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'