 
 there remains only the action of the ascending vertical com- 
 ponent on the horizontal arms. That component, supposing 
 the horizontal currents to be directed eastward and westward, 
 will meet the circular currents of the eastern arm on the south 
 
616 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 side, and those of the western arm on the north side. Thus a 
 motion of rotation will be produced from east through north 
 towards west. If the currents proceed from the ends of the 
 arms to the axis of rotation and so down the vertical column, 
 the rotation will be in the opposite direction*. 
 
 The directive action of terrestrial magnetism on a magnet 
 supposed to be moveable in any direction about a fixed central 
 point, may be explained as follows on the same principles. 
 Let us suppose, at first, that the axis of the magnet coincides 
 with the direction of the terrestrial current; and let the 
 positive pole of the magnet, that is, the one from which its 
 streams diverge, be southward. Hence the stream of the 
 magnet along its axis will have exactly the same direction as 
 the terrestrial stream. Now let the axis of the magnet be 
 caused to deviate a little from its original position in any 
 direction whatever, and let its streams and the earth's stream 
 be both resolved parallel and perpendicular to the new 
 position. The portions resolved parallel to the axis will 
 evidently not tend to produce any motion of rotation of the 
 magnet. Attention being given to the divergent courses of 
 the streams at the positive end of the magnet, and the con- 
 vergent courses at the other end, it will be seen that the 
 respective portions resolved perpendicular to the axis will be 
 everywhere opposed to each other on the side of the magnet 
 which is farthest from the first position of the axis, and will 
 everywhere coalesce on the opposite side. Hence the pres- 
 sure will be in excess on the more remote side, and will urge 
 
 * In Atkinson's Edition of Ganot, Art. 733, the rotation is stated to be "from 
 east to west" in the former case, and from "west to east" in the other; 
 which gives no information. In Jamin, Tom. in., p. 222, the direction, as in- 
 dicated by a diagram, is from west through north towards east. But in M. De La 
 Rive's Treatise (Tom. I., p. 256), I found the following explicit statement : 
 " When the current is directed from the centre of rotation to the free extremity, 
 the direction of the motion is such that, supposing the free extremity to be west- 
 ward, it is at first towards the sonth, continues its course towards the east, then 
 towards the north, to return to the west and recommence as before. The contrary 
 is the case if the current is directed from the circumference to the centre." 
 This agrees with the theory. 
 
THE THEORY OF MAGNETIC FORCE. 617 
 
 the magnet towards the position from which it was displaced. 
 As the displacement was supposed to be in any direction 
 whatever, this argument proves that the terrestrial stream is 
 directive, tending always to place the axis of the magnet in 
 coincidence with its own course. 
 
 By like reasoning it might be shewn that if the negative 
 end of the magnet were southward, and its axis were placed 
 in coincidence with the direction of the terrestrial stream, just 
 the contrary effects would result from a small displacement, 
 the magnet being caused to deviate more and more from the 
 first position, and settling finally with its positive end south- 
 ward. As the foregoing results agree with observation, it 
 follows as a Corollary from the theory that the mean direc- 
 tion of the proper currents of the dipping needle always coin- 
 cides with the direction of the terrestrial current, and the south 
 end of the needle is the positive pole. 
 
 The degree of magnetization of a magnet being given, the 
 directive force of the earth's magnetism, as inferred from the 
 time of small oscillations of the magnet about its mean posi- 
 tion, is a measure of the velocity of the terrestrial current, 
 or of the total intensity of the terrestrial magnetic force at the 
 place of observation. Instead of taking observations for 
 directly determining the total intensity and dip, these ele- 
 ments might be deduced, as is usually done, from observations 
 of horizontal magnetic force and vertical magnetic force, the 
 total intensity being the resultant of these two forces, and the 
 ratio of the latter to the former being the tangent of the Dip 
 or Inclination. 
 
 The directive action of the earth's magnetism on a solenoid 
 is analogous to that on a magnet. We have seen (p. 597) 
 that by the combination of the circular motions transverse to 
 the axis of the rheophore throughout the turns which con- 
 stitute the solenoid, a current is produced along its axis, with 
 divergence of streams at one end, and convergence of streams 
 at the other. In virtue of these streams two solenoids act on 
 each other like two magnets, and exclusively by these streams 
 
618 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 a solenoid acts upon a magnet like a magnet (see in p. 599). 
 By acting on the same streams just as on those of a magnet 
 the earth's current gives direction to the axis of the solenoid*. 
 On taking account of the exterior circular streams about the 
 axis of the solenoid (discussed in page 597), it will be found 
 that the resulting action due to their coexistence with the 
 earth's current tends to give a motion of translation to the 
 solenoid, but not to change the direction of its axis. 
 
 Before entering upon the subject of magnetic induction, 
 it is proper to state that I regard the explanations which the 
 theory has already given of magnetic phenomena as affording 
 very strong presumptive evidence that the magnetic condition 
 of a body is always such as it was defined to be by the initial 
 hypothesis ; namely, that it is a regular gradation of its in- 
 terior density, but not depending, as in the case of an electri- 
 fied body, either on second-order molecular force, or on a 
 particular state of the superficial stratum of atoms. The ques- 
 tion as to the modes of action by which this gradation of 
 density is generated and maintained, as it relates to a specific 
 property of the body depending on its atomic constitution, 
 I do not profess to be able to answer completely : but some 
 views on the subject will here be proposed, with the intention 
 of employing them in the subsequent exposition of the theory 
 of magnetic induction. 
 
 According to the theories of heat and molecular attraction 
 given in this work, the atomic repulsive forces, and the forces 
 of molecular attraction, which by their action on any atom 
 in the interior of any substance keep it at rest, are separately 
 equal in opposite directions, the density being supposed to be 
 uniform, and no extraneous force acting. (See p. 465.) But 
 it is also conceivable that the atom may be in equilibrium 
 by the action of the interior forces, when the density, instead 
 of being uniform, varies uniformly with the distance from 
 a fixed plane in the substance. For in that case it suffices 
 
 Jamin, Torn. TIL, pp. 226229; Ganot, Arts. 737739. 
 
THE THEORY OF MAGNETIC FORCE. 619 
 
 for the equilibrium, if the resultant attraction towards the 
 denser parts be equal to the resultant repulsion towards the 
 rarer parts, and both act perpendicularly to the above-men- 
 tioned plane. 
 
 Let us, therefore, assume that this kind of equilibrium 
 can take place in certain substances within very restricted 
 limits of the gradation of density, and that such substances 
 are by that circumstance endowed with the property of being 
 magnetic. We may then conceive of magnetization as being 
 an operation by which a transition is made from the ordinary 
 equilibrium of the atoms which exists when the density of the 
 substance is uniform, to an extraordinary equilibrium con- 
 sistent with a uniform variation of its density. It appears 
 from experiment, as might perhaps have been anticipated 
 from this view, that the change from one state to the other, 
 in whatever degree, takes place simultaneously at all parts of 
 the substance. 
 
 The various modes of practically magnetizing a body 
 susceptible of the magnetic state seem all to be means of 
 locally altering the relative positions of its atoms. In the 
 " method of touch" by powerful magnets, the poles of the 
 magnets are applied to successive points of the surface of the 
 body to be magnetized, apparently because the efficacy of 
 the process depends on the magnetic streams being divergent, 
 or convergent, and, therefore, accompanied by gradations of 
 the density of the aether. The " method of single touch," 
 in which the magnetizing pole is passed along the bar al- 
 ways in the same direction, gives but a feeble result. In 
 Mitchell's "method of double touch" the effect is produced 
 by the passage of a stream from the positive touching pole 
 through a portion of the bar into the negative one, and the 
 two magnets may be moved in conjunction in either direc- 
 tion. (Epinus added greatly to the efficiency of this method 
 by inclining the two magnets at small angles to the bar in 
 opposite directions, and making at the same time its ends 
 rest on the contrary poles of horizontal magnets. By this 
 
620 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 arrangement two sets of streams, of increased intensity, pass 
 in the same direction, each through an inclined and a hori- 
 zontal magnet and the intermediate portion of the bar. The 
 positive pole of the magnetized bar is found to be at that 
 end out of which the magnetizing streams issue*. 
 
 When a bar of steel, or soft iron, is placed in the direc- 
 tion of the dipping needle, it is found to be magnetized by 
 the earth's magnetism t. In this case, the terrestrial stream 
 passes through the bar with an accelerated velocity on ac- 
 count of the contraction of the channel by the atoms, the 
 acceleration being produced by gradations of density due to 
 the entrance of the stream in converging courses at the 
 north end. At the south end the stream issues in diverging 
 courses, and the consequent retardation there is equal to the 
 previous acceleration. Hence, as in the foregoing case, the 
 bar is traversed by a magnetic current, and at the same time 
 is subject to local action from the convergence and divergence 
 of the lines of motion ; so that the process of magnetization 
 may be considered to be the same in the two cases. 
 
 Again, a steel bar is strongly magnetized by being placed 
 along the axis of a helix through which a galvanic current 
 is sent. (Jamin, p. 252 ; Ganot, Art. 742.) The effect ap- 
 pears in this instance to be attributable to the circular motions 
 transverse to the path of the helix, which, if the turns be 
 very close, produce by their composition a current along the 
 axis, with converging courses at one extremity and diverging 
 courses at the other (see p. 597). Thus in all the three 
 methods the magnetizing streams have the same character, 
 and it may be presumed that they all disturb the atoms of 
 the magnetized body in the same manner. 
 
 The fact discovered by Faraday (Experimental Researches 
 in Electricity, Series XI x), that the plane in which a sub- 
 
 * See Jamin, Tom. HI., pp. 524527; Ganot, Arts. 606609. If in Ganot's 
 Fig. 429, the letter A be at positive poles, the direction of the currents is from 
 the left to the right hand, and a marks the positive pole of the bar. 
 
 f Jamin, Tom. i., p. 529; Ganot, Art. 610. 
 
THE THEORY OF MAGNETIC FORCE. 621 
 
 stance polarizes light may be changed by the influence of 
 powerful artificial magnets, may be appealed to as giving 
 evidence that the relative positions of the atoms of bodies 
 are alterable by the action of galvanic streams. But the 
 precise mode of the alteration, as depending on the magni- 
 tudes and arrangement of the atoms, does not admit of being 
 exactly investigated in the present state of theory ; and for 
 the same reason we cannot account theoretically for the fact 
 that steel and soft iron, although capable of being magnet- 
 ized by the same means, differ widely as to the power of 
 retaining the magnetism. It seems that the atoms of steel, 
 after being put by a disturbance into positions of extraordinary 
 equilibrium, are kept there by the intrinsic forces of the sub- 
 stance, whilst the atoms of soft iron retain such positions 
 only so long as the disturbance lasts. 
 
 Neither can a theoretical reason at present be given for 
 the fact that the direction from the negative to the positive 
 end of a magnetized bar of steel, or soft iron, that is, the 
 direction in which its proper streams flow through it, is the 
 same as that of the magnetizing current. That this fact de- 
 pends on the particular constitution of iron may be inferred 
 from the circumstance that there are certain substances whose 
 proper streams generated under the same magnetizing circum- 
 stances are in the opposite direction*. Such substances are 
 called diamagnetic^ . Although the exact processes of mag- 
 netizing and diamagnetizing are alike unknown, on the 
 principles of the present theory it may be pronounced that in 
 both cases a regular gradation of interior density is produced, 
 differing only as to direction. It is also to be noticed that 
 the diamagnetism of bismuth, like the magnetism of soft iron, 
 is temporary ; on which account a bar of bismuth, suspended 
 horizontally between the poles of a sufficiently powerful 
 magnet, takes a transverse position, since if it deviated in 
 
 * An account of experiments by Tyndall which fully establish this fact is given 
 by Jamin, Tom. HI., pp. 262264. 
 
 + Jamin, Tom. in., pp. 258261; Ganot, Art. 776. 
 
622 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 either direction from this position, the generated poles would 
 be repelled by the poles of the magnet. 
 
 In magnetizing steel bars, certain irregularities which 
 have been named consequent points, are found to occur, espe- 
 cially if the bars be long. They present themselves as poles 
 intermediate to the extreme poles, and alternately positive and 
 negative. (Jamin, Tom. I., p. 525.) Such effects might, 
 according to the foregoing theory, be expected to result, if the 
 gradation of density, instead of being uniform from end to end 
 of the bar, have alternate maximum and minimum values at 
 certain intermediate positions. It is known from observation 
 that consequent points slowly disappear. This mode of dis- 
 appearance is such as might be accounted for by a gradual 
 distribution of the irregularities of density throughout the 
 length of the bar. 
 
 The preceding theory of magnetic induction gives the 
 means of accounting for the attraction of iron filings by a 
 magnetized bar. The streams of the magnet passing through 
 the iron filing magnetize it inductively in the manner de- 
 scribed in page 619, and by reason of the gradation of density 
 thus induced the filing becomes a small magnet having its 
 own streams. By what is shewn in page 620, these streams 
 flow in the same mean direction as the inducing stream, and 
 consequently if the latter proceeds from the positive pole of 
 the magnet, the more remote pole of the small magnet is also 
 positive. Hence opposite poles of the two magnets are turned 
 towards each other, and, as usual, attraction ensues. If the 
 inducing stream flows towards the negative pole, the nearer 
 pole of the small magnet is positive, so that opposite poles 
 are oppositely directed, and, as before, attraction is the result. 
 The theory thus accounts for the adherence of the iron 
 filings equally at both ends of the magnet. It also explains 
 the adherence of iron filings in succession one to another; 
 for all the filings are converted in greater or less degree 
 by the magnetized bar into small temporary magnets, with 
 like poles all turned in directions towards the bar, so that 
 
THE THEORY OF MAGNETIC FORCE. 623 
 
 they are prepared for mutually adhering by the contact of 
 unlike poles. 
 
 It is here appropriate to remark that the magnetic streams 
 through a magnetic bar are (by hydrodynamics) accompanied 
 by a decrease of astherial density towards both ends, and it 
 might at first sight be supposed that the accelerative forces 
 thence resulting, would account for the attraction of the iron 
 filings. But if this were true, there would be no reason why 
 bodies of all kinds should not be attracted, as is the case in 
 electrical attractions. Since magnetic streams act on bodies 
 as being subject to the induction of magnetism or diamag- 
 netism, it follows that the gradation of density of the aether 
 from point to point of space, so far as it is only due to the 
 original stream, is much less effective than the gradation of 
 density due to the coexistence of the secondary stream of the 
 magnetic or diamagnetic induction. We have here a physical 
 circumstance analogous to that adverted to in page 529 relative 
 to electrical attractions and repulsions, which, according to the 
 theory explained in pages 544 552, are the results of streams 
 having their origin in gradations of the densities of the 
 electrified bodies induced by second-order forces, although 
 the immediate translatory action of these forces is compara- 
 tively very small. 
 
 As the point just referred to is of primary importance, I 
 shall here adduce, for the purpose of elucidating it, some addi- 
 tional mathematical investigations respecting the dynamical 
 effects of composite steady motion. (See Proposition xviii. 
 p. 240, and Example vili. p. 313.) The motion being sup- 
 posed to be steady, the general equations for a compressible 
 fluid become 
 
 du dv dw dp dp dp 
 
 ~r + -j- + -J- + u ~ r + v j + W J = 0, 
 ax ay dz pdx pdy pdz 
 
 a*dp du du du a?dp s a?dp p 
 
 r L + U-T- + V-J- +w =0, /- + &c. = 0, - r + & c>= o. 
 pdx dx ay dz pdy pdz 
 
 The last three equations shew that the last three terms of the 
 
624 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 first equation are of the third order with respect to the velo- 
 city. Hence omitting these terms, that equation is reduced to 
 its first three terms. (This is the reasoning referred to in the 
 Note in p. 547.) If now u^v^wj u z , v a , w 2 ; &c. be the 
 resolved velocities of several sets of steady motions due to 
 separate disturbances, we shall have 
 
 and by adding the several equations, 
 
 d. (M 1 + M a + &c.) t d. (i? t + . . 12 . 
 
 dx dJT ~~^~ 
 
 That this last equation may be applicable to the motion of a 
 fluid, it is only necessary that the resultant of the several 
 steady motions, assumed to coexist, should be steady motion. 
 But plainly this will be the case, because the components 
 being constant as to magnitude and direction, the resultant is in 
 like manner constant. It is thus shewn that steady motions 
 may coexist consistently with satisfying the condition of con- 
 stancy of mass. Hence the general hydrodynamical equation at 
 the top of page 241 is applicable to composite steady motion if 
 
 = 0, and Hb- = 0, and V be the resultant velocity at any 
 dt dt 
 
 point. The equation in that case may be put under the form 
 
 _ Z! / y\ 
 p = p Q e >* = p Q {l- 2 J nearly ; 
 
 p Q being the constant value of p at a given point along a given 
 line of motion. In the application of this equation to magnetic 
 streams it cannot generally be assumed that udx + vdy + wdz 
 is an exact differential. 
 
 Let us now take the case of a large magnetized bar, sym- 
 metrical about an axis, attracting an iron filing of the shape of 
 a small prismatic bar, the latter being magnetized inductively 
 by the large magnet. For the sake of simplicity let their 
 axes be in the same straight line, which, consequently, will be 
 a line of motion. Then if V l and V 2 be the velocities of the 
 
THE THEORY OF MAGNETIC FORCE. 625 
 
 aether due to the large and small magnets respectively at any 
 point within the latter, F/ and F 2 ' the corresponding velo- 
 cities at a point equally distant from the neutral plane of the 
 small magnet on the opposite side of it, and if the densities at 
 the two points be p l and />/, we have (as in page 550), 
 
 3 
 
 and since by symmetry F 2 ' = V 2 , 
 
 , ,:' ' ~ ft' = - ^ (FI ~ F;) (2 F * + Fi + 7;) - 
 
 If F! and /! apply to the point which is nearer than the 
 other to the large magnet, so that Fi is greater than F/, it 
 will follow that the right-hand side of the last equation is 
 negative, and p^ is less than /?/. Hence the symmetrical 
 arrangement of density which would subsist relatively to the 
 neutral plane of the small magnet so far as depends on its own 
 streams, is destroyed by the composition of the motions, and on 
 this account that magnet is moved by the differences of the 
 opposite accelerations of its individual atoms on the opposite 
 sides of the neutral plane. This effect is greater the greater 
 F^ F/. F 2 , and F x + F/ are. In order to determine more 
 precisely the manner in which the iron filing is attracted, I 
 shall now calculate the acceleration of an atom in each of the 
 two above mentioned positions. 
 
 For this purpose the expression for the accelerative force 
 which was obtained in page 315 may be employed. Let s and 
 s be the distances of the atoms from a fixed point on the line 
 of motion, reckoned positive in the direction from the large to 
 the small magnet, and let this also be the direction of the 
 streams of both. Then the accelerations of the nearer and 
 farther atoms in the positive direction are respectively 
 
 But by reason of the symmetry of the motion proper to the 
 
 40 
 
626 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 small magnet, F a '= F 2 , and-~=- - 2 . Hence, putting 
 
 (JiS CLS 
 
 fj, for the mass of each atom, it will be found that the sum of 
 the moving forces in the positive direction is 
 
 The same reasoning is applicable to any number of pairs of 
 atoms equidistant from the neutral plane. 
 
 Respecting this expression it is first to be remarked that 
 
 the term (V l F/) ^ is positive, because V l is greater than F/ 
 
 CvS 
 
 dV 
 and - is positive, the maximum value of F 2 being at the 
 
 neutral plane. Hence so far as this term indicates the iron filing 
 
 T fdV! dV,\ . 
 
 is repelled. I he term F 2 ~ + ^ is negative because 
 
 2 ds as J 
 
 , . ,^ . , _ . 
 
 both , , and ~ are negative ; and tor the same reason 
 ds ds 
 
 the two remaining terms are negative. These three terms, 
 therefore, indicate that the iron filing is attracted. We have 
 next to consider what circumstances may determine the 
 amount of attraction to be greater than that of repulsion. 
 
 If V 2 = 0, that is, if the small body be non-magnetic, the 
 last two terms within the brackets will express that attraction 
 (spoken of in page 623), by which the streams of the magnet 
 might act on the atoms of a substance of any kind. Experi- 
 ence shews that this attraction is very much smaller than that 
 exerted on a magnetic body ; but for a reason which will be 
 presently stated, it cannot be pronounced in any case to be 
 of no amount. 
 
 The chief exponent of magnetic attraction must, therefore, 
 be the term containing F 2 as a factor. Eespecting this quan- 
 tity, which represents the velocity of a given getherial particle 
 at any point of its course from one end to the other of the 
 small magnet, we may affirm that it depends, cseteris paribus, 
 
THE THEORY OF MAGNETIC FORCE. .627 
 
 on the degree of induced magnetization of this magnet ; but its 
 amount cannot be theoretically calculated. It is, however, 
 reasonable to admit that the given particle rapidly acquires 
 velocity as it enters the magnet, and loses it as rapidly on 
 quitting it, and that the maximum velocity, acquired at the 
 neutral plane, may be comparatively of large amount. On 
 this ground it might be assumed that the mean value of F 2 
 for the whole length of the small magnet is much larger than 
 the mean of the co-existing values of F x , because the latter 
 apply to parts of the stream of the large magnet which are 
 exterior to it and remote from its neutral plane. But for the 
 very reason that F 2 has a large maximum value within the . 
 
 iron filing, =-* will have small values, and on this account 
 
 the positive term containing this quantity as a factor may be 
 insignificant compared to the sum of the negative terms. 
 These appear to be sufficient reasons for concluding that the 
 observed attraction of iron filings by a magnet is in accord- 
 ance with the theory. Although we have only considered 
 what takes place along the axial line of motion, it is easy to 
 see that by reason of symmetry the result would have been 
 of the same kind if all the lines of motion had been taken into 
 account. 
 
 If the iron filing were acted upon by streams converging 
 towards the negative pole of the magnet, F x and F/ would be 
 negative, and at the same time F 2 would change sign. Hence 
 the terms above discussed will have the same signs as before, 
 and the action of the magnet on the filing will for the same 
 reasons be attractive. 
 
 In order to pass from a magnetic to a diamagnetic effect, 
 it is only necessary to suppose the direction of the induced 
 streams to be contrary to that of the inducing streams, or 
 to change the sign of F 2 . In that case the term involving V z 
 becomes positive, and the remaining terms within the brackets 
 will be negative. If the positive term exceeds the sum of 
 the negative, the action of the magnet will on the whole be 
 
 402 
 
628 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 repulsive, and the substance acted upon might be called dia- 
 magnetic. In short, theoretically considered substances are 
 magnetic, diamagnetic, or indifferent, according as the quan- 
 tity in brackets is negative, positive, or of insensible magni- 
 tude. Faraday's experiments (Series XX.) have shewn that 
 most, if not all, substances are more or less magnetic or 
 diamagnetic. It is for this reason that, as was remarked in 
 p. 626, it does not appear possible to separate the direct action 
 of magnetic streams on the atoms of bodies generally, from 
 the indirect action due to their having the magnetic or dia- 
 magnetic property. 
 
 The theory also explains why the earth's magnetism, 
 although it determines the direction of the magnetic needle 
 (p. 616), has no tendency to give it a motion of translation. 
 For the expression for the magnetic accelerative force, ob- 
 tained in page 626, is applicable in this instance, the earth 
 being the large magnet, and the needle the small one ; and 
 since the terrestrial magnetic current may be regarded as 
 moving uniformly in parallel courses, we have F x = F/, 
 
 =-i = 0. and TT" = 0. Hence that expression vanishes, and 
 as as 
 
 there is neither attraction nor repulsion of the needle. 
 
 It was shewn (p. 620) that the earth's current is capable 
 of magnetizing a bar of soft iron placed in the direction of 
 the dipping needle, and the effect was attributed to the 
 convergence and divergence of the lines of motion resulting 
 from the disturbance, by the contraction of the channel, of the 
 regular stream in its passage through the bar. The secondary 
 stream thus generated is distinguished from that due to 
 gradation of density, by the circumstance that it is not 
 maintained by a uniform acceleration from end to end of the 
 magnet; and its maximum velocity will most probably on 
 that account be very much less than that of the other. For 
 the same reason as in the case of the latter, it will produce no 
 sensible motion of translation by coexistence with the earth's 
 current. It seems also that when the lines of motion of the 
 
THE THEORY OF MAGNETIC FORCE. 629 
 
 primary stream are convergent or divergent, this secondary 
 stream (all its values of F 2 being probably very small) gives 
 rise to no motion of translation such as that resulting from 
 the magnetic secondary stream ; for otherwise bodies that are 
 non-magnetic would be attracted by a magnet. 
 
 Faraday states as a result of his experiments (Series 
 xxr., 2356), that " even the solutions of the ferruginous 
 salts, whether in water or alcohol, were magnetic." The 
 theory would, therefore, lead to the inference that the iron 
 contained in the solution imparts to it in a certain proportion 
 its own property of compelling the atoms, under the influence 
 of a magnet, to take new positions of equilibrium consistent 
 with a regular gradation of interior density. This seems 
 more likely than that the atoms of the iron alone take such 
 positions, while the other atoms remain quiescent. In either 
 case, however, the solution after being put into a glass vessel 
 and placed between the poles of a magnet (2363), would be 
 attracted towards both poles in the same manner as any solid 
 magnetizable body of the same form and magnitude is at- 
 tracted in the same position. The amount of attraction will 
 plainly be proportional, caeteris paribus, to the quantity of 
 the iron in the ferruginous solution. 
 
 Faraday also found (2367, 2368) that closed tubes, con- 
 taining the ferruginous solution in different degrees of 
 strength, when they were suspended either vertically or 
 horizontally between the poles of the magnet so as to be 
 capable of moving freely in the above-mentioned solution, were 
 attracted or repelled by the poles according as the strength of 
 their solutions was greater or less than that of the solution 
 contained in the glass vessel. This fact may be explained on 
 the principle that the amount of the induced magnetism of 
 the solution, that is, the rate of the gradation of its density, 
 is proportional to the quantity of iron which a given portion 
 contains. Thus if the solutions in the tube and the vessel be 
 of the same strength, their densities in the same positions 
 would vary at the same rate ; so that the solution in the tube 
 
630 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 would be attracted just as if it formed a part of that in the ves- 
 sel, the effect of the separation of the fluids by the containing 
 tube being inconsiderable. But if the solution in the tube be 
 stronger than that in the vessel there will be an excess of the 
 gradation of the density of the former which will account for 
 its being attracted relatively to the fluid in the vessel. And 
 if the solution in the tube be the weaker, the gradation of its 
 density may be conceived to consist of a gradation the same 
 as that of the surrounding solution, together with a gradation 
 in the opposite direction, equal to the deficiency. By reason 
 of the latter the magnet will act on the solution in the tube 
 just as if it were a diamagnetic body, and the tube will conse- 
 quently be repelled from either pole.* 
 
 Another fact of a very singular character has also been 
 established by Faraday's experiments (Series xx., 2283, 
 2302 2305), namely, that bismuth and other diamagnetic 
 bodies, whether they are in one piece or in very fine powders, 
 exhibit the diamagnetic property very nearly in the same 
 degree under the same circumstances. To account for this 
 fact, which at first sight might be thought to be out of the 
 reach of explanation by the present theory, the following 
 considerations are offered. Assuming that the theoretical 
 lines of motion, or curvilinear courses of the setherial stream 
 pertaining to a given magnet, are identical as to direction 
 with the lines of magnetic force which Faraday deduced from 
 his experiments (Series xxvm.), it will follow, that the 
 primary current of the magnet induces magnetism in mag- 
 netizable bodies, as iron filings, in the direction of tangents to 
 these lines ; that is, the induced gradations of density in the 
 pieces of soft iron are transverse to the lines of motion. As 
 the iron filings in which this induction is observed to take 
 place cannot be supposed to have initially any special rela- 
 tions to the inducing magnet, it may be concluded that the 
 
 * I consider this explanation to be preferable to that which I have given 
 in Art. 20 of the Theory of Magnetic Force in the Philosophical Magazine 
 for February, 1861. 
 
THE THEORY OF MAGNETIC FORCE. 631 
 
 gradation of density is induced independently of the condition 
 of magnitude, or form, or position of the iron filing. (At the 
 same time it is true that the mutual action between two 
 filings by means of the secondary streams consequent upon 
 the induced gradations of density does depend on such con- 
 ditions). Also it may be admitted that the rate of the 
 induced gradation of density of a given substance is always 
 the same where the velocity of the primary stream and the 
 degree of the convergence or divergence of its lines of motion 
 are the same. These considerations may be extended to the 
 induction of diamagnetism, which is only negative magnetism. 
 This being understood, it will be seen that if a piece of bis- 
 muth be placed with its centre of gravity in a certain position 
 relative to the poles of the magnet, first in a compact state, 
 and then in a state of fine powder, there will be induced in 
 the two cases very nearly the same amount of gradation of 
 density in the same directions, the quantity of matter being 
 the same in both. In the powdered state the resulting 
 secondary stream, generated as usual by gradation of density, 
 will be very nearly the same, on account of the small spaces 
 between the minute particles, as if the body were compact. 
 Now in the mathematical investigation respecting the 
 acceleration by a given magnet of magnetic or diamagnetic 
 substances (in pages 624 and 625), it was shewn that the 
 attractive or repulsive force at a given position depends only 
 on the velocity (F 2 ) in the secondary stream. And although 
 that investigation was restricted in its conditions, it is not 
 difficult to see that a general investigation would conduct to 
 a result expressible in the same terms. Hence as the whole 
 and the powdered bismuth have nearly the same secondary 
 streams, they will be nearly equally repelled by the magnet. 
 
 Iron filings are attracted in large quantities by a wire 
 immersed in them, so long as it is traversed by a galvanic 
 current, but become detached as soon as the current ceases. 
 (Jamin, in., p. 252 ; Ganot, Art. 742). This fact, discovered 
 by Arago soon after the publication of Oersted's experiment, 
 
632 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 discloses something that was not indicated by that experi- 
 ment, namely, that a galvanic current is capable of inducing 
 magnetism. A theory of magnetism which does not account 
 for this significant phenomenon fails essentially. The present 
 theory furnishes the following explanation of it. 
 
 Although the centrifugal force resulting from the spiral 
 motion of the galvanic current causes the density of the aether 
 to increase with the distance from the wire, and thus pro- 
 duces accelerative forces of the fluid tending towards the axis 
 of the current, these are not the forces which sensibly 
 attract the iron filings ; for if they were, non-magnetic bodies 
 would be equally attracted. The observed attractions are 
 attributable to a different and more effective operation, as, I 
 think, may be thus shewn. Conceive, for the sake of dis- 
 tinctness, one of the pieces of iron to be a small prismatic bar, 
 with its axis in a direction perpendicular to the axis of the 
 wire, and to be separated from the wire by a very small 
 interval. Then the axis of the bar will at all points be cut 
 at right angles by spiral lines of motion. Now it is evident 
 that the streams which pass transversely through the bar will 
 be accelerated at entrance and equally retarded at exit on 
 account of the contraction of channel by the occupation of 
 space by the atoms. Thus the velocity at any point within 
 the bar will be greater in a certain ratio than that at a point 
 outside equally distant from the axis of the wire. As the 
 circular motion f(r) and the longitudinal motion F(r) will 
 be increased in the same proportion, the same equation 
 
 f(r) = -F(r) applies both within and without the bar, and 
 
 the motion within conforms to the law of spiral motion 
 (p. 565). Let k be the ratio of the inside to the outside 
 
 f ff r \ ]2 
 
 velocity. Then for the fluid outside we have dp = **f^ dr 
 
 7.2 f ft\ ]2 
 
 (p. 564), and for that inside dp'= IJ v n dr. Hence if 
 there were no cause of disturbance at the ends of the bar, the 
 
THE THEORY OF MAGNETIC FORCE. 633 
 
 difference of the pressures at two points within would be to 
 the difference of the pressures at two points without the bar 
 at the same distances from the axis, in the ratio of k* to 1. 
 The pressure within the bar is at all points less than that at 
 the same distance from the axis without, on account of the 
 greater velocity within. Let us, for the moment, leave out of 
 account causes of disturbance at the ends of the bar, and let 
 e 2 be the excess of the exterior above the interior pressure at 
 the farther end, and e l that at the nearer end. Then if A/? be 
 the excess of the exterior pressure at the farther end above 
 that at the nearer, it is easily shewn that e 2 e^= (& 2 1) A^>. 
 Hence since Ap is positive and Jc greater than unity, e z is 
 always greater than e t . But evidently the excesses of pres- 
 sure gj and <? 2 will tend to produce confluent streams towards 
 the farther and nearer ends respectively by the action of 
 lateral pressures, and the tendency is in greater degree at the 
 nearer than at the farther end. Eventually, therefore, there 
 will be a stream whose mean direction will be along the axis 
 of the bar from the nearer towards the farther end, and whose 
 lines of motion will be on the whole divergent because it has 
 to overcome the inertia of the fluid beyond. Also, as in all 
 such cases, it will be circulating. Thus a current is generated 
 proper for inducing magnetism in the prismatic iron filing, 
 which is in consequence attracted towards the wire by the 
 pressures resulting from the co-existence of the inducing with 
 the induced current. In short, the filing is very nearly in the 
 same case as one that is attracted towards the positive pole 
 of a magnet. (See in page 622). The effectiveness of this 
 action will depend on the value of Jc, but probably still more 
 on that of Ap, which will increase rapidly with diminution of 
 distance from the rheophore. 
 
 If the axis of the bar had originally been inclined to a 
 perpendicular to the axis of the wire, streams of the same 
 kind would have been generated, but of less intensity. It 
 may be presumed that these magnetic streams, as in the case 
 of an iron filing attracted by a magnet, will have a directive 
 
634 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 effect, and ultimately place the axis of the magnetized body 
 at right angles to the axis of the wire. 
 
 According to the foregoing reasoning all the turns of a 
 solenoid ought to attract iron filings equally, and in this 
 respect a solenoid would differ from a magnetic bar, which 
 attracts them very little at its middle part. I am not pre- 
 pared to cite experimental evidence on this point, but 
 theoretically the difference would be accounted for by saying 
 that the motion along the axis and at the ends of a solenoid 
 is composite motion, resulting from the coexistence of the 
 circular motions transverse to the axis of the spiral rheo- 
 phore; whereas in the case of a magnet the motion is in 
 re-entering courses, without being accompanied by transverse 
 circular motions. (See in pages 597 and 617). 
 
 " By the discharge of a large Leyden jar or battery, a 
 steel wire may be magnetized if it is laid at right angles to 
 the conducting wire through which the discharge is effected, 
 either in contact with it, or at some distance. With less 
 powerful discharges a steel bar may be magnetized by 
 placing it in a tube on which is coiled a fine insulated copper 
 wire." (Ganot, Art. 674). This magnetization ~by frictional 
 electricity may be accounted for as follows on the principles 
 of the antecedent Theory of Electric Force. According 
 to that theory the atoms in a superficial stratum of the 
 conductor of the charged jar are in abnormal relative po- 
 sitions (p. 512), till the discharge suddenly brings them back 
 to their natural ' positions by combined translatory and vibra- 
 tory motions in directions transverse to the surface. These 
 vibrations excite vibrations of the aether, which act differen- 
 tially on the atoms of the steel bar, and thus have the effect 
 of magnetizing it; for this kind of disturbance suffices for 
 inducing the magnetic state, whether the relative displace- 
 ments of the atoms be caused by the translatory action of 
 vibrations or of currents. The differential effect will plainly 
 be greatest when the bar is transverse to the conductor. 
 This explanation is confirmed by the fact, noticed by M. 
 
THE THEORY OF MAGNETIC FORCE. 635 
 
 Poggendorff, that a metallic wire which transmits the dis- 
 charge of a battery presents a luminous appearance, especially 
 where the previous electricity was greatest. (De La Kive, 
 Tom. i., p. 418). The luminosity is most probably due to 
 the continuance of the vibrations of the atoms for a short 
 interval after the discharge. 
 
 " When the prime conductor of an electrical machine in 
 action is connected with one end of the wire of a very 
 sensitive galvanometer, the other end being connected with 
 the ground, a deflection of the needle is produced." (Ganot, 
 Art. 674). In this case the wire is actually traversed by a 
 current (like a galvanic or thermo-electric* current), which 
 has its origin in the gradation of density due, in the manner 
 described in page 510, to frictional electricity. The prime 
 conductor and the conducting wire are a channel for this 
 current, just as if it were a galvanic one (p. 563), and through 
 the wire it is conveyed by spiral movement to the ground. 
 At the same time the wire, acting as an electrode, conveys 
 to the ground the electricity which the machine is con- 
 tinually generating, on which account the electric tension is 
 always small, and the current, although continuous, is very 
 feeble. 
 
 It does not appear from experiment that frictional electri- 
 city, although it is capable of inducing magnetism, can itself 
 be induced either by a galvanic or a magnetic current. For 
 this negative fact the theory assigns the probable reason, that 
 the particular state of the superficial atoms which constitutes 
 frictional electricity is necessarily induced by the translatory 
 action of the waves of second-order molecular force (p. 522), 
 and that acceleration by a current is incapable of producing 
 the same effect. It is true, however, that electrical tension 
 is generated by a galvanic current at the terminals of an open 
 circuit ; but, as is explained in page 579, the forces which 
 
 * I have used the term "thermo-electric" here and in page 603, in deference 
 to common usage, but according to the principle of nomenclature stated in page 
 507, the term " thermo-galvanic " would have been more appropriate. 
 
636 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 maintain it are different from those concerned in frictional 
 electricity. 
 
 There is still another class of facts which a theory of 
 magnetism might be expected to account for. Gambey made 
 the discovery that a magnetic needle, oscillating under the 
 influence of the earth's magnetism, comes to rest much more 
 rapidly when suspended above and very near a considerable 
 mass of copper, than when it is removed from it. From this 
 experiment Arago immediately drew the inference that a 
 force exists between the needle and the metal which acts as 
 a kind of friction in opposition to their relative motion, and 
 that consequently a rotating mass of copper might be ex- 
 pected to draw a magnetic needle in the direction of its own 
 rotation. This anticipation was verified by experiment. 
 (Jamin, Tom. III., p. 300 ; Ganot, Art. 7,57). 
 
 To account for these phenomena and others of the same 
 class, Faraday went through an extensive course of experi- 
 mental research, by means of which he succeeded in demon- 
 strating that as galvanic currents are capable of inducing 
 magnetic, so also magnetic currents can induce galvanic*. 
 This law was established by numerous and varied experi- 
 ments, from which I have selected the following one (Series I., 
 101), as including all that is required for my present purpose. 
 A piece of copper plate, one fifth of an inch thick, one inch 
 and a half wide, and twelve inches long, was placed with its 
 faces horizontal between the poles of a powerful magnet, 
 the negative pole of which was vertically over the other, so 
 that the magnetic stream flowed through the plate upivards. 
 The terminals of the wire of a galvanometer being applied 
 transversely to the longer edges (which for facility of descrip- 
 tion will be supposed to be parallel to the meridian), no 
 current was perceptible so long as the plate was stationary. 
 When it was moved horizontally northward, immediately the 
 galvanometer indicated a stream flowing from the western 
 
 * Experimental Researches in Electricity, Series I. and II. and Series xx. 
 23092341. Philosophical Transactions, 1832 and 1846. 
 
THE THEORY OF MAGNETIC FORCE. 637 
 
 edge of the plate, and entering at the eastern; and when 
 moved southward, the current flowed in the contrary direction. 
 Also by reversing the positions of the magnetic poles, all 
 other circumstances remaining the same, the direction of the 
 induced current was reversed. This experiment is sufficient 
 of itself to prove the reality of the above mentioned law. 
 
 Faraday also found that a galvanic current was produced 
 when in the above experiment the earth's current was sub- 
 stituted for that of the magnet (Series IL, 149 153) ; so that 
 curvature of the lines of the inducing, magnetic force is not 
 a necessary condition. When a copper disk was made to 
 revolve about an axis coinciding with the direction of mag- 
 netic dip, the galvanic current induced by the earth's mag- 
 netism flowed from the centre to the circumference, or in the 
 contrary direction, according as the revolution was like that 
 of the hands of a watch, or in the opposite direction, to a 
 person looking along the axis northward. That this would 
 be the case, might easily be inferred from the facts of the 
 elementary experiment described above. 
 
 It was, besides, ascertained by Faraday, by employing 
 plates of different metals, that the intensity of the induced 
 galvanic current was in proportion to the galvanic con- 
 ductivity of the metal. Copper stands at the head of a list 
 of twelve metals possessing the property (Series xx., 2340). 
 Since, therefore, the induction of galvanism by magnetic force 
 is different for different substances, being dependent on their 
 particular atomic constitutions, it is not possible, with our 
 present knowledge, to ascertain the exact modus operandi. 
 It may, however, be remarked that as the character of the 
 galvanic current may be assumed to be the same, whatever 
 be the mode of generation, the following law is deducible 
 from the facts of magneto-galvanism* cited in the foregoing 
 
 * It has been proved by experiments that galvanic currents may be generated 
 in three different ways; by chemical analysis, by heat, and by magnetism. Ac- 
 cording to the nomenclature before proposed, these currents might be distinguished 
 as chemico-galvanic, thermo-galvanic, and magneto-galvanic. The first kind is 
 
638 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 description of the elementary experiment : The circular mo- 
 tion about the transverse rheophore at the parts of the circles 
 farthest from the positive pole of the magnet is always in 
 the same direction as the motion of the plate relatively to the 
 magnet. 
 
 It is to be understood that this law, although expressed 
 in terms borrowed from the present hydrodynamical theory 
 of a galvanic current along a wire, rests wholly on Faraday's 
 experimental results, no attempt being made to give an 
 a priori reason for it. I am not able to cite an experiment 
 expressly intended to shew that the effect is the same when 
 the magnet is moved while the plate is at rest, as when 
 the plate is moved and the magnet is at rest, but experiments 
 subsequently adduced prove indirectly that this must be 
 the case. 
 
 Now, whatever may be the true account of the generation 
 of elementary circular motions by magnetism, their existence 
 in the composition of magneto-galvanic currents may be as- 
 sumed for the same reasons as those adduced in page 598 
 relatively to the ordinary galvanic currents resulting from 
 chemical action. In short, elementary circular motions ap- 
 pear to be essential to the origination and maintenance of 
 all galvanic currents. From an argument contained in pages 
 569 572, I have concluded that a galvanic stream may flow 
 in a rheophore of irregular form and capacity, without suffer- 
 ing dispersion or diminution, as water flows in a confined 
 channel of any shape. But besides the reason there assigned 
 for this law, drawn from the reaction of the exterior mass 
 of the sether against the tendency of any portion of the fluid 
 to enter into the rheophore, or to issue from it, it is necessary, 
 as I now conceive, to take into account the effect of the com- 
 position of elementary circular motions in the stream, as it 
 passes through the rheophore. This effect will consist of 
 circulating motions of another order, flowing chiefly in the 
 
 generated between a metal and a liquid, the second between two metals, and the 
 third from a single metal. 
 
THE THEORY OF MAGNETIC FORCE. 639 
 
 parts of the interior near the boundary, and necessarily ac- 
 companied by centrifugal force, which, however, will not be 
 perceptible, as when the rheophore has the form of a fine 
 wire, at considerable distances from the boundary, if the 
 curvature of the circulating current be small at its effective 
 limits. The centrifugal force will be opposed by an ac- 
 celerative force due to gradation of density decreasing towards 
 the interior, which will tend to keep the current within its 
 irregular channel. 
 
 These preliminaries being admitted, an explanation can 
 be given, by the application on hydrodynamical principles of 
 the law above enunciated, of the phenomena already men- 
 tioned as having been observed by Gambey and Arago ; and 
 also of phenomena of the same class which Faraday found to 
 be exhibited by copper under the influence of a powerful 
 magnet. (Researches, Series xx., 2309 2325). The latter 
 will be considered first, because the theoretical considerations 
 required for their explanation will assist in accounting for the 
 others. 
 
 The experiments were chiefly made by suspending between 
 the poles of the magnet, (the axis of the magnetic field being 
 horizontal,) a bar of copper in such manner that it could 
 revolve about a vertical axis with its longest edges either 
 horizontal or vertical. The length of the bar was 2 inches, 
 the width one-third of an inch, and the thickness one-fifth of 
 an inch. Instead of the bar a copper cube or globe was also 
 suspended. For the sake of simplicity, let us first take the 
 case of a globe having its centre on the axis of the field, and 
 let it be supposed to revolve in such direction about a vertical 
 axis that to a spectator looking from the positive towards the 
 negative pole, the part nearest the former turns from right to 
 left. Then by the foregoing general law of the generation of 
 magneto-galvanic currents, the motion at any point of the 
 hemisphere nearest the positive pole, resolved perpendicularly 
 to the magnetic stream, will generate a vertical galvanic stream 
 flowing upwards. By the same rule at all points of the opposite 
 
640 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 hemisphere like currents will be formed flowing downwards. 
 These directions are such that the elementary currents do not 
 tend to neutralize each other, but to produce a circulating re- 
 sultant. But at the moment of generation the circular motion 
 of each elementary ascending current opposes, on the left side 
 of its axis, the magnetic current, and flows with it on the right 
 side. Hence the atoms are individually urged by accelerative 
 forces of the aether tending from left to right, and therefore 
 opposed to the impressed motion of the globe. Similarly the 
 action of the magnetic stream on the descending currents of 
 the other hemisphere produces accelerative forces tending from 
 right to left, and again, therefore, opposed to the motion of the 
 globe. If the globe had been started in the opposite direction, 
 the generated forces would still have opposed the motion. In 
 this way the theory accounts for the "sluggishness" of the 
 globe as respects rotation about its axis (2325). 
 
 The same kind of reasoning applies to a cube, or to a bar 
 such as that whose dimensions are given above, because in 
 each of these bodies there will be the same incipient genera- 
 tion of elementary currents favourable to a circulating re- 
 sultant as in the case of the globe, and it has already been 
 argued that circulating interior currents are compatible with 
 any form of a galvanic rheophore. Faraday has noticed 
 (2324) that when a bar is suspended so as to vibrate in the 
 field of an artificial magnet at the same time that it rotates 
 about an axis, as soon as a current is sent through the field 
 the rotation ceases, but the oscillation goes on without in- 
 terruption. The explanation which the theory gives of this 
 fact is, that the elementary galvanic currents which the 
 oscillation in the magnetic field tends to generate, are in the 
 same direction in opposite halves of the bar, and being on 
 that account incapable of having a circulating resultant, they 
 neutralize each other. It is evidently for the same reason 
 that there is no sluggishness of the bar, or the cube, or the 
 globe, if the axis of rotation coincides with the axis of the 
 magnetic field. (Series XX., 2328). 
 
THE THEORY OF MAGNETIC FORCE. 641 
 
 When the magnetic stream begins to act, the bar, whether 
 its axis be horizontal or vertical, has an incipient motion in 
 the direction in which a magnetic bar would move, but is 
 quickly arrested, and after describing a small angle remains 
 stationary. At the instant of interrupting the stream, there 
 is a strong revulsion carrying the bar through several revolu- 
 tions in the opposite direction. These phenomena do not 
 occur with a cube or a globe. (Faraday, 2309, &c.) Their 
 explanation seems to be such as follows. The magnetic 
 stream has a directive action on a bar the horizontal section 
 of which has unequal sides, just like that of the terrestrial 
 current on the dipping needle (p. 616). This action, which 
 is continuous, is counteracted, after a brief interval, by that of 
 steady interior galvanic currents, the generation of which 
 takes time; and the same currents, taking time also to vanish 
 after the cessation of the magnetic current, act for a brief 
 interval on the bar so as to cause the revulsion. It is, 
 however, to be understood, since copper is not magnetic, 
 but slightly diamagnetic, that the directive action above 
 supposed results from the composition of the current of the 
 powerful magnet with the secondary stream spoken of in 
 page 620, which is the consequence of contraction of channel 
 by the atoms, modified, it may be, in the case of copper by 
 its particular atomic constitution*. 
 
 The experiment of Gambey is the reverse of that of 
 Faraday. Let a horizontal magnetic needle be placed above 
 and very near a fixed mass of copper, which, for the sake of 
 distinctness, will be supposed to be a horizontal circular disk 
 of some thickness, its axis coinciding with that of the motion 
 of the needle. Then the re-entering magnetic lines of motion 
 will pass through the copper, so that if the motion along these 
 lines be resolved parallel to the position of the needle sup- 
 posed fixed, the resulting motions will extend across the disk 
 
 * The theory of the phenomena of copper which I proposed in Art. 23 of a 
 communication on Magnetic Force to the Philosophical Magazine for February 
 1861, does not recognise the existence of magneto-galvanic currents, and on that 
 account is unsuccessful. 
 
 41 
 
642 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 in that direction. This experiment is, therefore, like Fara- 
 day's in so far as the copper is traversed by a magnetic stream. 
 Now let the needle rotate, carrying with it necessarily the 
 lines of motion. The consequence will be that the magnetic 
 current will rotate relatively to the copper, instead of the 
 copper rotating, as in the other experiment, relatively to the 
 current. But, by hypothesis, magneto-galvanic currents are 
 generated in both cases. Hence, by the former argument, the 
 action of the resultant pressures 011 the atoms of the copper will 
 tend to move it in the direction of the motion of the needle. 
 The motion does not take place because the copper is fixed. 
 We may, if we please, suppose it to be fixed by the impression 
 at each instant of an angular acceleration equal and opposite 
 to that it would have if free to move, the needle itself being 
 at the same time impressed with the same angular accelera- 
 tion. These impressed motions, since they produce no relative 
 motion between the needle and the copper, will not affect the 
 generation of magneto-galvanic currents by the actual relative 
 motion. Hence the needle, tending always to draw the 
 copper in the direction of its own motion, will be subject to a 
 continual angular retardation, proportional evidently to its 
 actual angular motion. Supposing, therefore, that it oscil- 
 lates by the influence of terrestrial magnetism, it will under 
 these circumstances be gradually brought to rest, as is known 
 to be the case from experiment. 
 
 The theory shews that if the copper be moveable, as well 
 as the needle, about the common axis, and if the needle be 
 made to rotate continually in one direction, it will cause 
 the copper to rotate in the same direction. This infer- 
 ence is verified by an experiment made by Herschel and 
 Babbage, in which a copper disk was started and kept in 
 rotation by a revolving horse-shoe magnet placed under it. 
 (See Jamin, Tom. in., p. 301, fig. 634.) In this experiment 
 the course of the magnetic stream between the poles of the 
 magnet evidently fulfils the condition of traversing the copper 
 across the axis of revolution. 
 
THE THEORY OF MAGNETIC FORCE. 643 
 
 In Arago's experiment, on the contrary, the copper by 
 revolving causes the needle to revolve in the same direction. 
 (Jamin, p. 301, fig. 633.) This fact can be accounted for 
 by the application as follows of the same principles. Sup- 
 posing the effect of terrestrial magnetism to be neutralized, 
 and the copper to revolve with a uniform velocity, the gene- 
 ration of magneto-galvanic currents, by the relative motion 
 between the needle and the copper will produce in the latter 
 a retarding force tending always to diminish the velocity of 
 its rotation. If an angular acceleration just equal and op- 
 posite to this retardation be impressed both on the copper 
 and the needle, by the same argument as that employed in 
 the case of the stationary copper, no change will be made in 
 the generation of magneto-galvanic currents, the copper will 
 continue to revolve uniformly, and the needle will be accele- 
 rated in the direction of the revolution in proportion to the 
 relative motion. Thus the relative motion between the two 
 bodies will continually decrease, the needle eventually re- 
 volving with the uniform velocity of the copper. 
 
 The following more general considerations embrace all 
 the cases of the phenomena of copper discussed above. As- 
 suming that circulating galvanic currents are generated in 
 copper by the passage through it of a magnetic stream when- 
 ever there is relative rotation between the copper and the 
 direction of the stream, we may regard this effect as a dis- 
 turbance of the steadiness of the motion, or as a resistance 
 tending to diminish the intensity of the magnetic current, 
 analogous to the resistance which is the inverse of intensity 
 in a galvanic current (p. 574). The formula for the intensity 
 of a steady magnetic current in page 607, viz. I\ = Mint?, 
 shews that if X the length of circuit, I the length of the 
 magnet, and m 2 its transverse section be given, I the intensity 
 can change permanently only in consequence of change of 
 the intrinsic magnetism M. But experience shews that M 
 is constant for a given magnet, and not subject to change by 
 any motion that may be given to it, the lines of magnetic 
 
 412 
 
644 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 force and the motions along them having always the same 
 relations to the magnet. Hence in the cases before us the 
 intensity does not change permanently. Consequently any 
 relative rotation between a magnet and copper, which tempo- 
 rarily produces unsteadiness and change of intensity of the 
 magnetic streams, must dimmish and cease altogether with 
 more or less rapidity. This result applies to all the cases. In 
 Faraday's experiment the destruction of the relative motion 
 is sudden ; in the others it is more gradual. 
 
 Having, now, gone through the explanations, on the prin- 
 ciples of this theory, of the different classes of magnetic phe- 
 nomena, I think I have reason to say that the applicability 
 of hydrodynamical laws has been shewn in these explanations 
 with even greater evidence than in those previously given of 
 galvanic phenomena. There remains still for consideration 
 the theory of the variations of terrestrial magnetism, which 
 have been reserved for separate treatment because they con- 
 stitute an order of phenomena distinct from those of common 
 magnetism. Hitherto we have regarded the earth as being 
 a magnet of uniform intensity, and differing from other mag- 
 nets only in its vast magnitude. (See page 613). But obser- 
 vation has shewn that terrestrial magnetism is subject to 
 changes both as to intensity and direction, which for the 
 most part go through regular periods, but seem occasionally 
 to be abrupt and irregular. In all probability observation 
 will eventually elicit the laws which govern the irregular as 
 well as the regular variations, and point to the causes by the 
 operation of which both kinds are produced. At present, 
 however, the observations cannot be considered sufficient for 
 this purpose, whether as regards the extent of the earth's 
 surface over which they are simultaneously made, or the length 
 of time during which they have been continued ; and conse- 
 quently any theory of the causes of the magnetic variations 
 must necessarily be incomplete. Still to some extent expla- 
 nations of the phenomena may be given on the principles of 
 the hydrodynamical theory, as I now proceed to shew. 
 
THE THEORY OF MAGNETIC FORCE. 645 
 
 It will be proper to consider, in the first place, the mean 
 solar-diurnal variation of declination*, the principal facts of 
 which are thus stated by General Sabine : " In the northern 
 magnetic hemisphere the deflection of the north end of the 
 needle has an eastern extreme about 8 a.m., and a western 
 extreme between 1 and 2 p. m., returning to its mean position 
 about 7 p. m. During the hours of the night the needle is 
 comparatively tranquil t." In the southern magnetic hemi- 
 sphere the deflection of the south end of the needle follows 
 the same law. Since the maximum and minimum deflections 
 of the north, or marked, end of the needle in the northern 
 hemisphere at Toronto, Kew, Nertschinsk (in Asiatic Russia), 
 and Pekin, and those of the south end in the southern hemi- 
 sphere at St Helena, the Cape of Good Hope, and Hobarton, 
 are found by observation to occur at the same local hours, it 
 would seem to be a necessary inference that these effects are 
 to be attributed to magnetic action resulting from variations 
 of temperature produced by the solar rays. The origination 
 and mode of this action are the subject of the following argu- 
 ment. 
 
 Let the Sun's path in the heavens be supposed, at first, 
 to be along the equator. Then the earth's atmosphere will 
 be heated by the solar rays in such manner that there will 
 always be a maximum of temperature at a point of the equator 
 following by about Ij hour the point to which the Sun is 
 vertical. In all directions from that point along great circles 
 there will be decrements of temperature, but in different de- 
 grees in different directions. As far as regards the directions 
 northward and southward from any point of the equator, the 
 
 For an account of the observations of magnetic declination subsequently 
 referred to, see the two Articles by General Sabine in the Philosophical Trans- 
 actions for 1847 and 1863, and Walker's Adams Prize Essay on Terrestrial and 
 Cosmical Magnetism, Chap. n. The law of the mean solar-diurnal variation for 
 different localities north and south of the equator is exhibited in Plate xiv. of the 
 Phil. Trans, for 1863, and by the black lines intermediate to the red and blue in 
 Walker's Plate 2. 
 
 t The Rede Lecture for 1862, p. 17. 
 
646 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 decrements will, by symmetry, be equal, local causes of irre- 
 gularity being left out of account. But there will not be the 
 same equality between the decrements eastward and westward 
 from the maximum point, because the parts of the atmosphere 
 towards the east have been heated by the day temperature, 
 and those towards the west have been cooled by that of the 
 night. 
 
 This being understood, I propose to account for the diurnal 
 variation of magnetic declination in the following manner. 
 According to the theory stated in pages 613 and 614, the 
 magnetism of the earth is generated by its motions relative 
 to the aether, but only in consequence of its solid and fluid 
 parts being unsymmetrical with respect to a centre. If the 
 density at all parts, inclusive of the atmosphere, were a func- 
 tion of the distance from a centre, there would be no magnetic 
 streams. Hence the disturbance of the symmetry of the at- 
 mospheric strata by the influence of the Sun's heat in the 
 manner above described, will have the effect of generating 
 new magnetic streams, which will modify those existing from 
 other causes*. The effect may be generally represented as 
 a small displacement of the whole system of the lines of mag- 
 netic force that are due to the fluid and solid parts of the 
 earth. Excepting for local causes of disturbance, the new 
 streams will be of constant magnitude, and have a constant 
 relation to the Sun's position, so that they will travel with 
 the Sun over the earth's surface. In order to calculate the 
 displacement at any point, it would be necessary to take 
 account of the perturbing action at all other points, and the 
 problem so treated might be one of great difficulty. Perhaps, 
 however, the following general considerations may serve to 
 
 * I was led by the indications of my theory, while this part of the work was in 
 course of printing, to refer the solar-diurnal variation of the declination to 
 atmospheric magnetism, without being aware that the same view had been taken by 
 Faraday on experimental grounds in his Series xxvi. and xxvii. (Phil. Trans, for 
 1851). The theoretical explanation, if the general theory of magnetism be well 
 founded, will shew what atmospheric magnetism is, and how it originates. 
 
THE THEORY OF MAGNETIC FORCE. 647 
 
 indicate the kind of solar action by which the observed mag- 
 netic variations are produced. 
 
 It is a general hydrodynamical law relative to the passage 
 of a steady setherial stream through any substance, that in 
 consequence of occupation of space by the atoms, the velocity 
 of the aether within any small portion of the substance is 
 greater, and its density less, than within a contiguous small 
 portion of greater density. Consequently the horizontal com- 
 ponents of the terrestrial magnetic streams (which alone are 
 considered at present), will everywhere be disturbed by forces 
 acting in the directions from the rarer to the denser parts of 
 the atmosphere, that is, in the directions of the decrements of 
 temperature. If these accelerative forces be resolved along 
 and transversely to the lines of horizontal force, the transverse 
 components will alone cause the variations of decimation, the 
 others only affecting the horizontal intensity. The impulses 
 communicated to the fluid by the transverse forces will, by 
 reason of its inertia, be counteracted by centrifugal force ac- 
 companied by changes of curvature of the direction-lines of 
 horizontal force. For the sake of simplifying the reasoning, 
 it will be assumed that these lines, if undisturbed, would be 
 lines of no declination, this assumption, as far as regards tro- 
 pical and middle latitudes, not materially affecting the cha- 
 racter and amount of the variations. Thus we have only to 
 consider the impulses eastward and westward. 
 
 In the night time, since the decrements of temperature 
 are for the most part eastward, the disturbing impulses will 
 be given in that direction, and will be effective principally 
 between midnight and the epoch of minimum temperature, 
 this interval embracing the time during which the tempera- 
 ture decreases fastest. That these impulses may be counter- 
 acted by centrifugal force it is necessary that the lines of 
 horizontal force should be curved so as to be concave towards 
 the east. Since the curvature will be alike on the opposite 
 sides of the equator, a line drawn on the earth's surface, 
 coincident in direction at every point through which it passes 
 
648 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 with the line of horizontal force, will be symmetrical with 
 respect to the equator and cut it at right angles. Hence the 
 variation of the declination is zero at the equator, and increases 
 with the latitude northward and southward, the disturbed line 
 being continuously concave eastward. Hence also the north 
 end of the needle deviates eastward in north latitudes, and 
 westward in south latitudes. These results agree with obser- 
 vation so far as regards early morning hours. (See the plates 
 referred to in the note in page 645.) 
 
 But observation shews that the mean easterly deviation 
 in north latitudes goes on increasing to 8 h a.m., and, therefore, 
 considerably after the epoch of minimum temperature, which 
 is generally about 2 hours before sun-rise. Possibly this fact 
 may be accounted for by considering that at a given locality 
 the deviation does not depend immediately or solely on the 
 impulses due to decrements of temperature at that locality, 
 (the directive power of streams depending on the square of 
 the velocity), but on the aggregate effect of impulses extend- 
 ing over a large area westward, and may, therefore, continue 
 to increase so long as this aggregate increases. This, how- 
 ever, is a point which can only be settled by exact mathe- 
 matical investigation. Whatever be the explanation of the 
 prolonged easterly deviation, there will evidently be two 
 causes tending to check it, viz. the force by which the needle 
 is always drawn to its undisturbed position, and that arising 
 from the increments of temperature eastward, which will have 
 the effect of diminishing that curvature of the magnetic lines 
 which was induced by the former decrements. By comparison 
 with observation we may infer that after the maximum 
 easterly deviation is attained, by the prevalence of these two 
 forces the needle is made to move westward, till about 
 10 h a.m. the easterly deviation is zero. From that time the 
 lines of horizontal force are caused by the increments of tem- 
 perature eastward to be concave towards the west, and ac- 
 cordingly the north end of the needle has a westerly de- 
 viation. This deviation increases rapidly with the increase 
 
THE THEORY OF MAGNETIC FORCE. 649 
 
 of temperature, till at l h p.m. it reaches a maximum, being 
 checked by the force which urges the needle towards its 
 neutral position, and by the eastward decrements of tem- 
 perature extending over a large area eastward. (These de- 
 crements, which were supposed above to prolong the epoch 
 of the easterly maximum, now appear to accelerate that of 
 the westerly maximum, which occurs somewhat 'before the 
 epoch of maximum temperature.) The operation of the two 
 causes produces a return of the needle eastwards after l h p.m., 
 diminishing at the same time the curvature of the lines of 
 horizontal force, till about 7 h p.m. the variation is again zero. 
 Afterwards, to past midnight, the changes of temperature are 
 gradual and produce but little effect on the needle. The 
 same considerations apply, mutatis mutandis, to the diurnal 
 variation south of the equator. (See the before mentioned 
 plates.) 
 
 Although in some respects the foregoing explanation must 
 be regarded as incomplete, it seems to account satisfactorily 
 for the principal features of the diurnal variation, namely, 
 that in 24 hours there are one easterly and one westerly 
 maximum, and that the interval from the easterly to the 
 westerly maximum is much smaller than that from the 
 westerly to the easterly. 
 
 At Toronto, the latitude of which is 43 N., the magnetic 
 declination is 1 J west, the dip + 75, and the intensity 1,83 ; 
 and at Hobarton, lat. 43 S., the declination is 10 east, the 
 dip 71, and the intensity 1,81. According to the above 
 theory, the amount of diurnal variation should, under these 
 circumstances, be nearly the same at these two places, but 
 in opposite directions at the same local hours. This was 
 proved by General Sabine to be very approximately the case 
 by a discussion of two years hourly observations at each of 
 the places*. 
 
 In high northern or southern latitudes, the type of the 
 
 * Phil. Trans., 1847, Plate iv., fig. 3, p. 55, and Walker's Plate i. The above 
 numbers were derived from Walker's Plates 6, 8, and 9. 
 
650 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 mean solar-diurnal variation will be much influenced by the 
 magnitude of the declination, while from the theory there is 
 reason to conclude that the magnetic streams generated by 
 the variations of temperature of the atmosphere will have the 
 same character as in lower latitudes. For instance, taking 
 two places not far distant from the north magnetic pole, and 
 supposing the decimation to be at one, and 180 at the 
 other, it may be presumed that the change of curvature of 
 the lines of horizontal force produced by the atmospheric 
 magnetic streams will be in the same direction at both. Con- 
 sequently, when the deviation of the marked end of the 
 needle at the more southern position is eastward or westward, 
 its deviation at the northern position will be westward or 
 eastward at the same local hour. According to the theory, 
 this would be the case with respect to any two places in high 
 latitudes, if their magnetic decimations differ by 180. It 
 happens that Port Kennedy and Point Barrow, (the latitudes 
 of which are 72,0 N. and 71,4 N. and the difference of the 
 longitudes 62), fulfil this condition very nearly, the difference 
 of the declinations being 176. Now, by a discussion of nu- 
 merous hourly observations taken at these positions, Sabine 
 has shewn that the extreme deviations of the marked end of 
 the needle occur at the same local hours in opposite directions 
 if regarded geographically, but in the same direction if re- 
 garded magnetically*. This instance, therefore, is confir- 
 matory of the theory. Between the cases of and + 180 of 
 decimation, the deflections change direction as to east and 
 west, but not as to the direction through the magnetic pole. 
 
 The Sun having been supposed to move in the equator, 
 the foregoing discussion applies only to mean variation. We 
 have now to consider how this variation is modified by the 
 actual motion in the ecliptic. Observation has shewn that 
 the solar-diurnal variation of declination is subject to a 
 semi-annual inequality obviously referable to the changes of 
 
 * Phil. Trans., 1863, p. 661; the Rede Lecture, note in p. 17; Walker, p. 330. 
 
THE THEORY OF MAGNETIC FORCE. 651 
 
 the Sun's polar distance. The mode in which the observed 
 inequality is produced, may be conceived to be such as 
 follows. From the preceding theory of the mean variation 
 it may be inferred generally that the maximum amount of 
 variation depends partly on the latitude and partly on the 
 diurnal range of the temperature of the air. Now at any 
 time during the six months that the Sun is north of the 
 equator, the diurnal range at a place in north latitude will be 
 greater than at a place in the same south latitude. Hence, 
 a line drawn at any time on the earth's surface always 
 in the direction of the declination at the points through 
 which it passes, (being assumed, as before, if undisturbed, 
 to be coincident with a meridian) will not be symmetrical 
 with respect to the equator, but will cross it at a certain 
 angle of inclination. The mean of all its courses during the 
 six summer months, for the daily epoch of the northern 
 maximum of easterly deviation, will be such that a meridian 
 will be a tangent to it at a certain point (which observation 
 shews to be at about 35 S.), and northward and southward 
 from this point it will deviate continuously from the meridian 
 in such manner as to be, throughout tropical and middle 
 latitudes, concave eastward. According to this theory, the 
 angular deviation from the undisturbed position should be 
 greater at a north position than at one equally south. This, 
 in fact, is the case with respect to Toronto and Hobarton. 
 
 The mean declination curve for the same six months, 
 similarly traced for the daily epoch of the northern maximum 
 of westerly deviation, is at that time concave towards the west, 
 and touches a meridian at about 20 of south latitude. Be- 
 tween these two epochs the line oscillates from the first to 
 the second position through the meridian, and in the re- 
 mainder of the 24 hours oscillates slowly back again. Ex- 
 actly analogous considerations apply to the diurnal variations 
 of the mean line of declination for the six winter months, 
 this line touching meridians at points about as much above 
 the equator as the other at points below. The differences 
 
652 THE MATHEMATICAL PRINCIPLES OP PHYSICS. 
 
 between the angular deviations of the two lines from the 
 meridian at the same hours at any place compose the annual 
 inequality of the diurnal variation of declination at that 
 place*. The theory has indicated that as the mean variation 
 is due to the distribution of solar heat corresponding to the 
 Sun's mean position, so this inequality is due to changes of 
 the distribution consequent upon the changes of the Sun's 
 polar distance. It might probably be shewn on the same 
 principles, by a more exact application of the theory, that 
 the inequality, as appears from observation to be the case, 
 is nearly the same for very different latitudes. 
 
 If the foregoing theory of the regular variations of decli- 
 nation be true, it necessarily follows that magnetic streams 
 will be generated by variations of atmospheric temperature 
 due to local causes, such as contiguity of seas and continents, 
 operating over areas of greater or less extent. These streams 
 would be likely to produce changes of the declination appa- 
 rently irregular as to their amount and the times of occurrence, 
 which on that account might be called disturbances. The 
 fact that such disturbances are found to prevail generally, 
 and to be in greater number and degree at some places 
 than at others, points to an origin of the kind here supposed. 
 For instance, magnetic disturbances and appearances of the 
 Aurora are stated to be more frequent at Point Barrow than at 
 any other locality hitherto known t. May not this be owing 
 to its position between the continents of Asia and North 
 America in one direction, and between two oceans of different 
 temperatures in another, these circumstances being favourable 
 to the production of magnetic currents by changes of tem- 
 perature, and consequent gradations of the density of the 
 
 * The law of the inequality for each half year for seven different places is 
 expressed by the ordinates of the lines in Plate n. of Sabine's Rede Lecture, 
 (derived from Plate xv. in the Phil. Trans, for 1863), and by the differences 
 between the ordinates of the dark line and those of the blue and red lines in 
 Walker's Plate 2. 
 
 f Art. xxiv. by General Sabine in the Phil. Trans, for 1857, and Walker, 
 p. 328. 
 
THE THEORY OF MAGNETIC FORCE. 653 
 
 atmosphere, extending over large areas? If it be thought 
 that the gradations of density of so rare a substance as the 
 atmosphere would not suffice to generate sensible magnetic 
 streams by the contraction of channel due to the atoms, in the 
 manner supposed in this theory, it should be considered that 
 on account of the very large value of a in the equation p a*p 
 for the aether, a change of p almost infinitesimal might pro- 
 duce a sensible change of^>*. 
 
 On the above theory of irregular variations of the declina- 
 tion an explanation may be given of the Aurora Borealis and 
 Austr ah's. It has already been maintained that, according to 
 hydrodynamical laws, magnetic currents are necessarily cir- 
 culating, or re-entering. Thus the principal streams, after 
 issuing from the northern regions of the earth, and performing 
 courses which attain their maximum distances from the earth's 
 surface in or near the plane of the magnetic equator produced, 
 return by like courses and enter at the southern regions. For 
 the same reason there will be return currents both of the 
 regular-variation and the disturbance-variation currents, the 
 former more steady and of larger extent than the latter. Both 
 sets will return at considerable distances from the earth's 
 surface, generally exceeding, it may be, the height of the 
 atmosphere. 
 
 Now under these circumstances it would seem reasonable 
 to attribute the phenomena of the Aurora to the crossing of 
 these streams by the principal terrestrial streams. But ac- 
 cording to the hydrodynamical theory of light contained in 
 this work, no luminosity would be thus produced if all the 
 streams were steady or nearly steady, because in that case the 
 
 term -5- in the expression for -~- would vanish or be ex- 
 
 * I venture to express the opinion that earth- currents, which, according to the 
 results of observations made at the Greenwich Observatory, appear to affect 
 simultaneously magnets and galvanometers, may be irregular magnetic streams 
 having a local origin of the same kind. See the Abstract in the Proceedings 
 of the Royal Society, No. 99, p. 249, of a communication on this subject by the 
 Astronomer Royal. 
 
654 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 tremely small. On this account it is probable that the ap- 
 pearance of Aurora is in no sensible degree attributable to the 
 regular- variation streams, but is in all cases due to those pro- 
 ducing the disturbance-variations. These, however, will not 
 give rise to the phenomenon unless they are in a state of 
 unsteadiness ; on which account Auroras will be oftenest seen 
 at places where, as at Point Barrow, the disturbances are fre- 
 quent, and the return currents are liable to continual changes. 
 It should also be observed that the mutual agitation of cross- 
 ing currents will be greater the more exactly the courses are 
 transverse to each other, and that for this reason, as well as on 
 account of greater variations of temperature, Auroras abound 
 in the polar, much more than in the middle and tropical 
 regions. The agitations of the magnetic needle which are 
 observed to accompany the appearance of an Aurora may very 
 well be ascribed to the production, by changes in the return- 
 ing disturbance-streams, of more or less unsteadiness of the 
 principal streams ; although, as appears from the observations 
 at Point Barrow, it is true that Auroras are often seen without 
 being accompanied by any prominent agitation of the needle. 
 (Phil. Trans, for 1857, p. 513.) In such cases the pheno- 
 menon is probably due to the minor streams of local origin. 
 But, apart from this kind of disturbance, an Aurora might 
 become visible by the crossing and perturbation of the prin- 
 cipal streams by streams of extraneous origin. This point will 
 be referred to farther on. 
 
 From appropriate discussions of observations at various 
 places, General Sabine has inferred that there exists a local 
 hour of maximum disturbance-variation of the declination, not 
 related in any general manner to the local hours of maximum 
 easterly and westerly regular- variation. The latter, as we 
 have seen, are in each hemisphere nearly the same for all 
 latitudes and longitudes, whereas the former appears to de- 
 pend on the locality and on extraneous magnetic action. (See 
 Phil. Trans, for 1857, p. 507.) The disturbances considered 
 above are solely due to irregular atmospheric magnetism. 
 
THE THEORY OF MAGNETIC FORCE. 655 
 
 The regular atmospheric magnetism which, according to the 
 theory, produces the solar-diurnal variation of declination, will 
 necessarily give rise to corresponding variations of the dip and 
 intensity. But there is no theoretical reason for asserting that 
 this magnetism acts entirely horizontally, so as to affect the 
 dip and intensity only by altering the horizontal force. On 
 the contrary, it seems to follow necessarily from the mode of 
 generation by solar heat of the unsymmetrical gradation of 
 atmospheric density which is supposed in this theory to pro- 
 duce the magnetism, that this effect takes place in vertical as 
 well as horizontal directions. In fact, the results of observa- 
 tions taken at Greenwich in the years 1842 1847 and 1849 - 
 1857 prove the existence of a diurnal variation of vertical force 
 having epochs of maximum and minimum approximately con- 
 stant throughout the year *. Hence consistently with the theory 
 it is found by observation that the three elements of declination, 
 dip, and intensity are all subject to regular diurnal variations, 
 having one maximum and one minimum value in 24 hours, 
 the epochs of which, it appears, are not the same for the 
 different elements f. 
 
 Also the same cause that produces the semiannual in- 
 equality of the diurnal range of declination, namely, the 
 change of the distribution of the solar heat by the change of 
 the Sun's polar distance, will account for a like inequality 
 of the diurnal range of the dip. For the daily changes of 
 this element, as well as those of the declination, will be 
 in some direct proportion to the diurnal range of tempe- 
 rature, which in tropical and middle latitudes is greater as 
 
 * Phil. Trans, for 1863, Art. xin., on the Diurnal Inequalities of Magnetism 
 at the Greenwich Observatory, by the Astronomer Koyal. See particularly 
 Plates xxn. and xxni. 
 
 f- See in Walker's Essay, pages 171 176 and 269 274, an account of the evi- 
 dence for these laws collected by General Sabine from observations at Toronto, 
 Hobarton, St Helena, and the Cape of Good Hope. Evidence of the same kind, 
 deduced from the above mentioned observations at Greenwich, is exhibited, as 
 far as regards the declination and horizontal force, by the curved lines in the 
 Platea xvm. and xix. of the Phil. Trans, for 1863. 
 
656 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the season is hotter. It has been found, in fact, by a dis- 
 cussion of observations taken at Toronto and at Hobarton 
 that the diurnal range of the dip is at each of these places 
 greater in its summer than in its winter months. (Walker, 
 p. 175.) 
 
 For the same reason, also, the diurnal range of intensity will 
 be greater at a given place in the summer than in the winter. 
 From observations taken at Toronto and the Cape of Good 
 Hope it has been inferred that at the northern position the 
 range is greater in the April-September than in the October- 
 March period (see the Table in Walker, p. 271), whilst at the 
 southern position " the diurnal range is rather greater in the 
 October-March than in the April-September period." (Wal- 
 ker, p. 274.) 
 
 It may be thought to be a necessary inference from the 
 foregoing explanation of the variations of terrestrial magnetism, 
 that, if so produced, they will all be greater, cceteris paribus, 
 from October to March than from April to September, because 
 the Earth is nearer the Sun, and receives from it in a given 
 time a greater amount of heat, in the former six months than 
 in the latter. It might, for instance, be expected that the 
 semiannual inequality of the diurnal variation of the declina- 
 tion would be greater at a south latitude than at an equal 
 north latitude, on account of a higher temperature in the 
 southern than in the northern summer months. But observa- 
 tion does not confirm this inference (see Plate II. of Sabine's 
 Rede Lecture}. For, in fact, there does not exist that differ- 
 ence between the effects of solar heat in June and December 
 which would result from a calculation made by supposing the 
 heat to vary inversely as the square of the distance from the 
 Sun. On the contrary, Professor Dove has concluded from 
 thermometric observations made at different seasons and in 
 different positions remote from each other, that the mean 
 temperature of the whole surface of the earth in June con- 
 siderably exceeds that in December; which he accounts for 
 by the excess of heating effect of the greater amount of land 
 
THE THEORY OF MAGNETIC FORCE. 657 
 
 in the northern than in the southern hemisphere*. The same 
 cause may be supposed to augment all effects of solar heat at 
 the earth's surface while the Sun is north of the equator, and 
 thus compensate for the opposite influence of the increased 
 distance of the earth from the source of the heat during that 
 interval. 
 
 Having considered the variations of terrestrial magnetism 
 so far as they seem to depend on variations of the temperature 
 of the atmosphere, i now proceed to another class, the laws of 
 which have been elicited by General Sabine by discriminative 
 discussions of observations taken at the Magnetic Observa- 
 tories of Kew, Toronto, Hobarton, St Helena, and the Cape of 
 Good Hope. "The phenomena," he says, "maybe briefly 
 stated to be an increase of the Dip and of the Total Force, 
 and a deflection of the north end of the Declination magnet 
 towards the West, in both hemispheres, in the months from 
 October to March, as compared with those from April to 
 September." (Phil. Trans, for 1863, p. 307.) These mag- 
 netic variations are distinguished from those previously con- 
 sidered chiefly in respect to their obeying the same laws at 
 the same epochs in both hemispheres, and being, consequently, 
 independent of temperature and the changes of the Sun's 
 polar distance. Although they are generally very small com- 
 pared to the variations of the first class, their existence and 
 laws appear to be satisfactorily established by the above- 
 mentioned observations t. The proposed hydrodynamical 
 theory of magnetism accounts as follows for these phenomena. 
 
 The earth's magnetism, according to this theory, is due 
 to its motion relative to the aether, and will consequently vary, 
 cceteris paribns, proportionally to the variations of its orbital 
 
 See Sir John Herschel's Outlines of Astronomy, Articles 368370- 
 f It should be noticed that the curves in the before cited Plates attached 
 to Mr Airy's communication in the Phil. Trans- for 1863, include the variations 
 of both kinds, and that the great difference between the summer and winter curves 
 in Plates xvm. and xix. is chiefly to be attributed to those variations which, 
 according to the theory, are due to the agency of atmospheric temperature. 
 
 42 
 
658 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 motion (p, 613)*. Hence there will be a maximum of inten- 
 sity when the earth is at its perihelion, that is, about the time 
 of the winter solstice, and a minimum about the time of the 
 summer solstice. And, generally, the mean intensity in the 
 winter months will be greater than that in the summer months. 
 This agrees with the above experimental law relative to the 
 Total Force. 
 
 Again, increase of the velocity of the setherial magnetic 
 streams will increase the extent of the circulating courses, so 
 that the course of a stream which issues from the earth, or 
 enters it, with a certain velocity, will be exterior to that of 
 one which issues or enters at the same place with less velocity. 
 Hence the inclination of the north end of the needle in the 
 north magnetic hemisphere, and that of the south end in the 
 south hemisphere, will increase with the intensity, and have 
 the same epochs of maximum and minimum values. In short, 
 the theory is in exact accordance with the inference from the 
 same observations, that " the annual variations of the dip and 
 intensity have epochs of maxima and minima coincident, or 
 nearly so, with the solstices." (Ibid. p. 307.) 
 
 It might, however, be urged that, if this explanation be 
 true, the ratio of the difference between the greatest and least 
 intensities to the mean intensity due to the orbital motion, 
 should be equal to the ratio of the difference -between the 
 greatest and least velocities of the earth in its orbit to its 
 mean velocity; whereas observation shews that the annual 
 variation is much smaller than it would be according to this 
 proportion, even when the motion of the solar system is taken 
 into account. This argument admits, I think, of the follow- 
 ing answer. 
 
 The Theory of Magnetic Force which I gave in the Numbers of the Phi- 
 losophical Magazine for January and February 1861, although fundamentally the 
 same as the present one, differs in some important particulars relative to terres- 
 trial magnetism, mainly because I had not then ascertained that the earth by 
 its motion does not carry with it any sensible amount of the a3ther. There is 
 no foundation for "gyratory motions" generated in the manner supposed in 
 Art. 27 of that Theory. 
 
THE THEORY OF MAGNETIC FORCE. 659 
 
 Adopting the results of the mathematical investigation 
 contained in pages 545 and 546, we have the two equations, 
 
 v-v \\ + k(n-D}} a * Ap - ^ 
 
 V )h P **~-\+k(D-Dj) A*' 
 
 The second equation gives an expression for the accelerative 
 force of the gether which produces those secondary streams 
 that are, in fact, the magnetic streams of the theory. If we 
 call this force F, we get by means of the first equation, 
 
 ka?V AZ> 
 F= __ _. Let D represent the proportional part of 
 
 space occupied by atoms, and V Q the velocity of the primary 
 stream, at a fixed point of the earth regarded as a magnetized 
 body, and let D and V be the same quantities at any point 
 situated on a line (z) drawn through the first point in the 
 direction in which the value of D increases. The motion of 
 the earth in its orbit, by changing the velocity of the virtual 
 
 primary stream, changes V Q and V\ but Z> , D, and -r 
 
 remain the same. Also it must be supposed that the factor 
 k (which is always positive, because V increases with D) 
 changes at the same time with F , although it may be pre- 
 sumed that the variation corresponding to that of F in the 
 course of the earth's orbital revolution, is very gradual and of 
 small amount. This being admitted, we shall have for the 
 change of F resulting from change of the earth's position in 
 
 its orbit, 
 
 2 A 4F. 
 
 or = a -r o . -=- . 
 
 kz 
 
 On substituting in this equation the foregoing value of F, it 
 will be seen that if k were absolutely constant, we should have 
 &F=0; that is, there would be no change of the earth's 
 magnetism due to motion in the orbit. It is found, in fact, 
 that there is but very little change ; so that the above assump- 
 tion of a gradual and small variation of k is supported by 
 observation. 
 
 Let us, therefore, assume k to be variable with change of 
 
 422 
 
660 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the earth's velocity; and let F ' and D ' be the values of V 
 and D at a certain point of the earth not far distant from 
 that to which F a arid D 9 were supposed to apply. Then since 
 
 V 
 
 k (D - D ') = 1 - -rf, it follows that k changes only by change 
 
 ^0 
 
 y 
 of the ratio -^-, . Let F be greater than F & ', and let us 
 
 'o 
 
 y 
 suppose the ratio 7 to change so as to increase in value as 
 
 "o 
 
 F increases. (The contrary supposition is inadmissible, be- 
 cause the equation - = kV -r shews that in passing at a 
 
 given epoch from F ' to F , the gradations of the value of F 
 for a given increment of z are greater as F is greater, and 
 vanish if F = 0, in which case there are no magnetic streams). 
 Hence, since the above expression indicates that k is greater 
 
 Y 
 
 as -p^ is greater, it appears that k also increases with F & , and 
 
 "o 
 
 JcV 
 that, consequently, the variation 8 . -== is greatest when F is 
 
 "o 
 
 greatest, that is, when the earth is at perihelion. The same 
 quantity has its minimum value at aphelion. Accordingly 
 &F, which is the exponent of the annual variation of the 
 magnetic intensity, has, as was supposed above, its maximum 
 and minimum values about the times of the solstices. 
 
 The westward deviation of the north end of the needle is 
 referable apparently to the same agency, since it may be- pre- 
 sumed that, together with the dip, the declination will be 
 affected by a change of the intrinsic intensity of the earth's 
 magnetism. From the facts of observation we may infer that 
 on tracing a line of motion of the setherial stream from its 
 issuing out of the earth in the northern hemisphere to its 
 entrance in the southern, the course is caused by an increase 
 of intensity to incline throughout eastward from that which 
 the stream would have followed if the intensity had not 
 changed. Consequently the line of motion strikes the earth's 
 
THE THEORY OF MAGNETIC FORCE. 661 
 
 surface at a point of entrance the more eastward from a mean 
 position the greater the excess of the actual intensity above 
 the mean. If the intensity be less than the mean the devia- 
 tion from the same point is westward. Now these facts indi- 
 cate that the courses of the magnetic streams, (which, so long 
 as the intensity is constant, have a constant relation to the 
 earth), are disturbed by a change of intensity so as to be shifted 
 in the direction of the eastward motion of the earth about its 
 axis while the intensity is increasing, and in the contrary 
 direction while it is decreasing. This result is apparently in 
 accordance with what might have been antecedently supposed, 
 namely, that the slower the stream is the more its course will 
 deviate westward, or fall behind relatively to the direction of 
 the earth's rotation. 
 
 The discussion of magnetic observations taken both in the 
 northern and southern hemispheres has established the exist- 
 ence of a periodical variation of small amount, which is called 
 the lunar-diurnal variation, because its periodicity at a given 
 place is found to depend on the transits of the moon over the 
 meridian*. From this fact it would seem to be allowable to 
 infer that the moon has either proper or induced magnetic 
 streams, or both kinds, extending as far as to the earth, and 
 that by these the earth's magnetism is modified. Although it 
 cannot be affirmed that such action is wholly excluded by the 
 observational results, the laws which they give most promi- 
 nently appear to be irreconcileable with it. The general 
 results may be thus stated: "The three magnetic elements 
 bear concurrent testimony to the magnetic influence of the 
 moon at the earth's surface. In the case of each we find a 
 
 * I refer more particularly to the results obtained by Sabine in the Phil. Trans. 
 for 1856, 1857, and 1863, from observations at Toronto, Hobarton, and Kew respec- 
 tively, and those in the Introduction to the " St Helena Observations," Vol. n., 
 from observations at St Helena, the Cape of Good Hope, and Pekin. The general 
 summary in pp. cxlvi cxlviii of the last named work should especially be con- 
 sulted. The Volume of Greenwich Observations for 1859 contains the " Reduc- 
 tions of the Greenwich magnetic observations referred to the Moon's place " for 
 the years 18481857. See also Walker, pp. 109 HG and 286-289, with Plate 4. 
 
662 THE MATHEMATICAL PRINCIPLES OP PHYSICS. 
 
 double progression in each lunar day. The declination has 
 two easterly and two westerly maxima, and the inclination and 
 total force have likewise each two maxima and two minima in 
 the same interval, the variation in each case passing through 
 zero four times during the lunar day." (Walker, p. 110). 
 
 To account for these facts I make the supposition, that 
 they are due to magnetism of the atmosphere resulting from 
 gradations of its density caused by the moons gravitational 
 attraction. This attraction, supposed to act on the atmosphere 
 at rest, would produce a maximum of height at the point to 
 which the moon is vertical and at the point opposite, and a 
 minimum of height at points distant from these by 90. In 
 the actual case of the revolution of the earth about its axis, 
 these maxima and minima are still produced, and have very 
 nearly the same positions relative to the point to which the 
 moon is vertical. (See the observations cited next page). In 
 fact, this point is the apex of a wave of elevation, which 
 is propagated uniformly over the earth's surface, and, except- 
 ing from incidental causes of irregularity, undergoes no altera- 
 tion. This being the case, it follows, according to a general 
 law of such propagation, that the effective accelerative force of 
 the fluid due to gradations of its density in directions parallel 
 to the earth's surface, has a constant ratio to the impressed 
 disturbing force of the moon in the same directions ; that is, 
 if 6 be the angle at the earth's centre subtended by the arc 
 joining the apex of the wave and any point of the surface, the 
 accelerative force of the fluid at that point varies as sin 20, and 
 tends always to produce a confluence of the fluid towards the 
 point of greatest elevation. As far as regards the production 
 of motion in directions perpendicular to the circle in which the 
 apex moves, (which, for simplicity, will be supposed to be the 
 equator), these forces neutralize each other. But because they 
 tend always towards the moving position of the point verti- 
 cally below the moon, they will have the effect of producing 
 motion of the fluid parallel to the motion of this point, and 
 causing the propagation of the wave with its apex to take 
 
THE THEORY OP MAGNETIC FORCE. 663 
 
 place in the same direction. The wave will thus be symmetri- 
 cal with respect to a plane passing through the apex and the 
 poles of the equator. On the opposite hemisphere there will 
 be a similar and equal wave symmetrical with respect to the 
 same plane produced. At points distant from the two apices by 
 90, there is no circumstance which determines either the 
 motion or the accelerative force to be in one direction rather 
 than the contrary, and consequently at such points the fluid 
 will be momentarily at rest. The accelerative forces towards 
 the above defined plane are greatest at octants, and vanish at 
 syzygies. The motion of the fluid is at all points in the 
 direction of the moon's apparent motion, and is greatest at 
 syzygies. 
 
 Hence the condensation depending on the square of the 
 velocity is least at syzygies and greatest at quadratures. But 
 it does not follow that the barometer will give a higher read- 
 ing in the latter positions than in the former. For we have 
 to take account of the disturbing force of the moon perpen- 
 dicular to the earth's surface, varying as cos 2 6, which lessens 
 at the same time the weight of the column of air and that of 
 the column of mercury. Hence if we suppose the density of 
 the air at the earth's surface to be the same at syzygy as at 
 quadrature, since the aerial column in the former position con- 
 tains more matter than in the other, it will require to be 
 counterbalanced by a proportionally longer mercurial column. 
 And again, if the two aerial columns contain the same quantity 
 of matter, but have different densities at the earth's surface, 
 the heights of the mercurial columns will be proportionally 
 different. It has been found by barometric observations near 
 the equator, that a maximum height occurs at the lunar hours 
 and 12, and a minimum height at the hours 6 and 18, and 
 that the difference is O in -,005*. This quantity, according to 
 the theory, may be the result of differences both of density 
 
 * This is the mean of results deduced by General Sabine in the Phil. Trans. 
 for 1847, p. 45, from observations at St Helena, and by Captain Elliot in the Phil. 
 Trans, for 1852, p. 125, from observations at Singapore. 
 
664 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 and column, and, therefore, does not prove that the density is 
 greater at syzygy than at quadrature, 
 
 We are, therefore, at liberty to suppose that by the effect 
 of lunar gravitation on the revolving atmosphere of the earth, 
 the density increases in all directions from the point to 
 which the moon is vertical, and from the point diametrically 
 opposite, and that it has a maximum value at all points 
 distant from these by 90. This hypothesis being admitted, 
 the generation of magnetic streams by the gradation of den- 
 sity is accounted for by reasoning so exactly analogous to 
 that employed in pages 647 649, relative to the effect of the 
 gradation of density due to solar heat, that it need not here be 
 given at length. It will suffice to state that by this reason- 
 ing it may be shewn that there will be a maximum westerly 
 deviation of the north end of the needle in the northern hemi- 
 sphere, and of the south end in the southern hemisphere, at 
 the moon's superior transit, like the mid-day solar-diurnal 
 maximum of variation, and that it will be similarly preceded 
 by a maximum easterly deviation. But between the lunar 
 and solar magnetic variations of declination there is this 
 difference, that, whereas the latter are altogether devoid of 
 symmetry and have only two principal maxima in a solar 
 day, the former, on account of the symmetrical action of lunar 
 gravitation, have four maxima alternately easterly and west- 
 erly, at intervals of six lunar hours. These theoretical infer- 
 ences, which assume that the moon moves in the equator, 
 agree with the mean laws of lunar-diurnal variation of decli- 
 nation stated in page 661, which were derived from observa- 
 tions taken during a large number of lunations. 
 
 Also the accelerative forces which cause the alternate 
 elevations and depressions of the aerial columns will give rise 
 to vertical gradations of the density of the atmosphere, which, 
 not being functions of the distance from the earth's centre, 
 may be expected to generate vertical magnetic streams. These 
 would produce variations of the inclination and intensity, 
 having their turning-points at the epochs of greatest magnetic 
 
THE THEORY OF MAGNETIC FORCE. 665 
 
 action, which will occur, not when the columns are highest or 
 lowest, but when they are changing fastest, that is, at octants. 
 The effects will be of the same kind, either maxima or 
 minima, at opposite octants, because the column at a given 
 place either ascends or descends at both, but of different kinds 
 at adjacent octants, because the column ascends at one and 
 descends at the other. These views agree in the main with 
 the facts stated in pp. 287 289 of Walker's Essay. 
 
 If the laws of lunar-diurnal magnetic variations, stated in 
 p. 661, be truly accounted for by this theory of gravitational 
 magnetism, it evidently follows that the magnetic influence of 
 the moon, so far as it acts according to those laws, does not 
 sensibly vary from year to year. It is, therefore, confirmatory 
 of the theory to find that General Sabine has arrived at the 
 same conclusion by discussions of four years' observations at 
 Toronto, and eight years' observations at Ho barton*. 
 
 From the foregoing theory of the phenomena of diurnal, 
 annual, and monthly variations of terrestrial magnetism, it 
 would seem to follow that the mean results remain the same 
 from year to year. But observation has shewn that the 
 annual means are subject to large variations. This is very 
 conspicuously indicated by the curves in the Plates xvi., xvir., 
 XX. and XXI. in the Philosophical Transactions for 1863, by 
 which the Astronomer Royal has exhibited the mean diurnal 
 changes in magnitude and direction of the horizontal force, 
 and in magnitude of the vertical force, from observations at 
 Greenwich during the years 1841 1857, the means of each 
 year being represented by a separate curve. These curves 
 change from year to year both in form and magnitude ; and 
 there is also reason to say that the changes are periodical^. 
 The present theory of terrestrial magnetism has thus far 
 
 Phil Trans. 1856, p. 499, and 1857, p. 1. Walker, pp. 113115, and 
 Plate 4. 
 
 f- The curves for the 17 years do not obviously indicate periodicity, but in a 
 Report read before the Greenwich Board of Visitors on June 6, 1868, Mr Airy 
 stated that "the forms of the curves for 1863 do not sensibly differ from those 
 in 1841." 
 
666 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 attributed the phenomena to operations immediately connected 
 with the earth, and it does not appear that there is any re- 
 maining influence of this kind to be taken into account. 
 Hence this new class of phenomena must be referred to ex- 
 ternal agency, which for distinction will be called cosmical, 
 and the theory is next required to point out the origin and 
 character of this agency. 
 
 But before the theory can well be thus applied, it is 
 necessary that the laws of these phenomena, and the lengths 
 of the periods of their recurrence, should be ascertained by 
 observation. In cosmical magnetism, just as the case was 
 with respect to the gravitation of the solar system, the obser- 
 vational determination of laws must precede the theoretical 
 explanation of them. The only result of this kind that has 
 hitherto been announced is, a decennial or undecennial period 
 of the disturbance-variation, which is considered by Sabine, 
 Hansteen, and other physicists to be deducible from observa- 
 tions of declination, dip, and intensity continued through a 
 sufficient number of years. The Astronomer Eoyal has, 
 however, stated*, that the curves above mentioned, extended 
 to the date of 1863, indicate no such period, and that it is not 
 exhibited by the large disturbances, or " magnetic storms " of 
 the years 1858 1863. With respect to those curves it is to 
 be noticed that they were formed by simply taking the means 
 of the hourly variations after abstracting the largest disturb- 
 ances, and that it seems possible to separate the remaining 
 diurnal variations into two systems obeying different laws, 
 one that of the regular variations already discussed, and the 
 other a system of disturbance-variations. Moreover it is not 
 unlikely that as far as regards disturbances riot of the largest 
 class, the curves may synchronise in part with periods of 
 different lengths, and that 1847 1848 may have been an 
 epoch of accumulative maximum. These, however, are points 
 which probably require a long course of observation for their 
 
 * In the Report referred to in the previous note. 
 
THE THEOEY OF MAGNETIC FORCE. 667 
 
 decision. In the meantime I propose, with some reservation, 
 the following theory of the cosmical variations. 
 
 In the first place it may be argued from the previous 
 theory that by the Sun's motion in space and revolution about 
 its axis, solar magnetic streams will be generated analogous 
 to those that result from the earth's translatory and rotatory 
 motions (p. 547). According to the law expressed in p. 607? 
 the intensities of the magnetic streams of two similar bodies 
 are proportional, cceteris paribus, to the squares of the linear 
 dimensions. Hence, although the Sun's motion of translation 
 in space is less than the sum of the earth's motion in its orbit 
 and that which it has in common with the Sun, on account of 
 the Sun's vast magnitude the solar magnetic streams may be 
 immensely more intense than the terrestrial. The same law 
 of circulation will govern both ; so that, on the hypothesis of 
 symmetry, the solar streams, after issuing at different inclina- 
 tions from all points of one hemispherical surface, will enter 
 at the same inclinations at the corresponding points of the 
 other. All the courses will be symmetrical with respect to 
 the plane which contains the Sun's equator, those of the 
 streams which issue and enter near the poles being diverted 
 so as to cut that plane at very remote distances from the Sun's 
 surface. (See p. 653). Accordingly the portion of the aether 
 agitated at a given time by these streams will have the form 
 of a double convex lens the middle point of which coincides 
 with the Sun's centre. 
 
 Now it seems allowable to assume that the 'Zodiacal Light 
 gives evidence of the existence of these streams, its form 
 being found by observation to be that of a double convex 
 lens having the Sun at its centre and its principal plane in- 
 clined at a small angle to the plane of the ecliptic. The same 
 plane appears also to deviate but little from the plane of the 
 Sun's equator. The luminosity is accounted for on the prin- 
 ciple (already adopted in p. 653) of the generation of luminous 
 vibrations by disturbances of the aether arising from the coex- 
 istence and mutual collisions of steady and unsteady motions, 
 
668 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 the unsteady motions in this case being the vibrations of sun- 
 light, and possibly also vibrations of other orders. That the 
 zodiacal light extends to the earth and still farther is proved 
 by the fact that it has been seen " simultaneously at both east 
 and west horizons from 11 to 1 o'clock for several nights in 
 succession*." 
 
 On this assumption, therefore, the magnetic needle will be 
 directly acted upon by solar aetherial streams, and possibly 
 also to some amount by terrestrial streams induced by these. 
 If we conceive the earth to revolve in a circle about the Sun 
 in the principal plane of the zodiacal light, the combined 
 effect of these two sets of streams will be to modify, in the 
 same degree throughout the year, the solar-diurnal variations. 
 But evidently these modifications will not obey the same law 
 as the variations due to solar heat, which, as was shewn in 
 pages 647 649, have one principal maximum and one prin- 
 cipal minimum in 24 hours, whereas the streams now con- 
 sidered, being equally effective by night and by day, will 
 produce two maxima and two minima in the same interval. 
 It may, therefore, be presumed that the phenomenon of a 
 subordinate maximum, called the nocturnal episode, results 
 from the coexistence of these streams with those to which the 
 regular diurnal variations are due. (See the remarks in 
 Walker, 72, p. 130). 
 
 Supposing, now, the earth to revolve in an elliptical orbit 
 in the same zodiacal plane, and the intensity of the solar mag- 
 netic streams to be a function of the distance from the Sun's 
 centre, there will be produced by change of distance a small 
 annual inequality, which will probably not be distinguishable 
 by observations from that which was considered (p. 660) to 
 
 * Respecting this and the other facts here stated, see an Article entitled 
 "A Theory of the Zodiacal Light" contained in the Philosophical Magazine for 
 February 1863. At the end of the Article mention is made of observations of a 
 luminous arch of uniform breadth and intensity, seen at night to stretch along the 
 ecliptic quite across the sky. This phenomenon appears to be analogous to the 
 zodiacal light, being probably due to the crossing of the terrestrial magnetic 
 streams by the gravitational and other vibratory motions emanating from the earth. 
 
THE THEORY OF MAGNETIC FORCE. 669 
 
 result from the varying velocity of the earth. And lastly, the 
 earth moving in its actual orbit in the plane of the ecliptic, the 
 inclination of the principal plane of the zodiacal light to this 
 plane will produce a small semi-annual inequality of the 
 diurnal variations, which may be supposed to be mixed up 
 with that which was attributed (p. 650) to the inclination of 
 the earth's equator to the plane of the ecliptic. The varia- 
 tions resulting in the above three ways from the solar mag- 
 netic streams will, consequently, not have the effect of altering 
 the annual magnetic means. To account for the changes 
 from year to year which, as already stated, observation has 
 detected, I make the hypothesis that they are caused by solar 
 magnetic streams generated ~by the gravitational attraction of 
 the Sun's atmosphere by the Planets. 
 
 That the Sun has a very large atmosphere is proved by 
 phenomena witnessed during a total solar eclipse, as well as 
 by the very sensible diminution of the brightness of the disk 
 at parts contiguous to the periphery, which may with reason 
 be ascribed to loss of light caused by the passage of the solar 
 rays through the lower strata of a thick atmosphere. The 
 attraction of such an atmosphere by the Planets would gene- 
 rate magnetic streams exactly in the manner already described 
 (p. 662) relatively to the attraction of the earth's atmosphere 
 by the moon; and on account of the large magnitudes of 
 some of the attracting bodies, and the vast extent of the solar 
 atmosphere, it may be allowable to suppose that the circulat- 
 ing streams thus generated are of such intensity as to have, 
 by propagation, a sensible effect at the distance of the earth 
 from the Sun. The fact that the zodiacal light is observed to 
 vary from time to time in brightness may be accounted for 
 by supposing that its luminosity is due in part to these 
 streams of planetary origin, which plainly must vary in in- 
 tensity with the varying relative positions and heliocentric 
 distances of the planets ; although it would be difficult to 
 ascertain in what proportions they and the streams before 
 considered contribute to the luminous effect. 
 
670 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 From what has already been shewn relative to the streams 
 which produce the lunar-diurnal variations (p. 663), the pla- 
 netary streams will be nearly symmetrical with respect to the 
 plane of the Sun's equator, and will circulate in the direction 
 opposite to that of the motion about its axis, and therefore to 
 that of the earth's motion in its orbit. Also the directions of 
 the before mentioned streams constituting the Sun's proper 
 magnetism, will cut the principal plane of the zodiacal light, 
 and, therefore, nearly the plane of the ecliptic, at right angles. 
 Hence the combination of the two acts of streams will produce 
 a resultant action the magnitude and direction of which will 
 vary chiefly by reason of variation of the intensity of the 
 planetary streams. The direction of the resultant will clearly 
 be always parallel to a plane perpendicular to the earth's 
 radius-vector, and be inclined to the plane of the ecliptic by 
 an angle which is constant so long as the ratio of the in- 
 tensities of the two streams is constant. This being the case, 
 it follows that the direct action of these streams on the needle 
 (even if we include the modifications they may receive by 
 their passage through the earth, which, probably, would be 
 of small amount) will appear as a solar-diurnal variation of 
 the magnetic elements. 
 
 If we call these effects disturbances, to distinguish them 
 from the regular solar-diurnal variations due, according to the 
 theory, to atmospheric temperature, the conclusions that may 
 be drawn from the above reasoning may be stated in the very 
 terms employed by Walker (p. 90) relative to the general 
 character of the disturbance-variation of declination as de- 
 duced from observation : (1) "The disturbances, when con- 
 sidered in their mean effects, are subject to a Law of Perio- 
 dicity, the period being a mean solar day. (2) This law is 
 totally different in character from that which governs the 
 diurnal motion of the needle when undisturbed." 
 
 To investigate mathematically the laws of the disturbance- 
 variations would require the solution of the following pro- 
 blem : To find the resultant action on the needle of two 
 
THE THEORY OF MAGNETIC FORCE. 671 
 
 streams, one the proper terrestrial stream having at a given 
 locality a given intensity and direction relative to the earth, 
 and the other the composite solar stream above defined. As 
 neither the intensity of the latter, nor its inclination to the 
 plane of the ecliptic is known, an exact solution of the pro- 
 blem is not at present possible. It seems, however, legitimate 
 to infer from the theory upon general considerations, as far as 
 regards the declination, that there will be at a given place 
 both easterly and westerly deflections of the needle in 24 
 hours, having each a prominent maximum, but in other 
 respects dissimilar in type both on account of the obliquity of 
 the ecliptic and the asymmetry of the lines of declination 
 relative to the earth's axis. These inferences accord with 
 observation. I think, too, it might be anticipated from the 
 conditions of the problem that the same types might occur at 
 different localities and different local hours. It is found, in 
 fact, that the disturbance-variations at Kew, Hobarton, St 
 Helena, and Nertschinsk present very similar types, although 
 the type of the westerly deflections in some instances agrees 
 with that of the easterly in others*. 
 
 I forbear to pursue this subject through more details, 
 because I consider the preceding arguments and explanations 
 to be sufficient for my present purpose, which is only to give 
 primd facie evidence that the proposed theory of magnetic 
 force is capable of embracing the intricate facts and laws of 
 Terrestrial Magnetism without having recourse to any other 
 principles than those on which the general physical theory is 
 founded. A more complete theory of the phenomena may be 
 expected to be attainable on the same principles by means of 
 an extension of the observations and a more exact application 
 of mathematical reasoning. A few additional inferences may, 
 however, be drawn from the theory in its present state, 
 which appear to me peculiarly to entitle it to confidence. 
 
 (1). Since the theory attributes the cosmical disturbance- 
 
 * See the Rede Lecture, p. 11 ; Proceedings of the Royal Society, Vol. x. 
 p. 633; and Walker's Chap, iv., On Disturbances, pp. 80100. 
 
672 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 variations to solar magnetic streams which originate, in part, 
 from the gravitational attraction of the Sun's atmosphere by 
 the planets, it follows that there will be changes of effect and 
 epochs of maximum depending on their synodic revolutions 
 and in some degree on the eccentricities of the orbits. By the 
 same argument as that in page 669, each planet will generate 
 two equal and opposite solar atmospheric waves, the crests of 
 which will be nearly in the plane passing through the planet 
 and the Sun's axis. Hence, in consequence of the Sun's 
 revolution about its axis, the waves will be propagated along 
 the surface in the opposite direction with the relative angular 
 velocity of the Sun and planet. This uniform propagation 
 involves the conditions (p. 664), that the velocity of the 
 atmospheric particles in the direction of propagation be a 
 maximum, and their density a minimum, at syzygies, and the 
 density be a maximum, and the velocity zero, at quadratures. 
 The gradations of density will tend, as always, to produce 
 setherial streams by forces directed towards the denser parts ; 
 and, just as in the case of the terrestrial streams due to the 
 moon's attraction, these forces will cause circular streams to 
 flow in the direction contrary to that of the Sun's rotation, 
 such that their velocity will be intermittent, the maximum 
 occurring at syzygies and the minimum at quadratures. 
 
 Now the theory assumes that the influence of the streams 
 thus generated extends to the earth, and that in conjunction 
 with the intrinsic solar streams, they produce the disturbance- 
 variations of the magnetic needle (p. 670). But since the vari- 
 able part of this effect depends on the square of the sum of the 
 velocities due to the different planets, and since each of these 
 velocities has maximum values at points in the direction of 
 the planet from the Sun and the opposite direction, it is evident 
 that the disturbances will fluctuate in magnitude according to 
 the -changes of the relative directions of these maxima. There- 
 fore, apart from an exact determination theoretically of the 
 periods of these fluctuations, which may not at present be 
 possible, the existence of periodicity, and of maxima and 
 
THE THEORY OF MAGNETIC FORCE. 673 
 
 minima in the magnetic effect of these streams, strictly follows 
 from the foregoing reasoning. Consistently with this in- 
 ference, General Sabirie has concluded from a large number 
 of observations in different localities that the disturbance- 
 variations go through a cycle the length of which he considers 
 to be about ten years. (Walker, pp. 95 100.) It is not 
 likely that the eccentricities of the orbits can sensibly affect 
 the duration of this cycle, which is probably a composite 
 synodic period, involving the periodic times of Jupiter and 
 Venus, inasmuch as the magnetic streams generated, in the 
 manner supposed, by gravitational attraction, and being 
 effective at the earth, must be mainly due to the attractions of 
 these two planets together with that of the earth. It is, 
 therefore, in this respect worth noticing that 13 semi-synodic 
 periods of Venus exceed 19 semi-synodic periods of Jupiter by 
 only six days. Consequently, as far -as regards the influence 
 of these two planets we might expect to find a cycle equal in 
 length to 9 \ synodic periods of Jupiter, or 10,4 years ; which 
 accords with the above-mentioned inference from observation. 
 (2). The observations of solar spots by M. Schwabe 
 during the years 1826 1864 indicate that in respect to the 
 number of groups and frequency of their occurrence, these 
 phenomena are subject to periodic variations, and that the 
 period is between ten and eleven years. (See the table in 
 Walker, p. 102.) In a <communication to the Royal Society 
 in March 1852, General Sabine, having previously discovered 
 the existence of a periodical variation in the magnetic dis- 
 turbances, pointed out its coincidence in period and epoch with 
 the variation in frequency and magnitude of the solar spots, 
 then recently announced by M. Schwabe*. This remarkable 
 coincidence established a probability that the disturbance- 
 variations and the solar spots might be related to each other 
 by some common cause. The foregoing theory of the dis- 
 
 See in the Philosophical Transactions, Art. v., 1851, and Art. Tin., 1832 ; 
 the Rede Lecture, pp 69; Walker, pp. 100103. 
 
 43 
 
674 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 turbance-variations gives the means of indicating, as follows, 
 a conceivable mode of such relation. 
 
 The solar spots are phenomena pertaining to the Sun's 
 photosphere, which appears from observation to be a stratum 
 of cloud-form matter, suspended and moveable in the Sun's 
 atmosphere, and subject, like terrestrial cloud, to accumula- 
 tions and separations of parts. The accumulations are seen as 
 faculce, and the separation of parts gives rise to the breaking 
 out of maculce, or spots, which are considered to be depressions 
 below the mean surface of the photosphere, generally pene- 
 trating through the photospheric stratum. Now whatever 
 may be the exact modus operandi of the forces producing 
 these changes, it is not unreasonable to suppose that the 
 magnetic streams which, according to this theory, are ge- 
 nerated in the Sun's atmosphere by planetary attractions, 
 and circulate about its surface, are mainly concerned in this 
 operation. The Sun's heat, inasmuch as it is symmetrical 
 about the centre, would have no such effect. By attributing 
 the solar spots to planetary influence we at once account for 
 the fact that they prevail for the most part in the solar equa- 
 torial regions. Also, according to this view, we have, in the 
 above-mentioned relation between the synodic periods of 
 Venus and Jupiter, a reason for the ten or eleven years' cycle 
 of the spots, as well as for its being the same as that of the 
 disturbance- variations, and having the same epochs of maxima 
 and minima. 
 
 (3). Besides the more regular disturbance- variations 
 hitherto considered, there are others of a violent and transitory 
 character, called magnetic storms, which have been found to 
 occur simultaneously at widely different localities. For in- 
 stance, on September 25, 1841, one was felt at Toronto, the 
 Cape of Good Hope, Prague, and Van Diemen's Land. 
 (Walker, p. 81.) These effects may, possibly, be attributable 
 to variations of the Sun's magnetic streams, (principally of 
 those due to the planetary attractions), resulting from the 
 operation of solar local causes, analogous, it may be, to those 
 
THE THEORY OF MAGNETIC FORCE. 675 
 
 which produce the earth-currents in the terrestrial atmosphere. 
 This explanation seems to receive countenance from a singular 
 phenomenon, which was witnessed independently by two ob- 
 servers, Messrs. Carririgton and Hodgson, namely, a sudden 
 outburst of light near a large spot, about noon of Sept. 1, 
 1859, at which time, it was subsequently found, the mag- 
 netic needles at Kew were considerably disturbed. Before 
 and after this occurrence, on August 28 and September 2, 
 there were magnetic disturbances of extraordinary violence 
 felt in Europe, America, and Australia, and accompanied by 
 magnificent displays of Aurora seen over a very wide extent 
 of the earth's surface*. The sudden appearance of luminosity 
 might, accordingly, be indicative of disturbances of the regular 
 solar magnetic streams by others due to local causes in con- 
 nection with the large spot. To account for the phenomena 
 of magnetic storms, such disturbances must be supposed to 
 be occasionally so violent as to be very sensibly felt at the 
 distance of the earth. Also it is presumable that Auroras of 
 the largest class may be produced by the crossing of the 
 proper terrestrial streams by these unsteady sun-streams, as 
 the Auroras of a more local character were supposed to result 
 from the crossing of the same streams by the returning earth- 
 currents. This extraneous cause of Aurora is alluded to in 
 p. 654. 
 
 From the fact that the magnetic storms have the same 
 local hours of maximum as the more regular disturbance- 
 variations, it may be inferred that the disturbances of all 
 kinds are propagated through space in the same courses. Also 
 as the perturbations of the largest class would be likely to be 
 in greatest degree when the planetary disturbance-variations 
 are greatest, it might be expected that the length of the cycle 
 of disturbances would be the same when both kinds are taken 
 into account as when the largest are excluded. This, in fact, 
 
 * See Walker, pp. 105 108; Monthly Notices of the Royal Astronomical 
 Society, Vol. xx., Nov. 11, 1859; also a paper by Mr Balfour Stewart in the Phil. 
 Trans, for 1861, p. 423. 
 
 432 
 
676 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 is considered by Sabine to be indicated by the observations 
 he has discussed with especial reference to this point*, and 
 the results stated in page 670 may accordingly be regarded as 
 embracing disturbances of all classes. On theoretical grounds 
 it is probable that if all disturbance- variations could be elimi- 
 nated from the solar-diurnal variation, the decennial cycle 
 would disappear. 
 
 On the mysterious phenomenon of the secular variations of 
 the magnetic elements I have only the following few remarks 
 to offer f. Considering the manner in which, according to 
 the theory, the earth's magnetic streams are produced (p. 613), 
 it was to be expected that, as is known to be the case from 
 observation, there would be two poles of maximum intensity 
 in the nothern hemisphere, determined as to position by the 
 two large tracts of land constituting the North American and 
 Asiatic continents, and only one in the southern hemisphere. 
 But why are the lines of equal declination, dip, and intensity 
 continually undergoing slow changes? If the materials of 
 which the earth is composed always retained absolutely the 
 same relative position, there would seem to be no reason for 
 such a change. A very gradual oscillation of the earth's solid 
 crust, accompanied by a corresponding change of distribution 
 of the interior fluid part, might, without altering in any 
 considerable degree the positions of the poles of maximum 
 intensity, very much change the courses of the above-named 
 magnetic lines, because such an oscillation could be effected 
 only by changes of those gradations of density of the materials 
 of the earth to which the theory attributes the generation of 
 the magnetic streams. 
 
 * See the " St Helena Observations," Vol. IT., p. cxxxii. 
 f The explanation I proposed in the Phil. Mag. for February 1861, art. 41, is 
 not tenable. 
 
GENERAL CONCLUSION. 
 
 I HAVE now completed to the extent of my ability a 
 general physical theory, consisting of explanations of the 
 principal facts and laws in the different departments of 
 dynamical physics, conducted by means of mathematical 
 investigations and results contained in the first and more 
 abstract part of the work (ending in page 316). The leading 
 idea of the theory is, that all explanations of phenomena, and 
 all quantitative physical laws, may be mathematically de- 
 duced from a few fundamental hypotheses, distinct conceptions 
 of which may be formed from sensation and experience. The 
 hypothetical facts on which the theory rests are, (1) that all 
 substances consist of minute spherical atoms, of different, but 
 constant, magnitudes, and of the same intrinsic inertia ; and 
 (2) that the dynamical relations and movements of different 
 substances and of their constituent atoms, are determined by 
 the pressures of the aether against the surfaces of the atoms, 
 together with the reaction of the atoms against such pressure 
 by reason of the constancy of their form and magnitudes. The 
 aether is assumed to be a uniform elastic fluid medium, per- 
 vading all space not occupied by atoms, and varying in 
 pressure proportionally to variations of its density. The 
 theory recognises no other kinds of force than these two, the 
 one an active force, resident in the aether, and the other a 
 passive reaction of the atoms*. 
 
 * For a synoptical view of the principles of Natural Philosophy which lonj; 
 before the composition of the present work I laboured to uphold, I may refer to 
 an Article in the Philosophical Magazine for June 1866, entitled, " The funda- 
 mental ideas of matter and force in Theoretical Physics." 
 
678 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 I am well aware that some of the explanations of physical 
 phenomena which I have given on these hypotheses are 
 incomplete, and too little supported by exact analytical rea- 
 soning, or numerical calculation. These deficiencies are 
 partly owing to difficulties not overcome, but not, I think, 
 insuperable, respecting the proper modes of applying the 
 solutions of partial differential equations in physical investi- 
 gation, and partly to an imperfect knowledge of the exact 
 physical conditions of the questions themselves. There are 
 various questions which for their complete mathematical 
 treatment according to the principles of the proposed general 
 theory, would require a knowledge of the arrangement and 
 relative magnitudes of the atoms of bodies, and the propor- 
 tionate numbers in which those of different magnitudes enter 
 into their composition ; in short, such data as the sciences of 
 crystallography and chemistry, when they have reached a 
 more advanced stage, might be expected to furnish. But in 
 the existing state of physics these data are not obtainable; 
 and it is, perhaps, not possible that they will ever be obtained 
 except by an appropriate combination of mathematical theory 
 with experimental research. I have not attempted in this 
 work to give theories of those two departments of physics, 
 thinking that antecedently to such attempt, definite conclu- 
 sions must be reached respecting the intrinsic character and 
 the laws of operation of the different kinds of physical force ; 
 to which object, therefore, my efforts have been mainly 
 directed. It is true, however, that the theory of Double 
 Refraction contained in pages 375 382 does involve con- 
 siderations respecting crystalline structure and atomic ar- 
 rangement; but this instance only serves to shew that the 
 undulatory theory of light, placed on an intelligible mathe- 
 matical basis, may be employed as an instrument of enquiry 
 into the interior constitution of substances. 
 
 But although for these reasons the proposed physical 
 theory must be regarded as incomplete, still a very strong- 
 presumption has been established that its hypotlwses are true, 
 
GENERAL CONCLUSION. 679 
 
 inasmuch as, few and simple as they are, they have been 
 applied in a very wide range of subjects without encountering 
 any irreconcileable contradiction from facts. The number 
 and variety of the explanations of physical phenomena which 
 in the second part of this work have been shewn, "either with, 
 or without, the support of symbolical reasoning, to be com- 
 patible with these hypotheses, seem already sufficient to justify 
 the conclusion that the ultimate parts of bodies are atoms such 
 as they have been assumed to be, and that the active forces 
 concerned in Light, Heat, Electricity, Galvanism and Mag- 
 netism, are modes of operation of a single elastic fluid. 
 
 It will be proper to notice here an objection which has 
 been raised against the idea of a universal elastic medium. 
 The objection I refer to is one advanced by Sir John 
 Herschel, in these terms : " Under no conception but that of 
 a solid can an elastic and expansible medium be self-contained. 
 If free to expand in all directions, it would require a bounding 
 envelope of sufficient strength to resist its outward pressure. 
 And to evade this by supposing it infinite in extent, is to 
 solve a difficulty by words without ideas to take refuge from 
 it in the simple negation of that which constitutes the diffi- 
 culty*." To this argument I reply that no objection can be 
 urged against assuming, simply as an hypothesis, the existence 
 of an elastic fluid medium the pressure of which varies as its 
 density. If on this hypothesis a large number of physical 
 tacts and laws can be explained, presumptive evidence is 
 given that the hypothetical fluid is a reality, and it will be so 
 much the stronger as the number of the facts explained is 
 greater. The strength of the evidence would be much in- 
 creased if it should appear that all the laws and modes of 
 action of all the physical forces admit of being referred by 
 mathematical reasoning to the supposed properties of this 
 
 * This passage occurs in an Article on Light in a periodical called Good 
 Words (Number for July, 1865, p. f00). 1 have thought it necessary to meet the 
 argument it contains, although I do not know that it was written with any refer- 
 ence to views I had previously published. 
 
680 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 medium ; in short, that all the known forces of the universe 
 may be conceived of as consequences of its immediate action. 
 But in that case it necessarily follows that the fluid is of 
 unlimited extent, because it could be limited only by forces 
 acting beyond its limits, whereas such forces would be non- 
 existent. I am not asserting, it should be observed, that the 
 theories of the physical forces proposed in this work have 
 conclusively shewn that all force is resident in the hypo- 
 thetical fluid, but I maintain that in proportion as this is 
 shewn the fluid is proved to be unlimited. Now there is no 
 antecedent reason why this conclusion, so arrived at, should 
 be regarded as inadmissible. By admitting it we should 
 only be compelled to think of the aether as we think of time 
 and space, that is, as extensible ad libitum. Regarded as the 
 seat of all physical power, it can hardly be otherwise con- 
 ceived of. This argument appears to me to be a sufficient 
 answer to the above objection. 
 
 The general theory is also liable to another objection 
 requiring an answer. It might be urged that although the 
 theory professes to account for all the laws of physical force, it 
 rests on the assumption of such a law relative to the asther, 
 namely, that its pressure varies as its density, for which, 
 therefore, it does not account. It is an answer, in part, to 
 this objection to say that the law is perfectly comprehensible, 
 experiment having shewn that an actual instance of it is 
 exhibited by air of given temperature. Hence, as explana- 
 tions of phenomena founded on the assumption of the law 
 are also intelligible, it is not necessary, as far as regards 
 physical research, to enquire what may be the reason that the 
 aether conforms to this particular law ; it suffices to accept it 
 as an ultimate fact. 
 
 But it may be farther said, that in the case of air of uni- 
 form temperature theoretical reasons for the law have been 
 actually given on the principles of the general physical theory. 
 (See pages 469 471.) These reasons not only rest on the 
 hypothesis of the law of the pressure of the sether, but sup- 
 
GENERAL CONCLUSION. 681 
 
 pose also that it acts dynamically by vibratory motions, the 
 maintenance of which, whatever may be the original causa- 
 tion, is due to the inertia and reactions of the atoms of the 
 grosser bodies. It is, therefore, conceivable that the law of 
 pressure of the aether may be caused by a like aether of 
 another order, similarly acting by means of vibratory motions; 
 and so on ad libitum. For here Newton's dictum, "nature 
 knows no limit" (neque novit natura limitem, Principia, 
 Lib. I. Scholium to Lemma xi.) may be applied with as good 
 reason as to the abstract relations of space. In accounting for 
 physical force in this way, we seem to reach the farthest con- 
 ception we can form of it, resolving it, in fact, into volition, 
 such as that we are conscious of when we exert power. This 
 inference accords well with an opinion expressed by Newton 
 in Query 28, attached to his Optics, where he says that the 
 part of philosophy is " to deduce causes from effects, till we 
 come to the very First cause, which certainly is not mechani- 
 cal." In short, force dissociated from personality and will, 
 must be for ever incomprehensible by us, because it would 
 be something contradictory to our consciousness. It is also 
 worthy of remark that the ultimate conception of physical 
 force to which the above argument conducted is analogous to 
 the idea which, as before said, we have of the extensibility 
 ad libitum of time, space, and the aether ; an analogy which 
 seems to shew that all these entities exist by antecedent will. 
 There is still another point that requires to be adverted to, 
 as it might be brought against the admission of the philo- 
 sophical principles maintained in this work. On those prin- 
 ciples it is argued that there can be no other kind of force 
 than pressure, because by the sense of touch we have a distinct 
 perception of such force, acquired in a unique manner, that is, 
 by the conscious exertion of power. But it has been admitted 
 in page 359, and is, in fact, a necessary consequence of the 
 proposed theory of matter and force, that when the hand 
 touches any s'ubstance there is no actual contact of the ulti- 
 mate parts of the hand and those of the substance. Let this 
 
682 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 be granted ; the fact still remains that we do in this way 
 acquire a distinct perception of pressure against a surface. 
 The question may consequently be asked, Can a true theory 
 of the universe be made to rest on such indications of sense, 
 so that, by reasoning on this foundation, we can come to know 
 the system of the world just as we know any thing? I think 
 I may say that the results of the theoretical investigations 
 contained in this volume have gone a long way towards 
 answering this question affirmatively. In the course of the 
 different theories I have frequently spoken of explaining facts 
 and laws. It may now be stated that this expression is to be 
 understood in the sense, that the fact or law is explained when 
 it is shewn by mathematical reasoning to be a necessary con- 
 sequence of certain ultimate facts, not capable of similar 
 explanation, but perfectly intelligible from sensation and 
 experience. 
 
 Admitting, for the sake of argument, that the foregoing 
 question has been definitively answered by physical investi- 
 gation conducted on the principles here advocated, physics 
 might then be' said to have done its part, and to have fur- 
 nished the data required by the metaphysician for reasoning 
 upon the essential character of human knowledge and of our 
 relation to the external world. The long-debated questions 
 respecting the real and the nominal, the subjective and the 
 objective, appear to me to admit of solution by no other means. 
 As it would be inappropriate to enter upon a metaphysical 
 discussion in a book devoted to physics, I shall limit myself 
 to a mere expression of the opinion, that metaphysical enquiry, 
 coming, as it ought, after the establishment of physical truth, 
 will eventually lead to the conclusion that external to our- 
 selves there exist personality, intelligence, and will, but that 
 sensations and perceptions, inclusive even of the perception of 
 space and time, are subjective, and such as they are by the sole 
 will of our Creator, operating thus immediately and constantly 
 for the purpose of bringing us into intellectual and moral 
 relations towards Himself and our fellow-creatures. 
 
GENERAL CONCLUSION. 683 
 
 It remains to deduce from the method of philosophy 
 adopted in this work some further inferences, which I consider 
 to be those of chief importance. It is well known that Newton 
 has repeatedly in his writings disclaimed entertaining the 
 opinion that gravity is essential to bodies. (See in page 359.) 
 The following passage from his Third Letter to Bentley shews 
 how decided his conviction was on this point : " That gravity 
 should be innate, inherent, and essential to matter, so that one 
 body may act upon another at a distance, through a vacuum, 
 without the mediation of anything else, by and through which 
 their action and force may be conveyed from one to another, 
 is to me so great an absurdity that I believe no man who has 
 in philosophical matters a competent faculty of thinking, can 
 ever fall into it." And yet not long after Newton's time the 
 view that he took was discarded, occult qualities were ad- 
 mitted, and force was considered to be inherent in matter. 
 This scientific heresy, as it may be called, arose apparently 
 from a misunderstanding of the distinction between law and 
 causation. Subsequently Hume added to the confusion by 
 his dogma that by experience we have no perception of causa- 
 tion, but only of a succession of events according to immutable 
 law. This view, which has obtained wide acceptance, was in 
 different ways advocated by Thomas Brown of Edinburgh, 
 Baden Powell of Oxford, and other writers, and gave occasion 
 to endless perplexities from which neither Scotch nor German 
 metaphysicians have been able to extricate themselves. 
 
 The method of philosophy which I have laboured to in- 
 culcate, which may be regarded as a legitimate extension of 
 Newton's views, meets the difficulty by making a distinction 
 between the induction of law from experiment and observa- 
 tion, and the accounting for law by a theory of causation, the 
 basis of which is intelligible from personal sensation and 
 experience. In physical theory, properly so called, not a step 
 can be taken without the employment of symbols, geometrical 
 or analytical, expressive of force, and such that by means of 
 them the effects of force can be calculated. By such calcula- 
 
684 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 tion founded on appropriate hypotheses (see pp. 356 359), 
 facts and laws are accounted for by being referred to their 
 dynamical causes, causation, as we are taught by personal 
 experience, necessarily implying force,. Natural Philosophy 
 is subordinately the knowledge of laws ; it is perfect only 
 when it embraces the causative reasons for laws. 
 
 The different physical theories discussed in this Volume 
 are conformable in their principles to the views just stated, 
 inasmuch as they wholly consist of calculations of the effects 
 of force acting as pressure under intelligible conditions. The 
 theory of Light comes under this description, because it re- 
 gards the phenomena of light as sensible effects of vibrations of 
 the aether acting dynamically. So also Heat and Molecular 
 Attraction, Gravity, Electricity, Galvanism, and Magnetism, 
 are all modes of force of the setherial medium. Now imperfect 
 as these theories may be in the condition to which my en- 
 deavours have brought them, it may yet be possible to state 
 the character of the scheme of the Universe the Systema 
 Mundi which they tend to unfold. It may, I think, be 
 asserted that the completion of these theories will demonstrate 
 the existence of a vast and wonderful mechanism, of which 
 not the least wonderful quality is, its being so constructed 
 that we can understand it. If it be objected that we do not 
 know what the sether is, nor what atoms are, it seems suffi- 
 cient to answer that by personal experience, as already ex- 
 plained, we can understand with what qualities they are 
 endued as elementary constituents of the system of the uni- 
 verse, and this is enough for enabling us to comprehend the 
 whole of the mechanism. It would be thought unreasonable 
 if an engineer, after explaining the construction of a steam- 
 engine, should be required to say what fire is, or what water 
 is. He would think that his explanation of the mechanism 
 and working of the engine should satisfy the enquirer if 
 it rested on properties of fire and water that are known by 
 ordinary observation. I think, however, we may go a step 
 farther in answer to the above objection, and assert that the 
 
GENERAL CONCLUSION. 685 
 
 sether and the atoms are entities to us of the same class as our 
 sensations and perceptions, that is (as was before intimated) 
 our conceptions of them may be determined by the imme- 
 diate volition of the Creator, while as external realities they 
 are simple manifestations of His presence and power. 
 
 The operations of nature being, according to these views, 
 intelligible to us because they are conformable to laws the 
 reasons for which we are able to enquire into and comprehend, 
 it does not hence follow that the universe is such as it is by 
 any antecedent necessity. Eather, it was so framed because 
 the Framer of it had this very end in view, that of making it 
 comprehensible by human understanding. And here we may 
 see a reason why our knowledge of the creation is riot a bare 
 knowledge of laws, but of the evolution also of the laws from 
 fundamental principles. The mathematical investigation of 
 the reasons for physical laws is not to be regarded as merely 
 an exercise of human intellect, although as such it is not with- 
 out value ; but it is of far greater moment as the means of our 
 attaining to a view, such as cannot otherwise be reached, of 
 the Power and Wisdom by which the Creator accomplishes 
 His purposes in the natural world. For all the dynamical 
 consequences of the qualities of the aether and the atoms, and 
 of their mutual relations, which we labour to ascertain by so 
 much toil in experimenting and calculating, must have been 
 present to the Divine Mind when these elements were called 
 into existence, and all the working of the complex machinery 
 must have been antecedently understood. It was, we may 
 presume to say, for the purpose of making His power and 
 wisdom known by this means, and thus bringing us into 
 intellectual relation to Himself, that the Almighty, 6 Havro- 
 /cpdroop, "ordered all things in measure, and number, and 
 weight." (Wisdom xi. 20.) 
 
 Those who conceive of natural operations as invariable 
 sequences of events according to fixed laws, may well feel 
 difficulty in admitting that there can be such deviation from 
 the course of nature as that which constitutes a miracle. The 
 
686 THE MATHEMATICAL PRINCIPLES OF PHYSICS. 
 
 same difficulty need not, I think, be felt when it is understood 
 that the Creator operates by laws for special purposes relating 
 to us as intellectual and moral beings, and that, being Author 
 and Upholder of the laws, He can, when His purpose is 
 simply to display His power, dispense with them, or alter 
 them, according to His pleasure. Possibly to some minds 
 the difficulty may be met by the consideration that the phe- 
 nomenal world around us, of which our bodies also form a 
 part, although to all human intents and purposes it is objective, 
 may yet intrinsically, and as regards divine operations, be 
 subjective, that is, consist of sensations and perceptions im- 
 mediately impressed (as I before said) by the ever present 
 Deity, and therefore changeable at His will. We are taught 
 that "He is not far from every one of us," and that "in Him 
 we live, and move, and have our being." I am not sure 
 that this view of the phenomenal universe is not necessitated 
 by the fact that the reality of miracles has been established 
 by observation and testimony. 
 
 All the laws for which I have endeavoured to assign 
 mathematical reasons have reference only to inorganic matter. 
 Organized bodies, while they obey the same laws, have, be- 
 sides, distinguishing qualities, respecting which scientific 
 research has already pointed to the conclusion that the laws 
 which govern them are developments from a few simple prin- 
 ciples. These principles and laws seem adapted to reveal to 
 us, even more than those of inorganic substances, the wisdom 
 in planning and the excellent working by which the various 
 parts of the creation are made to fulfil their destined purposes. 
 And beyond all this there are the mysterious relations of 
 organized beings to animal and rational life, the dependence, 
 apparently, on such relations of thought, feeling, and affec- 
 tion, and the multitude of moral and social consequences 
 thence arising. Through all this scheme law reigns, in order 
 that we may thereby be educated morally as well as intellec- 
 tually, and be taught the end of our creation. In proof of 
 this conclusion, it need only be said that when the Eternal 
 
GENERAL CONCLUSION. 687 
 
 Word, 6 Ao709, through whom the worlds were made, visited 
 His creation, by assuming a body like ours and submitting to 
 its laws He shewed us how we must act and suffer, that our 
 spirits, through faith and patience, may " partake of the 
 divine nature," and be formed for immortality. 
 
 In making the foregoing remarks I have followed the 
 example set by Newton at the end of the Principia, thinking 
 that they might not be considered to be an inappropriate 
 conclusion to a work devoted to the advancement of Newton's 
 Principles of Natural Philosophy. 
 
APPENDIX. 
 
 HAVING discovered that some questions relating to principles 
 of mathematical reasoning were omitted in the body of the work, 
 I have supplied the discussion of them in an Appendix. 
 
 I. Since in what is said on numerical equations in pages 
 31 37 no general proof is given that every such equation has as 
 many roots, real or impossible, as it has dimensions, I propose to 
 exhibit here the principles on which, as I conceive, the proof 
 under any form must rest, and also to indicate a process by which 
 all the roots may be found. 
 
 By ordinary algebraic multiplication we prove that the pro- 
 duct of any number (n) of binomial factors x + a, x + ft x + y, &c. 
 is a polynomial of n dimensions such that 
 
 (x + a) (x + fi) (x + y) &c. I x* + px*~ l + qx n ~* + &C. + SX + t, 
 
 the symbol x indicating that the two sides of the equation are 
 identical in value for any arbitrary value of x. We shall assume 
 that each of the quantities a, ft, y, &c. is an algebraically ex- 
 pressed numerical quantity, and consequently included under the 
 general form m + njl, m and n being real numerical quan- 
 tities, positive or negative, zero values being included. It will 
 also be supposed, as may be done without affecting the generality 
 of the reasoning, that p, q, &c. are real positive or negative 
 numerical quantities. In that case, if the value of one of the 
 quantities a, (3, y, &c. be m l +n l j\ i n l not "being zero, another 
 must have the value m^ n^J- 1 . 
 
 The formation in this manner of a polynomial by binomial 
 factors is a simple direct operation, in which a, ft y, &c. are given 
 quantities, and p, q, &c. are functions of these which the operation 
 itself determines. But every direct operation suggests the reverse 
 one, and it might accordingly be proposed to resolve a given 
 
APPENDIX. 689 
 
 polynomial, having real positive or negative coefficients of entirely 
 arbitrary values, into binomial factors. In this problem, assuming 
 the binomial factors to be x - a, x - /?, x y, &c. p, q. r, &c. are 
 the given quantities, and a, /?, y, &c. are unknown quantities to 
 be determined. 
 
 "We may obviously proceed to solve the problem by performing 
 the multiplication of the factors on the left-hand side of the 
 equality, and, on the principle of identity of value, equating the 
 coefficients of the same powers of x on the two sides. When this 
 is done, we get by elimination, as is known, either 
 
 a n +pa n ~ l + qa n ~ 2 + &c. + sa + t = 0, 
 
 or the same equation with each of the other quantities /?, y, &c. 
 in the place of a. It is here to be noticed that the sign of 
 equality = has a different signification from that of I before 
 employed, and that, in virtue of that signification, the foregoing 
 operation has brought the unknown quantity a into algebraic 
 relation with the given numerical quantities^, q, &c. Whether 
 that relation be a possible or impossible one at present we do not 
 know, and must, therefore, give to a the most general form that 
 an algebraically expressed numerical quantity can have, which is 
 determined by antecedent algebraic principles to be '% + y*J-l, 
 z and y being real, and either positive or negative. The proof of 
 the resolvability of the given polynomial into binomial factors de- 
 pends, therefore, on shewing, after substituting z + yj - 1 for a 
 in the above equation, that the numerical values of z and y can 
 be actually found. It is clear that what has been said of a applies 
 by parity of reasoning to J3, y, &c. 
 
 If, for the sake of brevity, we call the given polynomial < (x), 
 we shall have, after the substitution for a in the above equation, 
 <}>(% + yJITl) = Q. Hence in any case in which y = 0, the equa- 
 tion ^(*) = 0, or <(o;) = 0, is satisfied by a real value of x, that 
 is, it has a real root. All such real roots may be found by a process 
 which is sufficiently indicated in page 33. Let z 19 * a , z 3 , &c. be the 
 roots thus found. Then by ordinary algebraic division it is shewn 
 that <p(x) is divisible by each of the quantities x-z^ x-z^, &c. 
 without remainder. Hence we have the identical equality. 
 
 4> (x) !/() x (a: - a,) (x - * 2 ) (* - * 3 ) &c., 
 * 44 
 
690 -APPENDIX. 
 
 f(x) being another rational polynomial of lower dimensions than 
 <(#) by the number of the roots z^ z 2 , &c. We have thus suc- 
 ceeded in partly resolving < (x) into binomial factors. It remains 
 to effect a like resolution off(x). 
 
 By exactly the same reasoning as before, this is to be done 
 by obtaining values of z and y which satisfy the equation 
 f(z + yj 1) = 0. This equation may be put under the form 
 
 and as we know already that y cannot be zero, we have by reason of 
 the impossible symbol, ty (z, y 2 ) = and x (z, y*) = 0. These equations 
 suffice to determine z and y 2 , which, according to the foregoing 
 principles, must have one or more sets of corresponding real 
 values which satisfy both equations, and therefore the equation 
 f (x) = 0. Each set furnishes a pair of roots off(x) = 0, because 
 + y and y are equally applicable. Hence if z f and y be one 
 of the sets of values, f(x) will be divisible by x z' y'<Jl 
 and x z' + y'J 1, and consequently by the quadratic quantity 
 (x-z)* + y' z . Calling the several quadratic factors Q lt Q 2 , $ 3 , &c. 
 we have thus found that 
 
 < (x,) i Q, . Q z . Q 3 , &c. x (x-^) (x - z 2 ) (x - ,) &c. 
 The proposed polynomial is thus resolved into rational binomial 
 and quadratic factors. The above equality proves (1) that the 
 number of the binomial factors on the right-hand side is equal to. 
 the dimensions of < (x), .for otherwise there could not be identity 
 of value for arbitrary values of x ; and (2) that, for the same 
 reason, there can be no factors different from these, for as no other 
 can divide the right-hand side of the equality, no other can be a 
 factor of the left-hand side*. 
 
 As the resolution of < (x) into binomial factors is evidently 
 equivalent to finding the roots of the equation <f> (x) = 0, the above 
 reasoning has shewn that an equation has as many roots as it has 
 dimensions. The proposition is sometimes proved by employing 
 sines and cosines of arcs ; but as in a question which is purely 
 abstract these functions can only stand for numerical ratios having 
 
 * See on this subject three Articles which I contributed to the Philosophical 
 Magazine in the Numbers for February and April 1859, and January 1860. 
 
APPENDIX. 691 
 
 no geometrical signification, the proof so conducted does not differ 
 in principle from that produced above. In the Berliner Jahrbuch 
 for 1841, p. 281, Encke has given a general method of finding all 
 the roots of an equation, on the principle of separating it (as 
 above) into two parts for determining the impossible roots. 
 
 II. In the course of making a few remarks on equations 
 of partial derived functions, I say (in p. 56), 'by proceeding to 
 partial derived functions of the second order, two arbitrary 
 functions may be eliminated ; and so on.' It had not occurred to 
 me to question the possibility of effecting the eliminations, because 
 they require only direct operations, and I took for granted that 
 such operations, whether in algebraical calculus, or the calculus of 
 functions, are always practicable. But I might have known from 
 the discussion of this question in Peacock's Examples (pp. 209 
 213) that although the elimination is always possible, a partial 
 differential equation of the second order is often hot obtainable, 
 and that whenever this is the case the elimination conducts to two 
 partial differential equations of the third order. I find also that 
 Mr Airy, at the end of his Elementary Treatise on Partial Diffe- 
 rential Equations (1866); has shewn that after performing the 
 partial differentiations to the second order, there will be six func- 
 tions and not more than six equations, and hence he infers that 
 " we cannot eliminate these six functions so as to leave an equa- 
 tion between z, a?, y, and the partial differential coefficients of z 
 with respect to x and y of the first and second orders." As this 
 argument carried no farther might seem to indicate that the elimi- 
 nation is in no case practicable, I propose to give here a method 
 of elimination, which, although it may not be the only one or the 
 best, will succeed in finding either a differential equation of the 
 second order, or two differential equations of the third order. 
 
 Let one of the arbitrary functions be obtained as an explicit 
 function of the other and of z, x, y, and after differentiating it 
 partially with respect to a?, and then with respect to y, divide one 
 result by the other. By this means that arbitrary function is 
 eliminated, and there remains an equation containing in general 
 the other arbitrary function and its first derivative. Next obtain 
 this derivative, as an explicit function of the other quantities in- 
 volved in the equation, and differentiate it partially with respect 
 
692 APPENDIX. 
 
 to x and y. Then on dividing one result by the other it may 
 happen that all the arbitrary functions disappear from the equation 
 thence ariaing, and that the required partial differential equation 
 of the second order is thus arrived at. But if not, there will, at 
 least, be an equation containing the arbitrary function and its first 
 derivative, by means, of which equation and the previous one con- 
 taining the same two functions, both may be obtained in terms of 
 z y x, y, and partial differential coefficients. Then the separate eli- 
 minations of these functions by partial differentiations will produce 
 two partial differential equations of the third order. In the two 
 following instances the process conducts to a differential equation 
 of the second order. 
 
 dz dz d 2 z d a z 
 
 .. -7-t ~r> :r~2> j / > j~ 
 
 * dx dy dx dxdy dy 
 
 Example I. Let 
 
 Then, 
 
 Differentiating first with respect to x, andihen with respect 
 
 Hence, by eliminating <^' (xy), 
 
 Consequently, differentiating with respect to y, 
 
 and differentiating with respect to x, 
 
 x 3 r sx z x* /\ x , 
 
APPENDIX. 
 
 693 
 
 By equating these two expressions for the same quantity the 
 result is 
 
 an equation which is satisfied by the given formula. 
 
 Example II. Let 
 
 z = (j> (y.+ ax) $(y- ax). 
 By the differentiations, as before, 
 
 p za\l/'(y ax) 
 
 Hence, putting for brevity, ^ and ^f for \j/(y-ax) and 
 i// (y - ax), we obtain 
 
 By partial differentiations of this equation, 
 
 Hence it is readily found that 
 
 (r-a a t)z + a s q*-p 2 = Q, 
 which equation is satisfied by the given formula. 
 
 It is observable, in these two examples, that on equating the 
 two expressions for the second derivative of the second arbitrary 
 function, all arbitrary quantities disappear from the resulting 
 equation. This appears to be a condition necessary for obtaining 
 a differential equation of the second order. On applying the rule 
 to the example z = x$ (x + y) + y*\j/ (xy\ that condition was not 
 fulfilled, and the elimination of the arbitrary functions led to one 
 or the other of two complicated differential equations of the third 
 order. 
 
694 , APPENDIX. 
 
 III. In the few words I have said on the Calculus of Varia- 
 tions in page 58, I omitted to mention that the functions which 
 the rules of that Calculus give may be discontinuous ; that is, if 
 the answer to a question be, for instance, a geometrical line, this 
 line may consist of continuous portions of two or more different 
 lines, joined end to end, and inclined to each other at any angles at 
 the points of juncture. There is the more reason for adverting 
 to this principle, because the verification of it is necessary for 
 establishing fully the property of comprehensiveness which I have 
 much insisted upon in this work as characteristic of analytical 
 calculation. All quantitative relations, whether continuous or 
 not, are deducible by appropriate methods of calculation, when the 
 necessary data are furnished. Hence if in the nature of things 
 there be discontinuous maxima or minima, these should be de- 
 ducible by analysis. 
 
 As far as I ain aware the first instance of the solution of 
 a problem in the Calculus of Variations of this kind is one which 
 I proposed in the PhilosopJvical Magazine for January 1834 (p. 33) ; 
 viz., To find the shortest course of a ship from one given point to 
 another, its velocity being supposed to be a function of the angle 
 which the direction of its course makes with that of the wind*. 
 The solution of this problem according to the usual rules, as given 
 in the paper just mentioned, leads to the result, that the course 
 must consist of not fewer than two straight lines inclined at sup- 
 plementary angles to the direction of the wind. The ship has to 
 tack, or pass abruptly from one straight course to another. I have 
 also treated of the same problem in the Phil. Mag. for September 
 1862, where I have discussed more particularly the rationale of 
 the discontinuity of course. 
 
 The following problem involving in a somewhat different man- 
 ner the principle of discontinuity was proposed by Mr Todhunter 
 in page 410 of his ' History of the Calculus of Variations' : 
 To find the greatest solid of revolution of given superficies, the 
 generating line consisting of ordinates perpendicular to the axis at 
 two given points, and of a curve connecting them. Adopting the 
 
 * This problem, although it involved a new and important principle, did not 
 obtain a place in Mr Todhunter's History of the Calculus of Variations. In an 
 Article in the Phil. Mag. for September 1866, he has apologized for the omission. 
 
APPENDIX. 695 
 
 principle of discontinuity, Mr Todhunter proved that the curve must 
 join on continuously with the ordinates, and that the ordinates 
 must be equal. Although an equation between the co-ordinates x 
 and y of the curve is not obtainable, a relation between the arc 8 
 and ordinate y can be found, from which it may be inferred that 
 the curve is that which is described by the focus of an hyperbola 
 which rolls on a straight line. (See an article which I contributed 
 to the Phil Mag. for October 1866.) 
 
 The principle of discontinuity is also involved in the well- 
 known problem of finding the greatest solid of revolution the sur- 
 face of which is of given extent and cuts the axis in two given 
 points. Particular interest is attached to this problem because 
 mathematicians have expressed the opinion that the Calculus of 
 Variations fails to give a solution of it. Not being able to admit 
 that there can be such a thing as failure in analysis, but rather 
 regarding every supposed failure as simply indicating that some 
 point of principle required to be cleared up, I made many at- 
 tempts to remove the difficulty that has been felt in this case. 
 One attempt was the following. Assuming the usual expression 
 for the function u which is to be a maximum or minimum, viz. 
 
 we have, as is known, for determining the required function the 
 two equations A = 0, Ap = 0. Although there are many instances 
 in which these two equations lead to exactly the same function (as, 
 for example, in the problem just referred to, which is, therefore, 
 distinct from the present one), it cannot be admitted that the 
 equations are always equivalent ; for then the symbolic dif- 
 ference between them would have no signification, which from 
 the nature of analysis is impossible. Adopting the principle of 
 discontinuity, I supposed the equation Ap = Q might be equivalent 
 to A = and p = 0, and thus found for the required solid a cylinder 
 with hemispherical ends. Mr Todhunter, in the Phil. Mag. for 
 September 1866, exposed the fallacy of this reasoning by proving 
 that the form in question does not give a maximum, other forms 
 which he pointed out giving larger solids. It resulted incidentally 
 from this discussion that as there are degrees of th magnitudes 
 
96 APPENDIX. 
 
 given by surfaces of different continuous forms, there must be one 
 form which gives the maximum solid, and which ought to be 
 discoverable by calculation. 
 
 After being convinced of my error, I perceived at length what 
 the explanation of the difficulty is which this problem has given 
 rise to. There are, in fact, two kinds of maxima which satisfy it, 
 one discontinuous and the other continuous, and the rules of the 
 Calculus of Variations, on the principle of the comprehensiveness 
 of analysis, necessarily embrace both. The discontinuous solution 
 is sufficiently discussed by the Astronomer Royal in the Phil. Mag. 
 for July 1861. He finds the generating line to be partly recti- 
 linear .and coinciding ultimately with the axis of revolution, and 
 partly a semicircle. This result is deduced from the equation 
 Ap = 0, treated as an ordinary differential equation admitting of 
 being exactly integrated. 
 
 The continuous solution, the reality of which is demonstrated 
 above, is to be deduced from the equation A = 0, which for this 
 instance becomes 
 
 X X?/tf 
 2?/4. zZ = 0, 
 
 VI +P' (!+?) 
 p being put for -~ and q for -y^ , and X being a certain constant. 
 
 This equation does not admit of exact integration, its integral 
 being obtainable only in the form of a series which is not the 
 development of a definite function. It cannot, I think, be ques- 
 tioned that by methods of approximation the form of the line 
 given by that equation might be elicited, and thus the required 
 form of the maximum solid be obtained. But I have not the 
 leisure to pursue this research, my present object having been 
 attained if I have given reasons for concluding that in respect to 
 this problem analysis has not failed, and that it is not true, as 
 some mathematicians have assumed, that Ap = and -4 = are 
 always equivalent equations. 
 
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