GIFT OF Prof. i. P. Lev/is A MANUAL OF PHYSICS BEING AN INTRODUCTION TO THE STUDY OF PHYSICAL SCIENCE. for the ibe af JSttitoeraitj) P E E P A C E . THE best advice which can be given to a student of physics regard- ing the books which he should read is to use separate works written on the various branches of the subject by the leading physicists of the day. Yet this advice has one evident disadvantage. The student who follows it may not get so complete a view of the essential unity and interdependence of the various branches of his subject as it is desirable that he should. And, besides this, there is no doubt that a small volume which gives, as far as is possible, a review of the elements of the whole subject, is a desideratum to the student while in attendance on University classes. I have under- taken the writing of this work in the hope that it may to some extent meet that want. I have throughout endeavoured to bring into prominence the necessity for, and the value of, scientific hypotheses a matter regarding which very hazy notions are only too common. - It has also been my aim to make the treatment of the mathe- matical portions of the text as simple as possible. In this connection I have not adopted the process which has recently been termed ' calculus-dodging,' for the reason that the elementary methods of the calculus are more simple, certainly are more natural, than the methods by which they are usually supplanted. At the same time it may be well to remark that any student, who desires to do so, may simply assume the results of the mathe- matical portions, and use the remainder (which is much the larger part) of the text in his study of experimental physics. In writing a text-book on general physics it is impossible, if justice is to be done to the subject, to avoid borrowing methods from the writings of the masters. In this respect I have to acknow- ledge my indebtedness to the works of v. Helmholtz, Clerk- IV PREFACE. Maxwell, Thomson, Tait, and others. This is perhaps most evident in Chapter XXVI., where I have made use of the very simple form in which Tait has presented the analytical treatment of the theory of Thermodynamics ; and in Chapter XI., where I have adopted his mode of discussing the compressibility and rigidity of solids. This apart, I have endeavoured, whether successfully or not, to present the various subjects in as fresh a manner as I could. My indebtedness to Professor Tait has also been very great in the matter of criticism, which he kindly afforded me on various points while the book was passing through the press. I have also to acknowledge with thanks the kindness of Mr. J. B. Clark, M.A., F.B.S.E., Physical Master in Heriot's Hospital, Edinburgh, in reading the work both in manuscript and in proof. His careful revision has resulted in the elimination of a number of defects which had escaped my notice. WILLIAM PEDDIE EDINBUKGH UNIVERSITY, November, 1891. CONTENTS. CHAPTER I. INTRODUCTORY: THE PHYSICAL UNIVERSE. PAGE Physical science. Test of reality : conservation. Matter : its conservation : its inertia. Energy : its transformation : its conservation : its degradation . . . 1 8 CHAPTER II. THE METHODS OF PHYSICAL SCIENCE. Observation and experiment. Cause and effect. Hypothesis and theory. Crucial test. Argument from analogy Principle of stable equilibrium. Errors of observation. Instrumental errors. Empirical laws. The graphical method ....... 917 CHAPTER III. THE THEORY OF CONTOURS, AND ITS PHYSICAL APPLICATIONS. Dimensions. Contour points of plane and tortuous curves. Examples of their applications in physics. Contour lines. Special physical applications. Use of trilinear co-ordinates . . . . . . 18 32 CHAPTER IV. VARYING QUANTITIES. Dimensions of physical quantities. Determination of rates of variation. The inverse problem . . . 38 47 VI CONTENTS. CHAPTER V. MOTION. PAGE Position. Displacement. Speed and velocity. Acceleration of speed. Curvature. Acceleration of velocity. Accelera- tion in, and perpendicular to, the direction of motion. Average speed and velocity. Resolution and composition of velocities and accelerations. Motion of projectiles. The hodograph. Moments. Acceleration perpendicular to the radius-vector. Simple harmonic motion. Composi- tion of simple harmonic motions. Wave motion along a line. Rotation. Alteration of co-ordinates because of rotation. Uniplanar motion of a rigid body. Motion of a rigid body in space. Composition of angular velocities. Displacement of the parts of a non-rigid body. Strain. - Homogeneous strain. Shear. Non-homogeneous strain. Motion of fluids. Fluid circulation. Vortex motion 48 78 CHAPTER VI. MATTER IN MOTION. Force. The laws of motion. The first and second laws. Examples. Dynamical similarity. The third law. Centre of inertia. Moment of a force and of inertia. Rotational equilibrium. Propagation of motion through a non-rigid solid. Motion of a perfect fluid. Equilibrium of a fluid. Propagation of surface-waves in liquid . 79 100 CHAPTER VII. PROPERTIES OF MATTER. Definitions of matter. States of matter. General properties. Special properties. Specific properties. Molecules and atoms. Molecular forces .... 101 106 CHAPTER VIII. GRAVITATION. Weight and mass. Kepler's laws with Newton's deductions. The law of gravitation. Newton's theorems regarding spherical shells. Mean density of the earth. Hypotheses CONTENTS. Vll PAGE framed to explain gravitation. The nebular hypothesis. Potential. ' Gravitational potential. Equipotential sur- faces. Lines and tubes of force. Special theorems 107 124 CHAPTEE IX. PROPERTIES OF GASES. Compressibility. Boyle's law. Compressibility of a perfect gas. Deviations, from Boyle's law. Compression of vapours. Elasticity. Viscosity. Diffusion. Effusion. Transpiration ..... 325133 CHAPTEE X. PROPERTIES OF LIQUIDS. Compressibility. Elasticity. Viscosity and viscidity. Diffu- sion. Osmose. Dialysis. Cohesion. Capillarity. Surface-tension ..... 134144 CHAPTEE XI. PROPERTIES OF SOLIDS. Compressibility and rigidity. Elasticity. Viscosity. Cohesion Tenacity . . . . . 145 153 CHAPTEE XII. THE CONSTITUTION OF MATTER. The hard atoms of Lucretius. Boscovich's centres of force. Vortex atoms. Heterogeneity of matter. Structure of crystals. Molecular nature of matter. Size of molecules and the range of molecular forces . . . 154 162 CHAPTEE XIII. THE KINETIC THEORY OF MATTER. Kinetic theory of gases. Gaseous pressure. Boyle's law. Avogadro's and Charles' laws. Diffusion. Thermal con- ductivity. Viscosity. Evaporation. Dissociation, etc. Difficulties of the theory. General kinetic theory of matter 163170 Vlll CONTENTS. CHAPTEE XIV. SOUND. PAGE Propagation. Intensity. Eeflection. Kefraction. Diffrac- tion. Interference. Pitch. Musical intervals. Vibra- tions of rods. Vibrations of plates. Vibrations of strings. Vibrations of air-columns. Speed of sound in a gas Partial tones. Resonance. Quality. Beats. Con- sonance and dissonance. Combination tones. Dissonance of pure tones .... . 171 191 CHAPTEE XV. LIGHT : INTENSITY, SPEED, THEORIES. Eectilinear propagation. Intensity. Speed. Theories. Colour ...... 192197 CHAPTEE XVI. LIGHT : REFLECTION, REFRACTION, DISPERSION. Laws of reflection. Eeflection from plane surfaces. Eeflection from curved surfaces. Caustics. Focal lines. Theoretical deductions of the laws of reflection. Laws of refraction. Eefraction through a plane surface. Focal lines. Caustics. Eefraction through parallel layers. Mirage. Prisms. Eefraction through spherical surfaces. Lenses. Oblique refraction. Formation of images by lenses. Dispersion. Aberration. Achromatism. Eainbows. Halos. Theoretical deductions of the laws of refraction. Hamilton's characteristic function . . 198 227 CHAPTEE XVII. RADIATION AND ABSORPTION : SPECTRUM ANALYSIS. ANOMALOUS DISPERSION. FLUORESCENCE. Equality of emissivity and absorptive power. Spectrum analysis. Law of absorption. Body colour. Dichroism. Surface colour. Metallic reflection. Anomalous dis- persion. Fluorescence. Phosphorescence. Theories of dispersion ...... 228240 CONTENTS. IX CHAPTEE XVIII. INTERFERENCE. DIFFRACTION. PAGE Principle of interference. Young's experiment. Fresnel's experiment. Lloyd's experiment. Fresnel's biprism. Coloured interference bands. Displacement of bands by refracting media. Interference bands in spectra. Colours of thin plates. Newton's rings. Colours of mixed plates. Colours of thick plates. Diffraction. Effect of a recti- linear wave. Effect of plane and spherical waves. Diffraction at a straight edge. Diffraction at a narrow slit. Diffraction at a circular aperture. Zone plates. Diffraction at an opaque disc. Coronae. Young's Eriometer. Diffraction gratings . . . 241 261 CHAPTEE XIX. DOUBLE REFRACTION. POLARISATION. Double refraction. Huyghens' construction. Special sections of the surface. Polarisation. Laws of polarisation by reflection and refraction. Direction of vibration in polarised light. Eeflection and refraction of polarised light. Plane, circular, and elliptic polarisation. Nature of common light. Metallic reflection. Double Tefraction in biaxal crystals. Fresnel's theory of double refraction. Conical refraction. Interference of polarised light. Colours of crystalline plates. Haidinger's brushes. Artificial production of the doubly refracting structure. Eotatory polarisation. Polarising prisms . . 265 295 CHAPTEE XX. THE NATURE OF HEAT. Eadiant heat. Its identity with light. Heat in material bodies. Hypothesis of molecular vortices . . 296 300 CHAPTEE XXI. RADIATION AND ABSORPTION OF HEAT. Prevost's theory of exchanges. Stewart's and Kirchhoff's extension of Prevost's theory. Laws of radiation of heat at constant temperature. Heat spectra. Eadiation at different temperatures. Solar radiation. Eadiation from moving bodies ... . 301 309 CONTENTS. CHAPTER XXII,. EFFECTS OF THE ABSORPTION OF HEAT : DILATATION AND ITS PRACTICAL APPLICATIONS. PAGE Temperature. Dilatation of solids. Dilatation of liquids. Dilatation of gases. Absolute zero of temperature. Measurement of temperature . . . 310 323 CHAPTER XXIII. EFFECTS OF THE ABSORPTION OF HEAT : CHANGE OF TEMPERATURE AND CHANGE OF STATE. Unit of heat. Specific heat. Thermal capacity. Specific heat of solids and liquids. Law of Dulong and Petit. Specific heat of gases and vapours. Change of molecular state. Latent heat. Fusion and solidification. Latent heat of fusion. Evaporation and condensation. Latent heat of vaporisation. Formation of dew. Continuity of the liquid and gaseous states. Critical temperature. Solution. Freezing mixtures. Dissociation and chemical combination ...... 324 343 CHAPTER XXIV. CONDUCTION AND CONVECTION OF HEAT. Conduction. Conductivity. Measurement of conductivity. Conduction through the earth's crust. Conduction in crystalline bodies. Conduction in liquids and gases. Convection ...... 344354 CHAPTER XXV. THERMODYNAMICS: HEAT AND WORK. Mechanical equivalent of heat. First law of thermodynamics. Carnot's complete cycle of operations. Carnot's rever- sible cycle. Reversibility the test of perfection". Second law of thermodynamics. Absolute temperature. The indicator diagram and its applications. Entropy. Total, available, and dissipated energy. Thermodynamic motivity ...... 355369 CONTENTS. CHAPTER XXVI. PAGE THERMODYNAMICAL RELATIONS 370 376 CHAPTER XXVII. ELECTROSTATICS. Electrification by friction. Conductors and non-conductors.^- Fundamental phenomena presented by electrified bodies. Positive and negative electricity. The gold-leaf electro- scope. Electrification by contact and by induction. Electric quantity. Continued production of electricity. The electrophorus. Law of electric attraction and repul- sion. Electric potential. Electromotive force. Capacity. - Condensers. Specific inductive capacity. Distribution of electricity on conductors. Electric density. Electric images. Lines of electric force. Electric induction. Tubes of induction. Electric energy. Electric absorption. Disruptive discharge. Atmospheric electricity. Pyro- electricity. Electrification by contact. The electrometer. Electric machines ..... 377 412 CHAPTER XXVIII. THERMO-ELECTRICITY. Thermo-electric phenomena. Laws of thermo-electric circuits. Variation of the electromotive force with temperature. The thermo-electric diagram. The Peltier effect. The Thomson effect. Further discussion of the thermo- electric diagram ..... 413 423 CHAPTER XXIX. ELECTRIC CURRENTS. Convection current between charged conductors. Flow of elec- tricity in metallic conductors. Ohm's law. Kirchhoff's laws. Electrolytic conduction. Faraday's laws. Polari- sation. Ohm's law in electrolytes. Production of electric currents. Primary cells. Secondary cells. Transforma- tions of electric energy in conducting circuits. Joule's law. Measurement of electromotive force, current strength, and resistance .... 424 440 Xil CONTENTS. CHAPTEE XXX. MAGNETISM. PAGE Fundamental phenomena. North and south magnetism. Paramagnetic and diamagnetic bodies. Magnetism a molecular phenomenon. The law of magnetic attraction and repulsion. Poles, axis, and magnetic moment of a magnet. Lines of magnetic force. Magnetic potential. Magnetic intensity. Magnetic induction. Permeability and susceptibility. Residual magnetism. Reteiitiveness. Coercive force. Eelation connecting magnetisation and magnetising force. Hysteresis. Effects of vibration and of temperature. Effects of stress. Magnetometric measurements. Terrestrial magnetism. Theories of magnetism . . . . . . 441 462 CHAPTER XXXI. ELECTROMAGNETISM, ETC. Oersted's discovery. Magnetic action of closed electric circuits. Electrodynamic action on an electric circuit. Linear circuits. Circular circuits. Solenoids. Ampere's hypo- thesis regarding magnetism. Continuous rotation under electromagnetic force. Electric motors. Electromagnetic induction. The dynamo. Electrokinetic energy. The galvanometer. The ballistic method. Electric and mag- netic units ...... 463480 CHAPTER XXXII. ELECTROMAGNETIC THEORY OF LIGHT. Magnetic rotation of the plane of polarisation of light. Hypothesis of molecular vortices. Hall's effects. Kerr's effects. Electromagnetic theory of light. Electromagnetic waves. ... . 481489 CHAPTER XXXIII. THE ETHER. Direct action at a distance. Action through a medium. Rigidity and density of the ether. Elastic-solid theory. Contractile ether. Electromagnetic ether. Dilatancy 421 496 INDEX 497501 E E E A T A. Page 17, line 11, for ' wrong ' redd ' discordant.' ,, 26, 9, for ' the closed curve at ' read ' a closed curve surrounding.' ,, 28, ,, 8 from foot, for ' here ' read l there.' ,, 49, ,, 1, for * relative position of P ' read ' position of P relative.' 149, Equation (2),/or ~0 read - 173, ,, (4), for * pd ' read * rf/o.' 183, line 12, for ' 63 ' read.' 64.' 198, ,, 13, /or ' refraction ' read ' reflection.' 207, I, for 'p> read 'If p.' 248, 5, for ' Fox,' read ' Fox-'. 257, 3, for * e. 2 , ? 2 ' read * and the unit of time inversely ; an acceleration involves a length directly, and the square of a time inversely. But, when dealing with extension, we have only to consider the unit of length. We say that the extension under consideration has one, two, or three, etc., dimensions, according as the unit of length is involved to the first, second, or third, etc., power. A line has only one dimen- sion, for only one number, with the proper sign attached, is required to completely specify the relative position of two points on the line. A surface has two dimensions, for two directed lengths define the position of a point on it with reference to any other point taken as origin. Thus we speak of one point on the surface of the earth as being so much north or south, and so much east or west of another. Three directed lengths determine the relative position of two points in space, that is, in extension of the third order. Thus we speak of the length, breadth, and thickness of a solid. It must be observed that three lengths are not necessary. One of the given conditions must be a length, but the others might be angles. For example, instead of saying that one mountain-top is so far north or south, so far east or west, and so much higher or lower, than another, we might give the distance between the two peaks and their relative altitude and azimuth. The intersection of any surface, which has a constant charac- teristic, with the surface of a solid is called a contour-line. A good illustration is furnished by the contour-lines in an Ordnance map. Any such line is the intersection of a surface, all points of which are at a constant height above sea -level, with the surface of the solid earth. An Ordnance map gives a very good idea of the dis- tribution of places on the earth's surface as regards height, as well THE THEORY OF CONTOURS, AND ITS PHYSICAL APPLICATIONS. 19 as with reference to latitude and longitude, and the information is more and more minute as the number of lines is greater. In other words, contours enable us to represent on a plane surface the mutual relations of three quantities. This gives the key to the great importance of the theory of con- tours in physical science. For, if the physical condition of a substance is completely denned when the simultaneous values of three of its properties are given, we can construct a solid the surface of which represents all possible conditions of the substance just as we can construct a model of the earth's surface in terms of latitude, longitude, and height above sea-level. 19. We may extend the above conception, and state, generally, that the contour of an object o/n dimensions, existing in extension of the (n+l) th order, is its intersection with an object of n dimen- sions at every point of which some quantity has a constant value. It is, therefore, of (n 1) dimensions. Since, in extension of the (n+l) th order, we may have objects of less dimensions than the n th , we might make the further develop- ment that in extension of the (n+l) th order we may have contours of all positive dimensions up to the (n - l) th inclusive. Indeed, there is no reason why we should not consider -contours of n dimen- sions in space of (n + 1) dimensions. Hence, in ordinary extension, we may have point, curve, and surface contours. The contours of a curve are points ; of a surface, curves ; of a solid, surfaces ; of a four-dimensional object, solids ; and so on. The properties of four- dimensional extension, or even of exten- sion of the n ih order, can be treated mathematically ; but, from want of experience, it is impossible to imagine the nature of such extension. 20. By means of contour-points the nature of curves may be exhibited in diagrams consisting of straight lines only. For we may intersect a given curve by curves, along each of which some quantity has a constant value, and then project the points of inter- section upon any straight line. Consider first a plane curve, and, for convenience, let its plane be taken as that in which two co-ordinate quantities, x and y, are measured in perpendicular directions from the same origin. Let fn ( x i y) foe the equation of the given curve, where the suffix denotes the degree of the equation. The equations to the curves along which some quantity, say c, is constant, may be written in the general form, n (x, y, c) = o. In the different curves of the system c has different values. As a particular example, the curves might be circles of different radii. Again, the equation might be 22 20 A MANUAL OF PHYSICS., c ) = o, which represents a series of lines parallel to the axis of sc. This is the simplest case which we can consider, and, at the game time, the most useful. The curves in Figs. 2 and 3 are inter- sected by lines parallel to the axis of x, and the points of inter- section are projected upon the axis, and are designated by numbers which give the various values of y corresponding to the given values of x. If the curve be continuous, a maximum or minimum value of y exists between two equal values. It is a maximum if y first increases and then diminishes as x increases continuously from its least value corresponding to the given value of y. It is a minimum if y first diminishes and then increases. The steepness of slope is shown by the closeness of the contours for equal increments of y, and its .direction is shown by the order in which the values of y occur as regards numerical magnitude when x increases. FIG. 2. FIG. 3. In the case of tortuous curves we may obtain the contours most conveniently by cutting the curve by surfaces over which some quantity is constant. In particular, these surfaces may be planes perpendicular to the z axis, in which case the equations are of the form/j (z, c) = o. The position of a moving point in space is obviously represent- able by a tortuous curve. Its position at any time can be got from the curve if the value of the time in terms of one of the co-ordi- nates, say z, is known ; for we should then only have to cut the curve by the plane \f n (z, t) = o, where t represents the time, and if n is a functional symbol, showing that the equation is of the first degree in z, but may be of any degree in t. [This condition is rendered necessary by the fact that the point must be in one definite THE THEORY OF CONTOURS, AND ITS PHYSICAL APPLICATIONS. 21 position at a given time, but may occupy the same position at different times.] If a number of such curves are simultaneously traced out in space by moving material points, we can obtain the diagram of configuration of the material system at any time by cutting the curves by planes corresponding to that time, and pro- jecting the points of intersection upon the parallel co-ordinate plane. It is evident that, in general, the plane which corresponds to a definite time will be different for each curve. The disadvantage sd entailed may be got rid of by the employment of trilinear co-ordi- nates to indicate the position of the point when the time is given; If the curve be cut by any plane, the distances of the point of intersection from three intersecting straight lines in that plane give the a?, y, z co-ordinates at the corresponding instant. The value of the time might be given by the distance of the plane from a fixed plane parallel to it. The curves which are, on this system, taken to represent the ; FIG. 4. positions of the points are not in general the actual curves traced out by the points in their motion through space. But the diagram of configuration obtained from them has the advantage of showing af once the values of all the co-ordinates of all the points, whereas, in the Cartesian system, this could not be done without projection on all the co-ordinate planes. The triangles of reference, though they may be similar, are not generally of the same magnitude. A; difference in magnitude is necessary, in order to represent varying values of the co-ordinates. In Fig. 4 the triangles are similar; and equidistant, while the point a is fixed ; hence the diagram re- presents the linear motion of a point. In general there must be a different set of triangles for each separate point whose motion is to be indicated. By the aid of such a diagram, the diagram of total displace- ments ( 40) in a given time may be constructed. And by taking the displacements in one n th part of the unit of time (n being 22 A MANUAL OF PHYSICS. indefinitely large), and magnifying them n times, the diagram of velocities can be got. Similarly, the diagrams of accelerations, forces, and so on, may be represented as the contours of curves. The curves, from which the diagrams of velocities, etc., are obtained, are, it is almost needless to remark, different from the original curves which represent the positions of the moving points. In the case of velocities, they are the hodographs ( 48) of the original curves on this trilinear system of reference. As another example of the use of tortuous curves, we may con- sider two quantities, x and y, connected by the equation if = ax, which gives ydy/dx = aj^ ( 30). We may now take dyjdx as a third co-ordinate quantity, and so obtain a tortuous curve. For example, if, in this case, y represents the time during which a body FIG. b. has been falling from rest under gravity, and if x represents the space described from rest, the velocity acquired is represented by the reciprocal of the third co-ordinate quantity. 21. If any curve be cut by planes parallel to that of (x, y), and if the various points of intersection be projected on any one of these planes, say z = o, the contour-points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break of con- tinuity by projecting every point of the curve upon the plane z o. But such a line may be regarded as the intersection, by the plane z = o, of a cylindrical surface whose generating lines are parallel to the -axis and are drawn from the given curve to meet that plane. Now this satisfies our definition of a contour-line, for it is the THE THEORY OF CONTOURS, AND ITS PHYSICAL APPLICATIONS. 23 intersection of a given surface by a surface over which z is constant (zero). A cylindrical surface supplies the simplest diagram of con- tour-lines. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is (Fig. 5) that in which the same values of the x and y co-ordinates of a point on a curve correspond to different values of z. In the case of a non- cylindrical surface, no part of the contours will be superposable in general. The contours of a hemisphere, for example, are concentric circles (Fig. 6). And, just as, in the case of contour-points, the steepness of slope of the curve is indicated by the closeness of the contour-points on the #-axis for equal incre- ments of ?/, so, in the case of contour-lines, the steepness of slope of the surface is indicated by th6 closeness of the contour -lines for FIG. 7. FIG. 8. equal increments of z. The contours are closer when their radii are large. The contours of a right circular cone are also concentric circles, but they are at equal distances apart for equal increments of . 22. As an example of a surface, the contours of which may be used to indicate certain physical properties, we may consider that one whose equation is ft****!/* If y represents the length of a simple pendulum, while x represents the square of its time of oscillation, we know that z represents the value of the acceleration due to gravity. The surface (Fig. 7) may obviously be supposed to be produced by the motion of a straight line which intersects the 2-axis and is always perpendicular to it, . 24 A MANUAL OF PHYSICS. and which rotates uniformly about that axis while it moves at a constant rate along it. The contours (by planes perpendicular to the axis of &) are straight lines passing through the origin and variously inclined to the a?-axis ( 17). Again, the intrinsic equation of the circle is S = &0, where a is the radius and $ is the angle between the radius vector -and the initial line. Hence the intrinsic equation of one involute is This involute is the one which meets the circle at the position from which is reckoned, and s' is measured along it from this point. If we consider a to be the mass of a moving body, and to be its speed, s and s' are respectively its momentum and kinetic energy. In Fig. S two circles, with their involutes satisfying the above condition, are drawn. The curves may be regarded as the contours of a right circular cone, together with an associated surface which is formed so that its intersection by any plane parallel to that of the diagram is an involute of the circle in which the cone is cut by the same plane. The acceleration and speed of a body falling under the action of gravity, and the space passed over by it, are given by the known equations ( 42) a = g v = V+gt s = c + Vt + $fft*. Hence these various quantities can be represented also by the contours of the surfaces just considered, 'taking the place of 0, an# g taking that of a in the former equations. The new terms which .appear present no difficulty. Again, in a thermo-electric circuit (Chap. XXVIII.), composed of two dissimilar metals, the electromotive force E is given, in terms of t the difference of temperature of the two junctions, by means of the formula Also the thermo-electric power e is given by the equation Hence these quantities are also representable by means of the same surfaces. 23. The contour lines which are most familiar to us are those THE THEORY OP CONTOURS, AND ITS PHYSICAL APPLICATIONS. 25 formed by the intersection of level surfaces with the surface of the earth. The line of sea-board is one such contour line. The numbers marked upon maps or charts which give the height above, or depth below, sea-level indicate contour-points. When the points are taken sufficiently close together, and continuous curves are drawn through points of constant height, we get contours, as in the Ordnance Survey maps. Such a contour coincides very closely with the contour-lines formed by level surfaces. They do not exactly coincide because of the non-spherical shape of the earth, and because of its rotation, etc. But the assumption that contour-lines of constant level are lines of constant height over sea-level will not introduce appreciable error, so long as the area on which they are drawn is FIG. 9. small in comparison with the whole surface of the earth. The kinetic energy, which is acquired by a body in falling freely from any point on one level line to any point on another level line is constant. Suppose the earth to be entirely submerged underneath the surface of water, so that we have only one region, and that a region of depression below the surface of the water, to consider. If we suppose further that the water is slowly absorbed by the solid matter of the earth, regions of elevation will be formed gradually, until finally we shall have again only one region, and that a region of elevation, Before a region of elevation is formed, we have a summit appearing above the water-level; and when the water subsides out of a region of depression, we have a lowest-point or imii appearing. 26 A MANUAL OF PHYSICS. The number of regions of elevation and depression may vary in two ways. Two regions of elevation may run into each other as the water sinks. The point where they first meet is termed a jimss or col (see Fig. 9 ; Pj, P 2 , etc.). Again, a region of elevation may throw out arms which run into each other, and so cut off a region of depression. The point where they first meet is termed a bar (B lt jB 2 , etc.). The contour-line for a level immediately underneath that corresponding to the bar has a closed branch within the region of depression cut off. Thus the closed curve at I 4 is part of the contour-line UV. In the map of such a country, a pass occurs at the node of a figure-of-eight curve (or out-loop curve, as Professor Cayley has termed it), while a bar occurs at the node of an in-loop curve. If, in the diagram, the Ps represented bars and the Bs represented passes, the map would be that of an inland basin ; so that, in the map of such a country, a pass is repre- sented by the node of an in-loop curve, and a bar corresponds to the node of an out-loop curve. If there were any advantage in having passes and bars always indicated by the node of the same kind of curve respectively, this could be attained by affixing the positive sign not constantly to the region on the same side of the level surface but to the region towards which (or from which) the surface is moving at any instant. As a particular case, two regions of elevation may run into each other at a number of points simultaneously. Of these points, one must be taken as a pass and the others as bars. Singular points may also occur, when, for example, three or more regions of eleva- tion meet. Such points are called double, treble, etc., passes. Multiple bars may similarly occur. Before a pass can be formed there must be two summits, and for every additional pass there is another summit. Thus the number of summits is one more than the number of passes. So also the number of imits is one more than the number of bars. Slope-lines are lines drawn at right angles to the contour-lines ; and, evidently, the steepness of a district is indicated 011 a map by the closeness of the contours. Two kinds of slope-lines are of special importance. These are the slope-lines drawn from summits to passes or bars, and from passes or bars to imits. The former can never reach an imit, and are termed water-slieds. The latter can never reach a summit, and are called water -courses. A perpendicular precipice is indicated on a chart by the running together of two or more adjacent contour-lines (F). An over- hanging precipice is indicated by the lapping of the upper-level line over a lower-level line. THE THEORY OF CONTOURS, AND ITS PHYSICAL APPLICATIONS. 27 24. Since we can represent the physical state of a substance with regard to three quantities by means of a surface, it follows that we can deduce from the contours of the surface the nature of the varia- tion of the properties of the substance the methods being identical with those of the preceding paragraph. Let us take, as a particular example, the thermo- dynamic surface which represents the state of water-substance with regard to volume, pressure, temperature, entropy, and energy (Chap. XXV.). If we construct the surface in terms of any three of these quantities, the value of the remain- ing two at any point of the surface may be given by contour-lines. The model of the surface, having volume, entropy, and energy measured along the axes, has been constructed by Clerk Maxwell, and is explained and figured in his ' Theory of Heat.' We shall FIG. 10. consider the surface representing directly volume, temperature, and pressure. This surface was first studied by Professor James Thomson. Suppose the surface to be cut by a plane of constant pressure, say P lf We thus get a contour-line, the general nature of which is indicated in Fig. 10. At a low temperature the volume is small, the sub- stance being in the solid state. As the temperature rises, the substance expands until liquefaction occurs. The volume then diminishes without rise of temperature, until the substance is completely liquefied. The temperature then rises while the volume diminishes, until the maximum-density noint is reached. After this, expansion accompanies rise of temperature up to the boiling-point. At this stage the volume rapidly increases, while the temperature remains steady until the substance is entirely in the gaseous state. Beyond this point both increase together. The contours for slightly less pressures (P 2 , P 3 ) are approximately parallel to P 15 but lie entirely A MANtlAt OF PHYSICS. above it ; the reason being that at a given temperature the volume increases as the pressure diminishes, while the freezing-point is lowered, and the boiling-point is raised, by pressure. The freezing and boiling points approach, and finally coincide as the pressure diminishes. At lower pressures the substance changes directly froni the solid into the gaseous state. The line AB in the figure above indicates the triple-point temperature, that is, the temperature at which portions of the substance in the three states solid, liquid, and gaseous can exist together in equilibrium. The increase of volume on vaporisation continually diminishes as the pressure is raised, until finally (at .C) the process of vaporisation ceases. The temperature at which this occurs is called the critical temperature'. There may also be a critical temperature for the solid-liquid con- dition. . That is to say, there may be a temperature belo^v which no amount of pressure will lower the freezing-point sufficiently to admit of liquefaction. The contour-lines which are obtained by cutting the surface by planes of constant temperature are called isothermals. If the tem- perature be above the triple-point, but below the critical-point, while the substance is in the gaseous condition, increase of pressure is accompanied by decrease of volume, until the liquefaction com- mences. At this stage the volume decreases without variation of pressure, until all the substance is liquefied. After this, a very great increase of pressure is required to produce a very small decrease of volume. Two such isothermals (not those of water- substance, see 278) are represented in Fig. 11. The form of an isothermal below the triple -point shows that the solid state is intermediate between the gaseous and the liquid states. As the pressure increases the volume decreases, until the point of sublimation is reached. The pressure then remains constant while the volume diminishes, until all the substance is solidified, Then the volume decreases slowly on increase of pressure, until liquefaction commences. Here the pressure becomes constant, while the volume diminishes until all the ice is melted ; after which the volume again decreases slowly on rise of pressure. Thus, here are two kinds of isothermals having their transition- stage at the triple-point temperature. As already indicated, the triple-point pressure occurs at the transition between two kinds of lines of equal pressure, for the liquid condition ceases to be possible at higher pressures. The form of the isothermals beyond the critical temperature is indicated in Fig. 11. It is quite possible that, as Professor James Thomson has suggested, the true form of the isothermals below the critical temperature does not include a THE THEORY OF CONTOURS, AND ITS PHYSICAL APPLICATIONS. 29 FIG. 11. 30 A MANUAL OF PHYSICS. part parallel to the volume-axis, but that it has a waved form, as shown in the diagram. Part of the waved portion represents an unstable state, since pressure and volume increase together. Thom- son made this suggestion in order to avoid discontinuity in the curve. The lines e lt e% are lines of constant energy, and those marked 0i 02 03 are lines of constant entropy. 25. It is only when the temperature considered happens to be one corresponding to a contour in the diagram that the relation of pressure and volume can be accurately found from the above diagram. This defect may be got rid of by the use of trilinear co-ordinates; and, in addition, the variation of a fourth quantity can be shown. In illustration of this we may take the case of a perfect gas, for which we have the equation vv = lit, where p, v, and t represent respectively the pressure, volume, and FIG. 12. temperature, and R is a quantity which depends on the nature of the gas. The triangle of reference is made equilateral in Fig. 12. The ratios of the distances of a point from the vertical and the inclined sides of the triangle are the ratios of the temperature, pressure, and volume respectively. The contours for different values of R are shown in the figure, and the equation shows that they are hyperbolas, with vertical and horizontal axes. No part of the hyperbolas outside the triangle of reference has any physical mean- ing; for, in that case, pressure, volume, or temperature (or any two, or all, of them) would be negative, which is impossible in the case of a gas. Evidently pressure, volume, and temperature are continuously represented for any one gas. THE THEORY OF CONTOURS, AND ITS PHYSICAL APPLICATIONS. 81 Of course, when we wish to find the absolute values of the co- ordinate quantities, we must use the equation which connects them. This is not necessary when we use Cartesian co-ordinates. The figure obviously represents the contours of a surface by planes parallel to the plane of the diagram, which may be looked upon as that corresponding to zero value of R. When li is zero, the hyperbola becomes a pair of straight lines which coincide with the sides AB, AC, of the triangle of reference. When E is infinite, the side BC is part of the corresponding hyperbola. All parts of lines through B and C perpendicular to the plane of the diagram lie upon the surface. These lines separate the parts of the surface which correspond to real physical states from those which do not. Outside the triangle the surface evidently overhangs the plane of the paper. The value of R being given, let P (Fig. 18) be the point which gives the proper ratios of p, v, and t. Draw PM , PN, parallel to the sides of the triangle. Since the asymptotes of the hyperbola are parallel to these sides, it follows that that part of the tangent at P, which is intercepted by the sides of the triangle, is bisected at the point of contact. Therefore AM = MQ, and AN = NR. Now the compres- sibility k of a gas is given by the ratio dv/vdp, where dv is the small alteration of volume produced by the small change of pressure dp. Vutdvldp = MQIMP = NPIMP = vlp. Hence k = 1/p, that is, the compressibility of a perfect gas is the reciprocal of the pressure. Similarly it can be shown that the expansibility is proportional to the absolute temperature. The work done during isothermal expansion can also be found from the diagram. The position of the point P gives the mutual 32 A MANUAL OF PHYSICS. ratios of p, v, and t ; but since t has a known constant value, the actual values of p and v are also known. Hence PN ( =p cosec BAG) is a known function of v. If P move to P', the area PNN'P' =/PNdv ( 34) = cosec. BACfpdv is a known multiple of the work done. ; 26. The applicability of, the method of contours to other physical problems is evident. Electric stream-lines and equipotential-lines may be regarded as contours of a surface, and the number of equi- potential lines which cross unit length of a stream-line may be used to indicate the strength of the current. So also air- current lines and isobars, isothermals and flow-lines of heat, etc., are rectangular systems of contours. CHAPTEE IV. VARYING QUANTITIES. 27. THOUGH every quantity, whatever be its nature, has magnitude, no quantity can be said to be large or small absolutely. When we speak of the size of any body we mean its size relatively to the size of some other body with which we compare it. A yard is large if we compare it with an inch ; it is small when compared with a mile. In the former case the number which represents it is more than 60,000 times larger than the number by which it is represented in the latter case. A mere number is therefore useless as regards the statement of magnitude, except when accompanied by a clear indication of what the thing measured is compared with. The quantity in terms of which the comparison is made is called the unit, and the number which tells how often this unit is contained in a given quantity is called the numeric. All dynamical quantities may be made to depend upon three units only. These are the units of mass (quantity of matter), length, and time. Thus speed, being measured by the distance traversed in a certain time, depends upon the unit of length directly, and upon the unit of time inversely. Hence by doubling the unit of length we double the speed unit, and therefore halve the numeric of any given speed ; whereas by doubling the unit of time we halve the speed unit, and therefore double the numeric of a given speed. Again, acceleration, being measured by the increase of speed in a certain time, depends upon the unit of speed directly and upon the unit of time inversely ; that is, it depends directly upon the unit of length and inversely upon the square of the unit of time. The manner in which the fundamental units are involved in any quantity determines the dimensions of that quantity. If M L and T represent the units of mass, length, and time, the dimensions of speed and acceleration are indicated by the symbols [LT" 1 ] and [L2 7 - 2 ] respectively, and the dimensions of energy ( 7) by [ML 2 T- 2 ]. 3 34 A MANUAL OF PHYSICS'. 28, When two quantities are so related that any change in the numerical value of one of them is accompanied by a change in the numerical value of the other, each quantity is called a function of the other. If to one value of the first there corresponds one, and only one, value of the second, the second is called a single-valued function of the first ; but if to a given value of the one there correspond more than one value of the other, the latter is said to be a multiple-valued function of the former. For example, a; 2 4-2a?-8 is a single-valued function of x; since, if we give x any value, there will be one corresponding value of a; 2 -f 2# 8. On the other hand, a? is a double-valued function of # 2 +2a? 8; since, if we give any value to # 2 -f 2# 8, we find that x has in general two distinct values. Again sin a? is a single-valued function of x, while sin" 1 ^ (the angle whose sine is x} is a multiple -valued function of #, The relation between the two quantities can be expressed by means of an analytical equation or by means of a curve, as is indi- cated in 17. The general expression of the relation may be given in the form =/(). where the quantity y is regarded as being dependent upon x, and the equation simply reads ' y is some function of #.' The function denoted by / may be of various kinds it may be algebraical (e.g., y = ax + bx 2 ), trigonometrical (e.g., y = sin a?-fcos x), ex- ponential (e.g., y = a*), etc. ; and under each kind there may be a number of different forms, e.g., a 'series of powers,' a 'product of tangents,' etc. In the above formula x is supposed to be that quantity the value of which is arbitrarily varied. It is therefore called the independent variable, while y is termed the dependent variable. 29. If y varies in value uniformly when x varies uniformly, the quotient of any increment of y by the simultaneous increment of x is constant, however large or small either of the increments may be. For the condition means that y is proportional to x, or to x + a constant, say y=~kx-\- c where k and c are constants. Hence if y changes from y 1 to y 2 when x changes in value from x l to x. 2 we have y l = kx i -}-c, y 2 =kx 2 +c, and therefore y 2 -yi = k (x. 2 x^, which proves the above statement, since a? 2 x\ may have any value we please. If the values of y and x be represented as the ordinates and abscissae respectively of a curve, we see that the constant c is represented by the length OA (Fig. 14) (since it is the value of y when x = o), that simultaneous values of x and y are represented by points on the line AB, and that k is represented by the tangent of the in- clination of that line to the #-axis ; hence, by this graphical process VARYING QUANTITIES. 3,5 also, the truth of the statement is evident. When they are thus related, y and x are said to be linear functions of each other. But when y is not a linear function of x, it is evident that the ratio of their simultaneous increments depends upon the absolute values of these increments. In the former case that ratio denoted the rate at which y varied when x varied ; but, in the case now supposed, it does not in general give the true value of that rate. It is of extreme importance that we should have a means of finding the true rate of variation whatever be the nature of the relation connecting the two quantities. Let A'B' (Fig. 14) be the curve which represents y as a function of x, and let P be the point at which we wish to find the rate of variation of y when x alters. Take another point P' on the curve, and let x' and y' be the values of its co-ordinates. It is obvious that in general the ratio of y' y to x' x does not give the true rate. It gives instead the rate of FIG. 14. variation corresponding to the line joining P and P'. But the line PP' coincides more and more nearly with the curve as P' approaches to P; and we can take P / closer to P than any assigned finite quantity however small, so that the difference in direction between the line and the curve can be made smaller than any assignable angle. Ultimately when P' is infinitely near to P the rates of variation corresponding to the curve and the line are identical. Hence, geometrically, the true rate at any point is given by the tangent of the angle which the line touching the curve at that point makes with the #-axis ; or, analytically, it is given by the ratio f y' ~ y to x' x when x' - x is made indefinitely small. This method is only applicable strictly to the case of quantities which have no sudden change in their rate of variation. But, in 32 86 A MANUAL OF PHYSUS. any such case, we can apply the method up to the point at which the sudden change occurs, and also, separately, beyond this point : and this is all that is required. It is usual to write dy and dx instead of y' y and x' x when these quantities are infinitely small, so that the symbol d means the infinitely small increment (or * differential,' as it is usually termed) of the quantity to which it is prefixed. In general. *-x*) _ ^a(x>+x)(x'-x) = ^ ^ '= ' (8) y = ax, f-x .'-j.) (x' H - 1 +a m ~ 9 g+ . . . X -X ** X VARYING QUANTITIES. 37 (4.) y ax n . This gives //" r= fall Jj" 1 . d(y n ) _ dx dx <%*) dy _ dy dx dx ny*- l ?y=a*mx-i .......... ty (3) x m-l m x m ~ 1 in ,_,_,.* m - i = a -, = ax m n =ax* , n m zl n n We see from (3) and (4) that when y is given as a positive power of x (whole or fractional), its #-rate of variation is got by the following rule : Multiply by the index of x and lower the index by unity. Example (7) will show that the rule is not restricted to positive powers of x, but holds also for negative powers. (5.) y = ax n -{-bx m djtj r which gives the rule for the rate of variation of a quotient of two functions. If u = xn, v = x m , we get dy _x n dx x* Again, if u is constant, we have dx -y 2 dx' Thus, if 2/ = aar, a takes the place of u, and xP replaces v. There- fore dx x^ n dx x^n VARYING QUANTITIES. 39 which proves the applicability, to the case of negative indices, of the rule given under example (4). In many cases y will be given, not in terms of x as hitherto assumed, but in terms of some function of x, say u. The ordinary principles of algebra then show (see the remark at the end of last section) that dy _ dy du dx du dx' For example, y = ( dy _ d(ur] _ ckt* du dx dx du dx te-iffc-^-^+^-i,-!^ dx dx dx 31. We shall now consider such rates of variation of trigono- metrical functions as may be of use later. (1) 2/ = sina?. di/T sin a;' -sin a; T sin (f ~ I) cos (f + I ) dx LJ x' x JU x' x x'=x x'=x But, when x' = x, sin (x f - x) = x' - x, and therefore the rate of variation is cos x. It follows at once from this result, by writing # + o instead of #, that if (2) 2/ = cos x ^=-sin*. dx Again, suppose (3) 7/ = sec x. Let cos x = u, and we get ^ dx du dx dx ' dx = sec 2 x . sin # = sec x tan x. From this result we readily obtain, when (4) ?/ = cosec x, -^-= -cosec x cot x. dx 40 A MANUAL OF PHYSICS. By means of the result of example (7) in 30 we can find the derived function of sin x (5) y = tan x That result gives cos x ax dx COS' X _cos 2 x-\-sm 2 x_ COS 2 X COS* X and, as formerly, by the substitution of x-\-- for x, we deduce, from (6) y = cot x, _=-cosec 2 z. 32. It will be instructive to deduce the results, which we have just obtained, by a geometrical process. Take two near points, P and P', on the circumference of the circle APB. Denote POA by 9 and POP' by d9, and suppose that the radius of the circle is FIG. 15. unity. Let OM = x and MP = ?/; then NP' = cfo/ and -NP = d.c. The angle d9 is assumed to be infinitesimally small (so that the arc PP' is practically a straight line), and it is measured by the ratio of PP' to OP; that is, by PP', since OP is unity. Also NP'P = 0. Hence NP'/PP' = cos 9 = dy[dO, and - NP/PP' = - sin 9 = dx/dO. But x = cos 0, y = sin 9 ; and therefore Also and therefore ^H^ = cos 0, ^^ = d9 d9 sec 9 = !/, - sin 9. d sec 9 = -dx x + dx x x(x -f dx) dx X* VARYING QUANTITIES. 41 ultimately. But -^ = 1 ( -^}d6 = i sin0 dO = sec 8 sinO d9 = seceta,nOdO. ar x 2 \ dO/ x~ Similarly, d tan 9 = y^ -U + dx x x(x-\- dx) cos . cos 9 d 9 - sin 9 ( - sin 0) d 9 ^ = 2-0 s ^(l+tan 2 ^) d0 = sec 2 d9. COS v 33. Lastly we may obtain the rate of variation of the exponential and logarithm of a varying quantity. (1) y = ex dj/ = T &f-e? = T ^a?~ LJ x'-x JLj x'-x x'=x But e, which is the base of the Napierian Logarithms, is by defini- tion the limiting value of (1 + u) u when u is a vanishingly small quantity ; so that we may write 1 X'=X Therefore the limiting value of e x '~ x is x'-x J fe a; a; and therefore By means of this result we can deduce the derived function of (2) y = log x. For this gives x e y , and therefore dx/dy = ey, so that dx Inverse Problem. 34. In the immediately preceding sections we have dealt with the problem to find the rate of variation of a quantity which is a given function of some independent variable. The inverse pro- blem to find the value of a quantity, the rate of variation of which is given is of quite as great importance in physical inquiries ; and, in order to solve it, we have merely to reverse the former process. 42 A MANUAL OF PHYSICS. In illustration we may consider the simplest possible case in which we have y = x. Here dyjdx is constant and numerically equal to unity ( 30, example (1) ). The relation y^x is indicated in Fig. 16 by the line bisecting the angle xoy, and the values of dyjdx are given by points on the horizontal straight line passing through the point y = l. If we choose any point P on the line y = x, and draw through it a line parallel to the 2/-axis cutting the line indicating the value of dyjdx in P', it is evident from the figure that the area of the FIG. 17. rectangle OP' is numerically equal to the value of y at P. For the ordinate of P is four units in length, and OP' contains four square units. Now take any curve (the particular curve shown in Fig. 17 is the VARYING QUANTITIES. 43 quadrant of a circle) representing graphically the relation ?/=/(#), and draw another curve representing the relation y'=f'(x). This may be called the ' derived ' curve of the former, since its ordinates give the values of dy/dx at the corresponding points, i.e., they give the values of the derived function. Take three near points, P^P^P.^ on the derived curve. The area included between the curve and the axes is greater than the sum of such areas as P 2 #i, but is less than the sum of such areas as PiX 2 . These sums differ by the sum of the rectangles PiP 2 , etc. But, as # 2 x^ etc., are made less and less and ultimately vanish, these rectangles become smaller and smaller and finally vanish. Of course any area PiX 2 becomes infinitesimally small as a? 2 approximates to coincidence with x ; but PiP 2 becomes infinitely small in comparison with P^ infinitely small though it be. The magnitude of each little area similar to Pj# 2 is y'dx, where dx represents as before an infinitesimally small increment of x. But y' dy/dx, and therefore y'dx = dy. So that, finally, the limiting value of the sum of the rectangles (the number of which increases indefinitely as their magnitude dimi- nishes without limit) up to a given value of x, is numerically equal to the value of y, at that given value of a?, in the original curve. This limit to which the sum approaches is indicated by the symbol fy'dx, and the quantity itself is called the integral of y' with respect to x. That is to say, while y' is called the ' derived function ' of y with respect to x, y is said to be the ' integral ' of y' with respect to the same variable quantity. If it were necessary to preserve similarity of nomenclature, we might term y the ' primitive function ' of y'. Analytically, the operation indicated by f is an operation which undoes the effect of the operation indicated by the symbol d. For dx The symbol d signifies an infinitesimal difference ; the symbol / signifies the sum of an infinite number of infinitesimally small differences. Indeed the symbol of integration is merely an exag- gerated form of the letter S, denoting a sum. In order to find the integral of y' we have to answer the question, What function has y' as its derived function ? A considerable know- ledge of derived functions is therefore essential. 35. We shall now consider some useful examples, and shall take them in an order similar to that which was observed when finding the derived functions. 44 A MANUAL OF PHYSICS. (1) Evaluate y=fadx. We have merely to write down that function the derived function of which with respect to x is a. In the direct process we had to lower the power of x by unity and multiply by the undiminished, power as a factor. Therefore, in the inverse process, we must raise the power by unity and divide by the increased power. But, in the above example, we may assume x as a factor since its value is unity ; so that ax is the quantity of which we have to obtain the integral. In accordance with the rule just given the result is ax. But we must remember that in the direct process all constant terms disappear ( 30, example (1) ) ; so, to the result as above obtained, we must always add on a constant. Thus : y = J adx = ax-\-b. The constant b is quite arbitrary unless some condition is laid down which determines it. Thus the condition might be that when x is unity, y is equal to a + 3, in which case we see that the value of 6 is 3. (2) Evaluate y = Jlaxdx. In accordance with the rule which requires us to increase the power of x by unity and divide by the power so increased and add a con- stant, we get y = 2az 2 + b = ax 2 -f b. Similarly from (3) y' = anx n ~ l we deduce y = ax n +b. Or, changing n into n + l from y' = ax n we obtain which is true whether n be positive or negative, whole or fractional. The proof, given in 30, that the rate of variation of a sum of powers of the independently varying quantity is the sum of the rates of variation of each term, gives at once, by inversion, the rule that the integral of a sum of powers is the sum of the integrals of each term. From (4) y'= we can at once write down x n + 1 , 7 v = a - + b n+1 m+1 In example (6), 30, it has been shown that, when y = uv, the VARYING QUANTITIES. 45 increment of y consequent upon alteration of the quantity of which u and v are functions is dy Hence, from (5) y' = we obtain y uv + a constant. Also from example (7) of the same paragraph we see that gves y = u + a constant. 36. In section 34 we found how we might represent graphically the value of y for different values of x by means of an area included between the a?-axis, the curve representing y' as a function of a?, and two ordinates of that curve. By means of this method we can obtain an independent proof of the result of example (3) above. Let the curve in the diagram represent the relation y' = x n . The value of y corresponding to x is represented by the area OxP ; and FlG. 18. OxP=OxPy'OPy'. Now, in precisely the same way that OxP represents fy'dx, 'the area OPy' represents the quantity fxdy', as is indicated by the horizontal rectangle. But fxdy'fxd x n = =fx. nx n ~ l . dx = nfx n dx. Also OxP fy'dx =fx n dx, and OxPy' = xy f = x n + 1 . Therefore x n + l =fx n dx + nfx n dx = (n + 1) fx n dx -= (n + l)fy'dx = (n + l)y ; that is / 46 A MANUAL OF PHYSICS. In Fig. 19 the rectangular area OP represents the product of u and v. "When u and v increase or decrease simultaneously, the increase of uv is evidently udv -f vdu (the little rectangle at P being neglected). And also uv =fvdu +/udv, which makes the result of example (5), 35, almost self-evident. If u decreases as v increases (as in passing from P / 1 to P' 2 ) the area uv increases by the quantity udv, but decreases by the amount vdu. But, in this case, dit,' is itself negative ; so that the result is still d(uv) = udv + vdu. M M u FIG. 19. If the curve P represents the value of v in terms of u, we may take its reciprocal curve P' which represents w in terms of u, where w = - The diagram gives (as above) uw fudw -\-fwdu. But when any quantity v increases by the amount dv, its reciprocal decreases by - = * ultimately; that is, the increase is - v 2 Therefore u _ fdu _ fudv _ f\ ~ -J-j J -tfT -J which is the result of example (6) preceding. 37. From the results of examples (1) and (2) of 31, we can at once write down /sin xdx = cos x and /cos xdx = sin x, because the derived functions of cos x and sin x are respectively sin x and cos x. VARYING QUANTITIES. 47 Since we know that d tan 9jd = sec 2 0, we can put tan0=/sec 2 0d0; and similarly /sec tan d = /sec 2 sin d = sec /cosec cot d 9= /cosec 2 cos d 9 - cosec 0, and /cosec 2 d0= -cot 0. 38. Lastly, since ( 33) de x ldx = e x and d log xldx = -> we get ' = 10 X. CHAPTER V. MOTION. 39. Position. The position of a point in space is completely determined when three independent conditions are given, each of which it satisfies. And its position can only be given relatively to that of another point, for we do not know any point of which we can assert that it is absolutely fixed. We may say that one point is so much to the north or south of another, so far to the east or west of it, and so much higher or lower ; or we may say that it is so far distant from the other, that the line joining the two is inclined at a certain angle to the vertical, and that the vertical plane through the two has a given inclination to the vertical north-and-south plane. The given quantities which determine the position are called the co-ordinates of the point. The first case furnishes an example of the ordinary Cartesian system of rectangular co-ordinates, the second illustrates the system known as polar co-ordinates. In the Cartesian system FIG. 20. the co-ordinates are usually denoted by the letters a?, y, z ; in the polar system, the length is generally denoted by r, and the angles MOTION. 49 by 9 and 0. Thus, in Fig. 20, the relative position of P to O may be given by # = AB, ?/ = OA, 2=BP; or by r=OP, 40. Displacement. When two points occupy different positions, we speak of the displacement of the one from the other ; and it is by means of the displacement that we determine their relative position. Two ideas are essentially involved in the term the idea of magnitude and the idea of direction. If we know only that one point is distant three feet from another, we cannot tell what position it occupies on the spherical surface all points of which satisfy this condition, Other two conditions are required to fix the direction also, that is, to determine the displacement. Addition of two displacements is effected when we find the single displacement, which produces the same result as the two do when FIG. 21. applied consecutively. The displacement from a to b may be denoted by ab, and that from 6 to c by be. Then ab-\-bc = ac. But the displacement denoted by be might have been performed first. The effect would be to transfer a to b', which is a point such that ab' is equal and parallel to be. This follows since a displace- ment does not involve the idea of position, but only the ideas of magnitude and direction ; in fact, ab' is the same displacement as be. And, similarly, bTc is the same as ab ; so that ab' + b r c=Tc + ab = ac = ab + be. Also, since a displacement is reversed when its direction is reversed, we have ab-ba ; and the ordinary laws of algebra apply to addition and subtraction of displacements. The lines ab, be, etc., may be used to represent the displacements ab, be, etc. ; for a line involves necessarily, and only, magnitude and direction. A line, given in magnitude and direction, is called a vector ; and any quantity which, like a displacement, requires for its com- plete representation a directed line is called a vector quantity. In the course of this chapter, and of Cha. Vl. We shall get various examples of such quantities. 4 50 A MANUAL OF PHYSICS. A quantity which is independent of direction which merely possesses magnitude is called a scalar quantity, that is, it is com- pletely determined by measurement on a suitable scale. When treating such quantities algebraically it is usual to denote scaiars by ordinary letters, as a, 6, x, y, etc., and vectors by Greek letters as a, (3, y, etc. Any displacement in space can be fully represented in terms of three distinct unit vectors and three independent scaiars. Thus, in the figure of last section, a may represent a unit length drawn in the direction AB, and the line AB may contain x units of length, so that the vector AB is xa. Similarly, if is a unit vector in the direction Oy, the vector OA may be denoted by yfl ; and the vector BP may be represented as zy. Hence - the vector OP is xa+y/3+zy. But this quantity denotes merely the position of P relatively to that of : and, consequently, if any other point P' is situated relatively to another point 0' in the same way in which Pis situated with^ respect to 0, the vector xa+y$ + zy represents the displacement O'F. If all the quantities x, y, z, are variable arbitrarily, the vector p = X a -f yfi + zy is the vector of any point in space. If x and y alone are variable arbitrarily, the point lies upon a plane parallel to the directions indicated by cr, (3. If one of the quantities alone varies, say x, the point lies on a line whose direction is indicated by a. If all three are fixed, the point is fully determined. More generally, if x, y, and are connected by one, two, or three relations respectively, the equation indicates respectively a surface, a curve, or a definite point. We have spoken of the displacement of one point relatively to another as that which determines the relative position in space of the first with respect to the second. Sometimes the two points may lie on a given surface or a given curve, and it is then frequently convenient to speak of the displacement on the surface or along the curve. This means that the magnitude of the displacement is to be measured along the surface or curve. 41. Speed and Velocity. The displacement of a point may be constant, or it may vary. If it varies, we say that the point is in motion : so that by motion we mean change of relative position. [The science which treats of motion, and which is generally called Kine- matics, therefore deals with the ideas both of space and of time.] The motion of a point is essentially a translation, for it has no separate parts which can rotate relatively to each other. Its posi- tion, we have seen, is not fixed unless three independent conditions MOTION. 51 are given. The removal of one restraining condition leaves the point more free to move than before ; and a point, the motion of which is unrestrained, is said to have three degrees of translational freedom. FIG. 22. If P moves to P' (Fig. 22), the change of displacement is represented by thejine PP\ or ( 40), by p' p, where p' and p represent the vec- tors OP' and OP. If this change occurs in time V - t, the time-rate of change is | /^^ or -jfe It is convenient to represent this quan- tity by p ; so that p is the time-rate of variation of p, or its first derived function with respect to the time as the independent variable ($28). FIG. 23. The mere magnitude of this time-rate is called the Speed of the moving point ; but, when the direction is considered also, the term Velocity is used. In illustration of this we shall consider the case of uniform motion in a straight line. Let QP (Fig. 23) be the line. Let Q be a fixed point on it, and let P be the uniformly moving point. Since P moves uniformly, the length of QP is proportional to the 42 52 A MANUAL OF PHYSICS. time t reckoned from the instant at which P occupied the position Q ; say QP at, where a is a constant. Let (3 be a unit vector in the direction QP, and let , p be the vectors from O to Q and P respectively. We have then P = + at/3. Hence, by the principles of Chapter IV., Here a is the speed of motion, and p is the velocity. The magni- tude a is constant, and the direction /3 is also fixed. Also p a =-. at(3. Hence the distance traversed is s = at. [This necessitates our defining unit speed as that speed with which unit distance is described in unit time.] 42. Acceleration of Speed. When the velocity of a moving point varies it is said to be accelerated, and the time-rate of its change is called the acceleration. Meantime we shall suppose that the change affects the magnitude only and not the direction : that is, we shall investigate non-uniform motion in a straight line. We may write (using the diagram of last section), P = a + #/3, where x is some function of t, and we get p = /3. Here x is the variable speed of motion. Forming the second derived function of p with respect to the time and denoting it by p, we obtain p = */3. If we assume the acceleration to be constant, that is, x = a con- stant = b (say), this becomes p = b(3. But, 35, this equation gives p = (a + bt) ft+y=xft+y where a is the value of the speed when t =- o (usually called the initial speed), and y is a constant vector which vanishes if we sup- pose the velocity to have the direction /3. And from this we deduce further p = ft/xdt = ft/ (a+bt) dt = (c + at + |6* 2 ) ft + a = xfl+a. If we suppose the point to be in the line of motion, a will vanish, x will be the distance traversed from Q, and c will be the distance MOTION. 53 from O to .Q. In other words, c is the numerical value of p (called the tensor of p, and denoted by the symbol Tp) when t = o. Thus. under uniform acceleration in the direction of motion we get x = b x a -\- bt If the acceleration is negative, 'i.e., opposite to the direction of motion, we must prefix the minus sign to the quantity b. If the motion begins from rest, the quantities a and c vanish. As a special case, when a body falls from rest under the action of gravity, in which case the acceleration is denoted by the letter g, we get x=g, x=gt, x = \g&. Again, if the body is thrown, upwards with speed V and we consider the upward direction to be positive, the equations become *=-<7, x = V-gt, x^t-\gt\ The second of these equations enables us to tell at once how long the body will take to rise to its greatest height above the ground. For, when the body is at its greatest height, the speed vanishes, that is, x = o, and therefore -Y. 9 From the third equation we can tell after what time it will reach the ground. Since we are supposing x to be measured from the surface of the earth, the condition is x = o. This condition gives either t = o, or * = 2V . g Hence the body takes as long a time to fall from its greatest height as it took to rise to it. Again, we have x 2 = V 2 -2V# + # 2 2 = V 2 - 2g(Vt - g&} = V 2 - 2gx. When the point is at its greatest height, x = o, and V 2 x= 2<7 Also, when cc = o, we get =V; that is, the point reaches the ground with speed V equal and opposite to the speed of pro- jection. 43. Curvature. Acceleration of Velocity. In the preceding section we have supposed the direction of motion to be unaltered. When the direction changes the path of the moving point is said to 54 A MANUAL OF PHYSICS. be ' curved.' The tangent to the curve gives the direction of motion at any instant, and the limiting value of the ratio of the angle between two tangents at near points to the length of path between these points as they are taken closer and closer together and finally coin- cide is called the Curvature of the path at that place. Thus the curvature is the rate of change of the direction of motion per unit length of the curve. If the tangent to the curve at any point makes an angle 9 with any fixed line in its plane, while s is the length of the curve, this definition gives as the measure of the curvature the quantity dO/ds. To obtain a measure of the angle between two lines in a plane (and we are here limiting ourselves to the case of plane curves) draw a circle (Fig. 24), of any radius r, from the point of intersection of the lines as centre. The angle 9 is measured by the ratio which the length, s, of the arc of the circle intercepted between the lines bears to the radius. It follows that the ratio of 9 to s is equal to 1/r, FIG. 24. and is therefore constant for a given circle no matter how large or how small 9 and s may be. Hence the curvature of a circle is the reciprocal of the radius. Now it is always possible to draw a circle the curvature of which is the same as that of a given curve at a given point. This circle is called the circle of curvature at that point ; its radius is called the radius of curvature ; and the reciprocal of its radius measures the curvature of the given curve at that point. In considering change of velocity as dependent on change of direc- tion, it will be convenient to assume first that the speed is constant, and also that the rate of change of direction is constant. In other words, we shall investigate the case of uniform motion in a circle. Draw any two diameters at right angles to each other (Fig. 25), and let o, (3 be unit vectors along them. Let p be the vector of the point P. According to the data OP revolves uniformly, that is, the angle through which it turns per unit of time (called the Angular Velocity) is constant. Let w be the angular velocity, so that, if P starts from the point A, the value of the angle POA is wt, where t is the time MOTION. 55 taken by the point to travel over the distance AP. Then we have vector ON = OP cos iot . , and vector NP = OP sin ut . , so that ifOP=a p = a (cos (at . a -j- sin at . (3) = a w sin cot . a + w cos at . 3 = u?a (cos w sn This result shows that the direction of the acceleration is inwards (for the negative sign is used) along OP, and that its magnitude is the square of the angular velocity multiplied by the tensor ( 42) of p. But the tensor of p is a, and the angle described per unit of time is the speed, v, of P in the curve divided by a. Hence w 2 is equal to FIG. 25. v 2 /a 2 , and therefore the magnitude of the acceleration is v 2 /a. This acceleration, it is to be observed, does not alter the speed, but only the direction of motion ; the reason being that it is perpendicular to that direction. If OP be constant in direction, but of varying length r, the acceleration is r ; and we have just found that, if r is fixed in length but revolves with angular velocity w ( = 9, if OP makes an angle 9 with a fixed line), the acceleration is-r0 2 . Hence, if both magnitude and direction vary, the acceleration along r is 44. Acceleration in, and perpendicular to, the Direction of Motion. When a point moves in any curve, the acceleration perpendicular to the direction of motion at any position may be found by drawing the circle of curvature at the given position. If r is the radius of curvature, and v the speed of motion, the accelera- tion perpendicular to the direction of motion is, by last section, v z fr towards the centre of curvature, and the acceleration of speed is s 56 A MANUAL OF PHYSICS. where s is the distance measured along the curve, in the direction of motion, from a fixed point to the moving point. The following method of deducing these results is exceedingly simple and instructive. Let p be the vector to any point of the path. Then dp dp ds dp . , N ' = 3hto"3i-5 '"'' (say) - Now, dp being a vector, p' is also a vector in the same direction that is, along the tangent (see Fig. 22, in which PP' must be supposed to be indefinitely small). But, in the limit, when ds vanishes, the length of dp is equal to ds. Therefore p' is a unit vector along the tangent. And, since the length of p' is constant, dp' must be per- pendicular to p'. Hence, dp'lds = p" is a vector inwards along the normal to the curve. And the quotient of the length of dp' by the length of p' is equal to the angle, dO, turned through by the radius of curvature. Hence the magnitude of p" is d9jds ; that is, ( 43,) it is the reciprocal of the radius of curvature. But p = a is the magnitude of the moment of a quantity containing a units about a point distant p units of length from its line of action. Thus the moment of AB about is twice the area of the triangle AOB. The moment of the resultant of two directed quantities is the sum of the moments of the components. Let AC, AB (Fig. 31), be the two components ; AD being the resultant. We have to prove that AOD = AOB+AOC. Draw OF parallel to CA to meet BA and DC produced in F and E. Then AOD = AOB+BOD - ABD = AOB+i FEDB-I ACDB=AOB+i FECA=AOB+AOC. The lines AC and AB in the figure have been so drawn that motion along them from A involves revolution in the same direction about 0. Had AC been so drawn as to involve rotation about opposite to that indicated by AB, it would have been necessary to regard one of the triangles as being negative. The same proof would then hold. It is usual to regard rotation opposite to that of the hands of a watch as positive. 60 A MANUAL OF PHYSICS. When the direction of the quantity passes through 0, its moment about vanishes. 50. Acceleration Perpendicular to Radius-vector. In, 43 we obtained an expression for the accleration along the radius-vector of a moving-point. We can now find an expression for the accelera- tion perpendicular to the radius-vector. If AB, in the last figure, represents the path of a moving point P, the moment of the velocity of P is twice the rate at which the area AOP is described as P moves along AB. For if 8s is a small length measured along the path, pSs=p St is twice the corresponding in- ot crease of the area ( = da, say). Therefore 77 PT1> and, in the limit when St = o, ~r,=p-T.=pv, where v is the speed, which proves the statement. It is evident that the proof still holds when the direction of motion varies, for it is true however small Ss and t may be. Let OP( = r), the radius-vector of the moving point P (Fig, 31), make an angle 9 with a fixed line in the plane of the figure. Let FIG. 31. P move to a point P', so near to P that PP' is practically a straight line. Draw PM perpendicular to OP'. Then PM = rd0, and 0PM = r . rdO. Hence the area traced out per unit of time is %r-0, so that r 2 ^ expresses the moment of the velocity. Now we have just seen that the moment of the velocity is equal to pv. But pv is equal to rw, where u is the resolved part of the velocity perpendicular to r. This is evident since, if p and r are inclined at an angle $ to each other, we have p = r cos 0, and u = v cos 0. Hence we have ru = r*V. Suppose now that the velocity is accelerated. The moment of the velocity represents the rate of increase of the area, and so the moment of the acceleration gives the rate of change of da/dt. MOTION. 61 The acceleration can be regarded as composed of two parts one part perpendicular to r, and the other along r. Of these components the former alone is effective in altering the rate of description of the area, for, 49, the direction of the latter part passes through 0. Hence the moment of the acceleration is ru t and so the acceleration perpendicular to the radius-vector is 51. Simple Harmonic Motion. When a point P revolves with uniform speed in a circle, the motion of the foot N (Fig. 32) of the perpendicular drawn from P to any fixed diameter is called simple harmonic motion. The velocity and acceleration of the point N can easily be found when the position and velocity of P is given. FIG. 32. The velocity of N is evidently the resolved part of the velocity of P in the direction ON. That is to say, its magnitude is v cos if v is the speed of P. But cos 9 is proportional to NP ; hence the speed of N is proportional to NP. The maximum speed is attained when N passes through O ; and it is then equal to v, the speed in the circle. Similarly the acceleration of N is the resolved part of the acceleration of P in the line ON. But the acceleration of P is, 43, - -y 2 /a, where a is the radius. Hence the acceleration of N is - v*la . sin 9 = -y 2 ON/a/ 2 . That is to say, it is inwards towards 0, the centre of the range of N ; and its magnitude is proportional to ON, the displacement from the centre. The ratio of the acceleration to the displacement is v 2 /a> 2 = w 2 , where w is the angular velocity of OP. But the angular velocity 62 A MANUAL OF PHYSICS. is 27T/7-, T being the time of a complete revolution in the circle. Hence acceleration 4?r 2 displacement ~~ ~r* ' If we call B the positive extremity of the range, the angle through which OP has turned since it coincided with OB is called the Phase of the simple harmonic motion. The phase may also be measured in fractions of a whole revolution. Thus we speak of the quarter phase, etc. The maximum distance to which N can get from is called the Amplitude of the motion. It is obviously the radius of the cor- responding circle. If the motion does not commence at the positive extremity of the range, the angle through which the radius has to turn until P reaches the positive end is called the Epoch. Thus the general expression for x, the distance of N from 0, is x = a cos (w+a). Here a is the amplitude, w is the (constant) angular velocity, t is the time, and a is the epoch. Simple harmonic motion is frequently exemplified in nature. It occurs in the vibration of stretched strings, the agitation of the luminiferous medium, etc. 52. Composition of Simple Harmonic Motions. (1) Motions in the same straight line and of equal periods. Let the motion of P (Fig. 33) correspond to one of the given motions. From P draw a line PQ, making an angle 0, equal to the phase of another of the motions, with the line OA ; and let the length of PQ be equal to the amplitude of this motion. Since and 9 increase at the same rate, the line OQ remains of constant length and revolves at the same rate as OP andPQ. But the foot of the perpendicular from Q on OB moves with a motion which is compounded of the two given motions. Hence the resultant of the two motions is another simple harmonic motion, of the same period, in the same straight line. And, in particular, when the amplitudes of the two components are equal, the phase of the resultant is the mean of the phases of the components. This proof is quite general, and applies to any number of such simple harmonic motions. Another proof may be obtained as follows. Let the separate motions be x l a 1 cos(w^ + aj, x c) = a.,cos(wt -f- a 2 ), etc. Then a? = a? 1 4-JJa+ etc. =a L co$(ut + ctj) -f- a.,cosU + a.J+ etc. But MOTION. 63 cos (u)t -f a 1 ) = cos u>t cos ttj sin wt sin a v Therefore x = (a l cos a x + a 2 cos a. 2 -^- etc.) cos w (% sin aj-|-# 2 sin a 2 4- etc.) sin wt. Now assume the multiplier of cos ut to be equal to a cos a, and the multiplier of sin at to be a sin , so that x = a cos w cos a - a sin wf sin a. This gives x = a cos (>ot-\-a). [The assumption made is obviously justifi- able ; for it only introduces two new quantities a and a, and gives two equations to determine their values.] Hence, in this case, the resultant is simple harmonic motion of the same period in the same straight line. (2) Two simple harmonic motions, of the same period and phase, in lines inclined at any angle. These obviously compound into a single motion of the same period and phase. For, let OA, OB be the two inclined lines, and draw any other line OP. From P draw PM, PN, parallel to OA, OB, respectively, to meet these lines in the points N and M respectively. If P moves along OP with simple N A FIG. 33. harmonic motion, it is evident that M and N move along OB and OA with simple harmonic motions of the same period and phase. And the motion of P is the resultant of their motions. (3) Two simple harmonic motions in lines at right angles to each other of the same period and amplitude, and differing in phase by one quarter of a period. A glance at the figure of last section shows that M' moves along OA' precisely as N moves along OB, since OP' is perpendicular to OP. Hence the motions of N and M differ in phase by one quarter of a period. And the motion of P is their resultant. That is to say, the resultant is uniform circular motion ; and the motion takes place from the positive end of the range in which the motion is one quarter of a period in advance to the positive end of the other range. (4) Two simple harmonic motions of equal period, and of phases differing by a quarter period, in lines inclined at any angle. By projection of the circle we obtain an ellipse in which conjugate diameters correspond to mutually perpendicular diameters of the circle. Hence the resultant is elliptic motion, with equal areas 64 A MANUAL OF PHYSICS. described in equal times (since this is so in the circle), and with a law similar to that given in (3) as regards direction of motion. (5) Any number of simple harmonic motions, in lines inclined at any angle to each other, and of any phase, but of equal periods. By a reversal of the first proof of (1), we see that the line OQ may be replaced by any two lines OP and PQ, which revolve with the same angular velocity. Hence any simple harmonic motion may be broken up into two of the same period, which differ in phase by any given amount, and one of which has any given phase. This appears also by a reversal of the second proof of (1). For if a cos ( an( i ^sinaj -}- a 2 sina 2 = a sin a. That is, there are only two conditions to be satisfied by the four quantities a v a v a lt a 2 ; so that two more may be imposed. Let Pj, P 2 , etc. (Fig. 34), be the points moving with simple harmonic motion. Let pi, p\, be two points, whose motions com- FIG. 34. pound into that of PI ; and let their phases differ by a quarter period. D.eal similarly with P 2 , etc., and let the motions of p^ p^ etc., be all of the same phase, while those of p\, p' 2 , etc., also agree in phase with each other, but, of course, differ in phase from the motions of pi, p^, etc., by a quarter of a period. Besolve all these motions into their components along two rectangular axes ox and oy. Then compound all of the same phase in each axis with each other. The result is, by (1), two simple harmonic motions in each axis differing in phase by a quarter period. Now combine each motion in oy with the motion in ox which is of the same phase, and we have ultimately two simple harmonic motions which differ in phase by a quarter period and take place in lines which are in general inclined to each other at a finite angle, These, as we have seen, combine into elliptic motion. MOTION. 65 53. Wave Motion along a Line. Let a point vibrate up and down the ?/-axis with simple harmonic motion about the origin, and let the paper be drawn along behind it at a uniform rate in the direction xo. The point will trace out a curve (indicated in the figure) which exhibits the simplest form of a wave. The quantities y and x can easily be seen to be connected by the relation y = a cos (ut - nx] in which a and M denote the same quantities as formerly. If x is constant, the equation is of the same form as the one which was given in 51, and shows that every point on the a>axis vibrates up and down with simple harmonic motion, and that y has one and the same value for all the values of t which differ by the amount 27r/w. This quantity STT/W is called the periodic time, or the period, of the motion. If t remains constant, y varies with x in precisely the same manner as it did when x was constant and t varied. The value of y is the same for all values of x which differ by Sirln. This quantity is the Wave-length. Lastly, y remains fixed in value if x and t vary simultaneously in such a way that (oit - nx} is zero, x being measured from any special position, and t from any definite instant. This gives x = w/ra, and shows that the wave runs along in the direction ox with speed w/n and without change of form. Similar reasoning shows that the equation y=a cos ((ut + nx)- represents a succession of precisely similar and equal waves which run in towards the origin with speed w/n. The resultant disturb- ance due to the superposition of this set of waves upon the former is represented by ya {cos (ut -nx) + cos (o)t + nx}\ =%a cos (tit cos nx. Whatever be the value of x, this vanishes when t is any odd multiple 66 A MANUAL OF PHYSIC8. of ir/2w ; and, whatever be the value of t, it vanishes when x is any odd multiple of 7r/2w. The motion, at any definite point is simple harmonic, of period 27r/w ; and the form of the wave at any definite instant resembles that shown in Fig. 35, the ordinates being all altered in the common ratio of 2 cos w to unity. The resultant is therefore a series of stationary waves, which oscillate up and down, parallel to the ?/-axis, in situ. This result is of importance in the theory of the vibrations of stretched strings, etc. 54. Rotation. While a mere point can have translational motion only, a rigid body (a body the parts of which cannot suffer relative displacement) is free to rotate also unless three points of it, which do not lie in the same straight line, are fixed. Three such points being fixed, the body is devoid of all freedom to move. If two points are fixed it can rotate about the line which joins them, and is said to have one degree of rotational freedom. If one point only is fixed the body may rotate independently about any three mutually perpendicular axes which pass through that point it has three degrees of rotational freedom. Finally, no point being fixed, it lias, in addition, three degrees of translational freedom. The greatest number of degrees of freedom which a rigid body can have is therefore six. [A non-rigid body has distortional freedom also.] , 55. Alteration of Co-ordinates because of Rotation. If a point P is rotating about the axis of z (drawn perpendicular to the plane of the paper through the point O, Fig. 36) with angular velocity w., the FIG. 36. FIG. 37. alteration of the x co-ordinate of P, in the time ct, is - w z r cos ^ct at z r sin St = - M z y$t. If P were revolving simultaneously about Oy with angular velocity w y , the alteration of x in the same time would be MyZdt. Hence the resolved part of the speed of P parallel to Ox is (dt being small) aj y z - w z ?/. 56. Uniplanar Motion of a Rigid Body. By ' uniplanar motion ' is meant motion parallel to one plane. The motion of a rigid plane figure is included as a special case. MOTION. 67 Let the motion be parallel to the plane of the paper; and let AB be the position of a line in the body before the motion occurs, while A'B' is its position at the end of the motion. Draw AA' and BB' ; bisect them, and erect perpendiculars at their points of .bi- section. Let these meet in O. We have OA = OA', and OB = OB'. Also AOB and A'OB' are congruent triangles, the angle AOB being equal to the angle A'OB'. Thus AB might have been moved into the position A'B' by a single rotation, about O as centre, through the angle AOA'. Hence any displacement of a rigid body parallel to one plane may be produced by rotation about a definite axis perpendicular to that plane. In general the body does not revolve in this way from its initial to its final position. On the contrary, each point usually describes a curve which is not the arc of a circle. In such a case we may regard the total displacement of any point as made up of a succession of indefinitely small displacements, each of which coincides with an indefinitely small arc of a circle. This circle is evidently the circle of curvature ( 43) of the path of the moving point. Its centre is called the instantaneous centre about which all points in the same plane are revolving. Thus, when a wheel rolls along the ground, the point of the wheel which is in contact with the ground is at rest for an instant it is the instantaneous centre about which the wheel is revolving for a moment as a rigid body. Let the point O, Fig. 38, be the instantaneous centre. The point p l revolving about will come into the position jp'*. Suppose now that p., revolves about p\ as the new instantaneous centre, and that this brings it into coincidence with p'. 2 about which the revolution next takes place, and so on. The points p lt p. 21 etc., are points, fixed in the body, which are successively at rest, and the points p\, p'%, etc., are points, fixed in space, at each of which in succession the in- stantaneous centre is situated. The instantaneous centre coincides for a short time with each point of both series, and passes suddenly 52 68 A MANUAL OF PHYSICS. from one to another. When the motion is continuous the polygons PiP-2 an d op'ip' 2 . . . become continuous curves, and the instantaneous centre moves continuously along them. In the case of a rigid body, moving parallel to the plane of the paper, the line through the centre perpendicular to that plane is instantaneously at rest, and is called the instantaneous axis ; and the curves in the figure are sections of cylindrical surfaces in the body and in space. Hence we see that the most general uniplanar motion of a rigid body consists in the rolling of a cylinder fixed in the body upon a cylinder fixed in space. An obvious example of this is given when a roller is drawn over the surface of the ground. A similar statement, modified merely by the substitution of the word ' curve ' for ' cylinder,' applies to the motion of a plane figure in its own plane. Mere translation is a special case in which the instantaneous centre is at an infinite distance. It may be considered to consist in infinitely slow rotation about an infinitely distant axis. 57. Motion of a Rigid Body in Space.- First, suppose one point of the body to be fixed, and consider a sphere in the body with its centre at the fixed point. Take any two points A,B, on the surface of this sphere which occupy the positions A',B', respectively, at the end of the motion. The reasoning of last section applies here also, great circles of the sphere taking the place of straight lines. We thus find that the displacement might have been produced by simple rotation of the sphere about a diameter passing through the point O (Fig. 37) on the surface. The actual motion consists in the rolling of a cone fixed in the body upon a cone fixed in space. This is at once evident if we suppose the curve op^.^ ... of last section to be drawn upon the surface of the sphere whose centre is fixed the points p it p z , etc., being successive positions of the extremity of the instantaneously fixed diameter. Now suppose that no point is fixed. The total displacement consists in general of both translational displacement and rotational displacement ; and it is clear that we may separate these, taking first one and then the other, and yet produce the same total effect. As we have just seen, the rotation leaves a set of planes (those perpendicular to the axis of rotation) parallel to their original position, and mere translation does not alter this parallelism ; so that, in the final position of the body, there is one set of planes which has been unaltered as regards orientation. The required rotational displacement can be produced by revolution about any axis perpen- dicular to these unaltered planes, but the final position of the body MOTION. 69 will depend on the particular axis chosen. And we can so choose the axis that, after the rotation has taken place, mere translation parallel to that axis will make the body take the required position. And this can only be done in one way. For let the plane of the paper be one of the planes which are unaltered in direction, and let AO be a line in that plane, the final position of which is to be A'O' in a parallel plane. Kotation through the angle between AO and A'O', FIG. 39. about any point in the plane of the paper, will make AO parallel to A'O' ; but AO will only be superposed upon A'O' if the point be chosen by means of the construction of last section. And, when this superposition is effected, a translation perpendicular to the plane of the paper will bring the body into its final position. Hence any displacement of a rigid body in space may be produced by rotation about, and translation along, a definite axis. When the rotation and the translation are simultaneous, the motion is called a twist about a screw. Such a motion is the most general kind of motion that a body which possesses only one degree of freedom can have. [The rotation and the translation do not occur independently, and so one degree of freedom alone is involved.] Any given motion of a rigid body in space consists in a twist about a screw, in which the axis and the linear and angular velocities are in general varying. The position of the axis at any instant is given by the line of contact of two ruled surfaces, one of which (fixed in the body) simultaneously rolls and slides upon the other, which is fixed in space. 58. Composition of Angular Velocities. An angular velocity is completely determined when its magnitude and direction are given, and these quantities may be indicated by the magnitude and direc- tion of a line drawn parallel to the axis of rotation. Angular velocities (and accelerations) are therefore vector quantities, and their laws of composition and resolution are identical with the laws for vectors, and therefore with the laws for linear velocities and 70 A MANUAL OF PHYSIOS. accelerations. Thus the resultant of two angular velocities or accelerations, which are represented by the two sides of a parallelo- gram, is represented, on the same scale, by the conterminous diagonal. An extremely important case is that in which a body with one point fixed is revolving uniformly about an axis and is subjected to uniform angular acceleration about a perpendicular axis. In the corresponding problem regarding linear velocity and acceleration ( 43), the magnitude of the linear velocity is unaltered while the direction of motion revolves uniformly, being always perpendicular to the direction of acceleration, which also revolves uniformly. So, in the present case, we can at once assert that the magnitude of the angular velocity will remain constant, but that its' axis will revolve uniformly, and will be always perpendicular to the axis of constant acceleration. Now the axis of acceleration is always horizontal when the rotating body is a top which spins uniformly with its axis of revolution inclined to the vertical. Hence the direction of the axis of the top will rotate uniformly, and will be always perpendi- cular to a horizontal line (not a horizontal plane) which rotates uniformly. The axis must therefore revolve at a uniform rate around the vertical. Precession of the equinoxes is due to angular acceleration of the earth about an equatorial axis, and the peculiar motions of gyrostats have a similar explanation. To compound two angular velocities about parallel axes, indicated in magnitude and direction by AB and CD (Fig. 40), it is merely necessary to find a point O such that the moments of AB arid CD about it are equal and opposite ( 49). Let p\,2hi be the lengths of perpendiculars from upon AB and CD respectively. Since the angular velocity of due to rotation about AB is numerically equal A B to AB, it follows that AB^ represents the velocity of perpendi- cular to the plane of the paper ; and, similarly, the velocity of O perpendicular to that plane due to rotation about CD is CT>p. 2 ; and this quantity is of the opposite sign to the former, for points moving MOTION. 71 along AB and CD revolve oppositely around O. Hence, if the areas AOB and COD are equal, the point O is at rest ; and thus the locus of that is, a line through parallel to AB and CD is the resultant axis of rotation. Similar reasoning shows that the resultant axis due to angular velocities indicated by AB and DC is a parallel line situated above AB, and that the direction of rotation coincides with that of AB, the larger of the two components. To find the effect of the superposition of an angular velocity w upon a linear velocity v, or of a linear velocity v upon an angular velocity w, we may resolve the linear velocity into its two compo- nents parallel to and perpendicular to the axis of rotation. The effect of the perpendicular component v' is to shift the axis parallel to itself through a distance rf, such that the speed dw is equal and opposite to the speed v'. The parallel component simply moves the whole body in the direction to the axis, so that the resultant is a twist. If a body is rotating simultaneously about three axes, the velocities being representable by the three sides of a triangle taken in the same direction round, the effect is that there is no rotation ; but the body is translated perpendicularly to the plane of the triangle with a speed represented by twice the area of the triangle. If one of the rota- tions is represented by a side of the triangle taken in the opposite direction round, there is no translation ; but the body rotates about an axis bisecting the two sides which were taken in the same way round. The rotation round this axis coincides in direction with the rotation about the side of the triangle parallel to it, and the angular velocity is twice as great as the velocity about the parallel side. Corresponding results obtain in the case of angular velocities representable by the sides of closed plane polygons. 59. Displacement of the Parts of a Non-Rig id Body. Strain. A non-rigid body may alter in form, or in volume, or both. Any such definite change of shape or bulk is called a Strain. Homogeneous Strain. When all parts of a body, originally similar and equal, are similarly and equally strained, the strain is said to be homogeneous. It follows that parallel straight lines in the unstrained body become parallel straight lines in the strained body ; but, in general, the direction of the lines and the distance between them is altered. And therefore parallelograms remain parallelograms, parallelepipeds remain parallelepipeds, and any figure or surface changes into a similar figure or surface. Thus a sphere becomes an ellipsoid. Such a strain is completely determined when we know what 72 A MANUAL OF PHYSICS. alteration in magnitude and direction has been produced in three originally non-coplanar lines. One number is required for each line to fix the change of length ; and two numbers are required to determine the change of direction of each line. In all, nine numbers are in general necessary. The simplest kind of homogeneous strain is that in which there is uniform expansion or compression in all directions. Any line in the strained figure preserves its original direction, and all lines are equally altered in length. Such strain occurs in the compression of fluids. One number completely determines it. Next in order of simplicity is a homogeneous strain in which lines in one definite direction are unaffected by the strain ; that is, the strain is c.onfined to planes perpendicular to a definite direction, and the alteration in any one of these planes is precisely similar and equal to the alteration in any other. It is usual, therefore, to call such a distortion a plane strain. A circle in the unstrained figure, drawn in one of the planes of distortion, becomes an ellipse ; and so a sphere in the body becomes an ellipsoid. (In the cases of equal expansion or of equal contraction in all directions in the planes of the strain, the ellipsoid is an oblate or a prolate spheroid respec- tively. All lines which' are not in or perpendicular to these planes are altered in direction.) All perpendicular diameters of the circle become conjugate diameters of the ellipse ; in particular, the principal axes of the ellipse were originally perpendicular diameters of the circle from which, however, they usually differ in direc- tion. Four numbers determine a plane strain of the most general kind. When the principal axes of the ellipse (called the strain-ellipse] into which the circle is deformed are not changed from their original directions in the unstrained body, the strain is called a pure (or non-rotational) plane strain. In this case the distortion consists in extension (or contraction) in two directions at right angles to each other. Any rotational, or impure, plane strain may (so far as the final effect is concerned) be produced by a pure strain superposed upon, or followed by, rotation about a definite axis perpendicular to the plane. In a pure strain every line except a principal axis has suffered rotation ; and it follows from this that the superposition of two pure strains generally produces an impure strain. Hence a body may be distorted by three plane strains in succession, and yet (the strains being properly chosen) be left unstrained, but rotated through an angle about a definite axis. Three numbers completely characterise a pure plane strain. A specially important case is that in which there is no alteration MOTION. 73 of volume. This implies elongation in one direction and equal con- traction in another. We may suppose that these directions are mutually perpendicular, giving a pure strain ; for, as we have seen, any other strain may be assumed to consist in a pure strain followed by rotation as of a rigid body. . Let oy be the direction of the elongation, and let ox be the direction of contraction. Let abed be y FIG. 41. a rhombus in the unstrained figure, which becomes the rhombus a'b'c'd' in the strained state, oa being equal to 06', oa' being equal to 06, and so on. We have then ab equal to a'b', with similar results for the other sides of the rhombus. It is, therefore, obvious that there are two sets of planes in the figure which experience no alteration, except as regards position ; so that (rotation excluded) the strain A' I ct & FIG. 42. might have been produced by holding fast one plane of either set say the plane through cd perpendicular to the plane of the paper and sliding all planes parallel to it through a distance proportional to their distance from the fixed one, until the originally acute angle 74 A MANUAL OF PHYSICS. of the rhombus becomes equal to its supplement. This motion is termed shearing motion, and the strain is called a simple shear. To make the result of this shear coincide with the result of the above pure strain, we must turn the body round in the direction of the hands of a watch through an angle b'db, so that b'd coincides with bd that is, through an angle equal to half the difference between the obtuse and acute angles of the rhombus. In the most general homogeneous strain, a sphere in the un- strained body becomes an ellipsoid in the strained state. Any set of three mutually perpendicular axes become mutually conjugate diameters of the ellipsoid. In particular, the three principal axes of the ellipsoid (which are called the principal axes of the strain) were originally perpendicular diameters of the sphere. These principal axes are usually rotated from their initial positions ; and, as in the corresponding case of plane strain, when this turning of the principal axes does not occur, the strain is said to be pure or non-rotational. So far as the ultimate result is concerned, any impure strain may be looked upon as due to a pure strain followed by a rotation. Any given strain may be produced by a simple shear followed by an extension (or contraction) perpendicular to the plane of the shear which in turn is succeeded by a uniform expansion (or compression). For the shear may be continued to such an extent as to give the proper ratio of the maximum and minimum axes ; and the perpendicular extension will then give the proper ratio of the mean axis to each of the other two ; while, lastly, the uniform expansion can be con- tinued to such an extent as to give the proper magnitudes of the axes. 60. Non-Homogeneous Strain. So long as rupture does not occur, all displacements in a portion of matter are essentially con- tinuous. Hence, however greatly the displacements may change throughout the body, we can always consider a portion so small that, within its limits, the strain is homogeneous. Let P and Q be two near points so near that the strain is homogeneous, and let 8x, dy, , where V is the volume of the mass m and p is the mass per unit volume, which is called the density of the substance. Again, the equation x = gt becomes mx ingt. But 'nix is the momentum produced in the body, which we thus see to be propor- tional to the time during which the body has been falling under the action of gravity. Also, instead of x-=- V- 2^.r, we have fynx 2 = fyn'V-ingx, or jra(V 2 a; 2 ) = mgx, which tells us that the loss of kinetic energy is proportional to the distance through which the body has risen. In 43 it was shown that the acceleration of a point moving in a circle of radius r with uniform speed v is v' 2 /r towards the centre. If the point have mass m, this corresponds to a central force mv-/r, which, e.g., in the case of a stone revolving in a sling, is supplied MATTER IN MOTION. 83 by the tension of the cord. The necessity for this central force gave rise to the erroneous idea of a ' centrifugal force,' which it was supposed to balance. In accordance with the First Law, the body tends to move along a tangent, and not from the centre, and the apparent force is really a result of inertia. 65. Further Discussion of the Second Law. The above examples involve the application of a single force, constant in magnitude and direction, to a material particle. But the Second Law enables us also to investigate the motion of a material particle under the action of any number of forces acting simultaneously, for it implicitly asserts that each -force acts independently of all the others, i.e., the effect produced by any force is the same as it would be if that force alone acted upon the particle when at rest. To completely specify a force we require to know its magnitude, the direction in which it acts, and the place at which it is applied. Hence a force is a vector quantity, and so the resultant of any number of forces acting simultaneously upon a material point is to be found by the ordinary law for the composition of vectors. Indeed this follows at once from the Second Law, since the forces are pro- portional to the accelerations which they produce. Hence, when a particle is acted upon by any number of forces, we need only consider it as moving under the action of the single resultant force. Since a mere particle has only three degrees of freedom all trans- lational three conditions completely determine its motion. If X, Y, Z be the components of the resultant force in the directions of the axes of x, y, z, respectively, these conditions are, by the Second Law, ma; = X; mv/^Y; mz = Z. In particular, the conditions of equilibrium are X = 0; Y = 0; Z = 0.- 66. Special Examples. We shall now apply these results to some special cases of motion of a material particle. (1.) A particle of mass m is projected with initial speed v a . It experiences a resistance which is proportional to its velocity. In- vestigate the motion. We may suppose that the motion is in the direction of the axis of c, so that we have only to consider the single equation mx = - kx, where k is constant. Multiplying each side by dt, this becomes mdx = kdx. The integral is mx = c-7cx. 62 84 A MANUAL OF PHYSICS. To determine the value of the constant c, we observe that v is the (given) value of x when x = o. Hence c mv . Therefore, finally, m(v a x) = kx. This result shows that the particle will come to rest at a distance mv a /k from the point of projection. We may write the equation in the form mdx mv u - kx This gives ( 38) t = T F m/k log (mv -kx), and so, since t = o when < = o, we have T = m/k log mv a . But when = T U we get xntv /k. That is r r,, mlk log mv tt is the value of the time which elapses until the particle comes to rest. (2.) A particle slides from rest down an inclined plane under the action of gravity. How long w r ill it take to move over a given dis- tance, and what will be its speed of motion when it reaches the given point ? Let m, , E, and F represent respectively the mass of the particle, the inclination of the plane to the horizon, the normal pressure, and the force of friction. We may take the axis of x in the direc- tion of motion, so that we get mx = mg sin a F. If /*' is the co-efficient of friction, this becomes mx = mg sin a - jt*'R. If the axis of y be taken perpendicular to the plane, the other equation is wy = = R mg cos a. Hence x = ff (sin n - ju' cos a) = A (say). {tru -of, FIG. 47. This quantity is independent of the mass of the particle. Multi- plying by dt we get dx = A.dt, from which MATTER IN MOTION. 85 where v is the initial speed, and therefore is zero by the given conditions. Hence, again, If we take the origin at the point from which the particle starts, we get a? u = 0, so that the time taken to move over the distance (sn a - // cos a) And the speed attained is (sin a fi' cos a). The kinetic energy gained by the particle is therefore mgx (sin a n' cos a). (3.) A material particle is attached by an elastic cord to a point on an inclined plane down which it would slide under the action of gravity if not so attached. Find the limiting values of the tension in the cord between which motion will not occur. T being the tension in the cord, and the other quantities having the same meaning as in (2), we get mx=Q=mg sin T[img cos . From this equation the two values of T may be found. The sign -|- corresponds to the case in which the cord has its greatest extension so that friction acts down the plane. The sign indicates that the particle is just on the point of sliding down, T having its smallest possible value. (4.) A particle of mass in is swung round in a vertical circle by means of a cord of length 1. What must be its angular velocity in order that the string may just be slack when the particle is at the highest point of its path ? The downward force is the weight of the particle, which must be balanced by the reaction to acceleration (the so-called ' centri- fugal force '). Hence, at the highest point, g = ^l, where w is the angular velocity. At the lowest point we have 67. Dynamical Similarity. We may write the expression for Newton's Second Law ( 64) in the form 8 A MANUAL OF PHYSICS. f where I is the distance which the mass m moves over from rest, in the time t, under the action of the force /. And we may further regard this equation as a dimensional equation ($ 27) ; in which case the sign of equality merely means that the dimensions of the quantity on the left-hand side of the equation are identical with those of the expression on the right-hand side of the equation. But, from a dimensional equation, we cannot make any deduction regarding the absolute magnitude of any of the quantities which are involved, for the equation simply asserts proportionality of magni- tude between its various terms. Still, by a suitable definition of units, we can pass from the dimensional to the ordinary equation. Thus, in the above equation, we may define unit force as the force which, acting on unit mass for unit time, causes the unit of mass to move over unit distance from rest ; or we might adopt the definition of 64. The idea of dimensions is of great importance in physics. It affords a useful check on the accuracy of algebraical work ; for the dimensions of all the terms in a physical equation must be the same. But its use is not limited to this extent. For example, we may write the equation in the form from which we see that if, in two similar material systems, the forces, masses, and lengths, are in the ratios a/1, /3/1, and y/1, respectively, and if the systems begin to move in precisely similar manners, the motions ivill continue to be similar, provided that we corn-pare them after the lapse of intervals of time which are in the ratio of \f to unity in the two systems. This principle, which was proved otherwise by Newton, has been called the Principle of Dynamical Similarity. Later on, we shall get various examples of its use. (See ^ 73, 76, 124, 159.) 68. The Third Law. Hitherto we have not discussed the motion of portions of matter between which there is mutual action of any kind. The first two laws of motion do not enable us to solve such problems. The requisite additional information is given by the Third Law of Motion : The mutual actions between any two bodies are equal and oppositely directed. MATTER IN MOTION. 87 A stress is defined as a system of equilibrating forces, and so we may put the above law into the form : The mutual action between any two bodies is of the nature of a stress. No one will question the truth of this law in the cases in which the various masses concerned are in equilibrium. Thus, when a book lies upon a table, we say that the table reacts upon the book j-rn C F| F , FIG. 48. with a pressure which is equal to its weight. But it is by no means so evident that a body, when' pulled along by means of a cord, pulls backwards with a force which is precisely equal to that by which it is dragged forwards. In order to see how this can be we must consider all the forces which are acting upon the moving body. Let B, Fig. 48, be the body and let F be the force with which it is pulled in the direction of F'F. Also let F" be some other force acting upon B in the opposite direction. The equilibrium of B is determined solely by the equality of the forces acting upon it, i.e., of the forces F and F", and is not at all affected by the force F' with which B reacts upon the pulling body. Hence the equality of the forces F' and F is a matter which is entirely independent of the equality of F" and F, and can only be proved by experiment. It is needless to add that all Newton's laws express the results of experiment or of observation. But (as Newton himself pointed out) we may regard ' action,' not merely as force but, as the product of force into the speed which it produces in the body upon which it acts. Now, the speed produced being the (time) rate at which the force moves the body, this pro- duct is ( 62) the (time) rate at which work is done by the force. [In modern terminology this is called the Activity.] Hence a second interpretation of the third law is that the (time) rate at which a set of forces do work upon a given system is equal and opposite to the rate at 'which the reacting forces do ivork. Had Newton been aware that heat was a form of energy this would (see Thomson and Tait's Elements of Natural Philoso2)hy) have been a complete statement of the modern principle of conservation of energy ; but, in his day, it was supposed that work spent in over- coming friction is unavoidably and entirely lost. Taken in conjunction with the Second Law, this law enables us to investigate the motion of bodies which impinge upon each other. 88 A MANUAL OF PHYSICS. To avoid unnecessary complications we may assume that two smooth spheres, of masses m x and m. 2 respectively, are moving in the direction of the line joining their centres with speeds v l and v. 2 respectively, and that after impact their velocities are v\ and i>' 2 . In most practical cases the time of impact is a very small fraction of a second, and the force is very large, so that it is impossible, without special appliances, to determine the values of these quantities. But the value of their product, called the Impulse, can generally be found without much difficulty. The third law tells us that the im- pulse is the same for each body, and hence 7>? 1 (v' 1 Vj) = m. 2 (v v'. 2 ). In addition to this we have the condition, determined experimentally by Newton, v\ v'., = e(v., Vi) , where e is a constant, less than unity, which is called the Coefficient of Restitution. This condition asserts that the relative speed of separation of the two bodies is less than, but is proportional to, their relative speed of approach. It ceases to be true if the distor- tion produced by the impact is too great. If the bodies, after impact, move together with a common speed V, the first of these equations becomes This principle is employed in the Ballistic Pendulum, which is used to determine the speed of a cannon ball or of a rifle bullet. In this case the mass of the pendulum, m. 2 , is very large in comparison with the mass m 1 of the bullet, and v. 2 is zero. The large relative value of m. 2 ensures that the two masses are moving together with the common speed V before the pendulum has been sensibly de- flected from the vertical. The value of V is found by observing the distance through which the pendulum swings. [From this, the height h through which the centre of inertia ( 69) is raised is obtained, and then ( 42) we get V = */%gliJ\ 69. Centre of Inertia. In a material system composed of masses m a , w , etc., we can always find a point such that the product of its distance from any plane into the sum of the separate masses is equal to the sum of the products of each separate mass into its own dis- tance from that plane. Let 2(m) denote the sum of the masses, and let 2(md) denote the sum of the products of each mass into its distance d from one given plane. We then can obviously find the distance D from this plane such that S(wd), ....... (a), MATTER IN MOTION. 89 for this is a single equation in one unknown quantity D. Let this be done for other two planes, neither pair of the three being parallel, and we get a fixed point, which satisfies the condition for these three planes. Let us suppose, for convenience, that the three planes are at right angles to each other, and that the lines of intersection are taken as the axes of x, y, and z respectively. We have then the three equations similar to the above, Now, multiplying these equations respectively by any quantities X, ft, i>, and adding, we get But X, //, and v may be the direction-cosines of the normal to any plane passing through the intersection of the three given planes, in which case the quantity \x+\iy+vz is the perpendicular upon this plane from any point whose co-ordinates are x, ?/, z. Hence equation (a) is true for any plane which passes through the intersec- tion of the three given planes. But equation (a) is still satisfied if we increase D and d by any constant quantity h, for this simply adds on 2(m)h to each side ; that is to say, (a) holds for any plane parallel to a given one for which it is true. Hence, it holds for all planes. The point so found is called the Centre of Inertia of the given set of material particles. By taking the time-rate of variation of the quantities in the above equation we obtain 2(m)D = 2(w dx dy Az which simply assert that the pressure has a constant value at all points of the liquid. If we assume that the origin is at the surface of the liquid, that gravity acts, and that the axis of z is drawn downwards, the equa- tions are dn dp dp ~i <- = > -T = > i = P9' dx dy dz The first two assert that the pressure is constant throughout a horizontal plane, and the last shows that it increases uniformly with the depth. The integral is This might have been obtained from (1) of last section by making v = o 76. Propagation of Surf ace-Waves in Liquids. We shall assume, MATTER IN MOTION. 99 for the sake of simplicity, that the waves are all similar, and that their ridges are parallel equi-distant straight lines. The principle of dynamical similarity then enables us to deduce easily the law of propagation. When the waves are propagated by gravity, the forces are propor- tional to the density of the liquid, to the value of gravity, and to the square of the wave-length ; and the masses are proportional to the density, and to the square of the wave-length. Hence the dimensional equation f=ml/t 2 , (67), becomes 7 o aV . I or g=ll&, which gives v 2 =gl. In these equations p represents the density of the liquid, and the other symbols have the usual significations. We see, therefore, that the speed of propagation is proportional conjointly to the square roots of the wave-length, and of the acceleration due to gravity. Such waves are called oscillatory or free waves. In the above case, it is assumed that the depth of the liquid is very large in comparison with the length of the waves. When the depth is very small in comparison with the wave-length, the above equations still apply, provided that I represents the depth of the liquid. For, when similar waves are propagated in liquids of different depths (the similarity having reference to the depth), we see that similar masses are proportional to the squares of the depths, while the ranges of vertical motion are proportional directly to the depth. Hence the speed of propagation of such waves, which are called long or solitary waves, is proportional conjointly to the square roots of the depth and of the acceleration due to gravity. In the propagation of ripples, surface-tension (Chap. X.) is much more effective than gravitation is. If T represents the surface- tension, while I represents the wave-length, the pressure per unit area of the surface is proportional to T/Z. Hence the pressure per similar area is proportional to T, for we are not concerned with lengths measured parallel to the ridges of the waves. The similar masses are proportional to p and .to Z", and so we get yu The speed of propagation of a ripple is therefore proportional 7 9 100 A MANUAL OF PHYSICS. directly to the square root of the surface-tension, and inversely to the square root of the product of the density into the wave-length. Thus we see that ripples run faster the smaller they are, while oscillatory waves run faster the larger they are. Hence there is a certain size of wave (about two-thirds of an inch in length in the case of water) which runs slowest. Smaller waves run more quickly because the effect of surface-tension preponderates ; larger waves run more quickly because of the increased effect of gravity. CHAPTEE VII. PROPERTIES OF MATTER. 77. Definitions of Matter. We are now in a position to give one or two provisional definitions of matter provisional, because we cannot yet say, possibly may never be able to say, what matter really is. It may be defined in terms of any of its distinctive characteristics. We may say that Matter is that which possesses Inertia. Or again, since we have no knowledge of energy except in association with matter, we may assert that Matter is the Vehicle of Energy. Another statement (which, from the results of Chap. I. and 62, we see to be an objectionable form of the latter) is that ' matter is that which exerts, or can be acted upon, by force.' Further knowledge would probably make it evident that these three definitions are merely differently worded statements of the same fact. 78. States of Matter. Matter is usually spoken of as existing in three different states the solid, the liquid, and the gaseous. A portion of matter in the solid state possesses a definite form of its own, and considerable force has to be applied in order to produce an appreciable change in the form. When in the liquid state, matter, on the other hand, possesses no definite form of its own, and can be made by application of the slightest force to change whatever form it happens to have. A similar statement holds in the case of a gas or vapour. But a gaseous body differs from a liquid in that its volume is limited only by the volume of the closed vessel in which it is contained ; while the volume of a given quan- tity of liquid is perfectly definite, under given physical conditions, however large the containing vessel may be. Still although these rules are of general applicability it must not be supposed that there is any hard and fast distinction between a solid and a liquid, or between a liquid and a vapour : possibly (but there is no experimental proof of the truth of this statement) there maj" be, 102 A MANUAL OF PHYSICS. under certain physical conditions, no sharp line of demarcation betwet-ji the, solid and.the vaporous states of matter. As the temperature of sealing-wax is gradually raised, the sub- stance slowly passes from the solid into the liquid condition, and, for some time, we cannot strictly call it either a liquid or a solid. The transition from ice to water seems to occur suddenly, but analogy would indicate that the process is really a continuous one. We shall also find (Chap. XXIII.) that the passage from the vaporous to the liquid condition may be made without break of continuity. [A very notable and extremely important example of the non- rigidity of the above distinctions occurs in the case of shoemakers' wax. This substance so far resembles a brittle solid that it will break into splinters under the blow of a hammer ; and yet, under the action of slight long-continued forces, it can be moulded into any shape we please.] The extreme form of the gaseous condition, which is known as the ' ultra-gaseous ' or ' radiant ' state of matter, will be discussed in Chap. XIII. 79. General Properties. Certain properties are common to all portions of matter in whatever state or physical condition they may be, and are, therefore, called general properties. Chief among them is the already-mentioned property of inertia, which requires no further discussion at present. Again, all matter occupies space ; and we consequently say that it has the property of extension. This subject has been considered in Chap. III. But the occupancy of space further involves the idea of form, and so we recognise form as a property of matter. There is little more to be said on this point except in the case of the form of crystallised bodies, to the consideration of which a considerable part of Chap. XII. will be devoted. So far as we know any one portion of matter occupies a given por- tion of space to the utter exclusion of all other matter. This cer- tainly holds in the case of any visible portion. Hence we look upon impenetrability as a general property of matter. But the quality of impenetrability does not interfere with inter-penetration of matter, the possibility of which depends upon the existence of pores in any finite portion of matter. The corresponding property is called porosity. All matter is more or less sponge-like in structure, the space in the so-called ' internal ' pores being in reality external to the material of the body. In the case of substances such as cork, wood, coarse sandstone, etc., the porosity is very evident. The porosity of metals is shown by the fact that gases can pass through them. Thus palladium has a remarkable power of absorbing or PROPERTIES OF MATTER, 103 ' occluding ' hydrogen ; carbonic oxide passes readily through red-hot iron ; and gases formed by the decomposition of electrolytes pass through the metallic electrodes. Bichromate of potassium passes into the pores of glazed earthenware, and, crystallizing inside, gradually breaks up the substance. The porosity of liquids is evidenced by their absorption of gases. Vitreous bodies alone have not yet been directly shown to be porous; but we may fairly conclude that we have not yet found the proper method of testing the point, and that, the proper method being found, they too will prove to be no exception to the rule. A very noteworthy example of interpenetration occurs in the alloying of certain metals, such as tin and copper. The bulk of this alloy is considerably less than the sum of the bulks of its constituents. This phenomenon does not occur when gold and silver are alloyed, and for this reason only was Archimedes' famous test of the impurity of Hiero's crown conclusive. A certain weight of pure gold had been given to a smith for the purpose of making the crown ; but it was suspected that he had abstracted some of the gold, replacing it by an equal weight of silver. Archimedes knew that, weight for weight, silver is bulkier than gold; and hence he concluded that the crown would be bulkier than the given amount of pure gold if silver had been used as an alloy. The problem which he required to solve was therefore the determination of the bulk of a solid which was so irregular in shape that no method of estimation by direct measurement was applicable. He determined this by measuring the volume of the water which the crown displaced. Had contraction taken place, the bulk and weight of the alloy might have been the same as those of the pure gold. Another property of all matter is divisibility. The question of the infinite divisibility of matter will be further alluded to in the chapter on the constitution of matter. In the meantime we are only concerned with examples of extreme division. Many such occur readily. Films of gold and of other metals may be made so thin as to be transparent. A film of gold precipitated by chemical means and burnished so that it forms a continuous sheet may be of no greater thickness than one ten-millionth part of an inch. A quartz -fibre may be made so fine as to be utterly invisible. The vapour from a particle of sodium will tinge a flame continuously of a deep orange colour for hours at a time. A single drop of a strongly coloured liquid will continuously tinge a very large quantity of water, and its presence may be made evident by chemical means long after the eye ceases to detect it. Further examples of the 104 A MANUAL OF PHYSICS. extreme smallness of portions of matter will be given in the chapter on the constitution of matter. All matter is capable of having its volume diminished under pressure to a greater or less extent, and so we speak of its compres- sibility. This subject will receive detailed treatment in Chaps. IX., X., and XI. Again, all matter is deformable. But it is often more convenient to speak of its rigidity than of its deformability, that is, the pro- perty in virtue of which it resists deformation. This question, too, will be discussed subsequently. We shall also afterwards consider more specially elasticity, which, in one or other of its two forms, exists in all kinds of matter ; and viscosity, that is, the property in virtue of which there is resistance to relative motion of the particles of a body. Expansibility will be dealt with in Chap. XXII. Weight which, though a universal property of matter, may be looked upon as a purely accidental property, seeing that it requires the existence of two separate masses in order that it may appear will be fully considered in the chapter on ' Gravitation.' (See also Chap. VI., 64.) 80. Special Properties. Many other properties might be enume- rated which are conspicuously present in some substances, and are as conspicuously absent from others, such as plasticity, ductility, brittleness, tenacity, etc. Again, many properties refer to matter in connection with special forms of energy ; for example, dispersive power, thermal and electric conductivity, magnetic permeability, translucency, opacity, etc. Indeed, all the properties of matter might be naturally investigated in a treatise on energy ; for we have no notion of what might be the properties of matter devoid of energy. Most probably it would not be matter at all in the sense in which we use the word. 81. Specific Properties. Many of the properties of a body depend upon the size of that body. Thus a portion of a given substance is more massive than another portion of the same substance in pro- portion as its bulk is greater than the bulk of that other. It is frequently very essential to define a property of a body in such a way as to make it independent of the size of any particular specimen. For example, the density (specific mass) of a substance is the mass per unit volume of that substance ; the specific gravity is the weight of unit volume, expressed in terms of that of water as the standard ; the specific weight might be defined as the weight per unit volume, expressed in absolute units. We also define rigidity, viscosity, etc. ( 108, 128) as specific properties. PROPERTIES OF MATTER. 105 82. Molecules and* Atoms. A visible portion of any chemically compound substance, e.g., water, may be divided into smaller parts each of which has the same chemical peculiarities as the whole had. These parts may even be so small as to be invisible to the most powerful microscope. But at last a stage is reached where further division cannot occur without the production of substances which are chemically different from the original one. These smallest parts, which are chemically similar to the whole, are called mole- cules. Every molecule can be divided further into what are termed its constituent atoms. These atoms are dissimilar when the molecule is complex like that of water : they are all precisely similar in a simple molecule, such as that of hydrogen. An atom is an indivisible part that is, indivisible by any means at present at our disposal. In all probability, it is really compound. A substance w r hose molecules are composed entirely of similar atoms is called an ' elementary ' substance, and the kind of matter composing it is called a ' chemical element.' We know of the existence in nature of only a comparatively small number of elements. One fact of the utmost importance is this that wherever in space it may be situated, or howsoever it may be circumstanced physi- cally, an elementary molecule or atom has absolutely unalterable properties. A molecule of hydrogen (for example) in the most distant nebula is precisely similar to a molecule of that substance on the earth's surface. We shall return to this point later on. 83. Molecular Forces. A considerable, frequently a very great, amount of work is necessary in order to separate the molecules of a body. When separated, we are accustomed to say that they possess potential energy of molecular separation the increase of potential energy being equivalent to the work spent in producing the separa- tion. If e, the energy, increases by the amount de when the distance s between the molecules is increased by the quantity ds, the work spent is represented also by the quantity de=ds. ds The quantity dejds the space-rate of variation of the energy is called the molecular force against which the work is done (see 29). That is to say, we figure the molecules to ourselves as held together by certain forces which maintain them in their relative positions. The work done in separating two molecules beyond the range of 106 A MANUAL OF PHYSICS. their mutual forces is usually great, and practically the whole of it is performed in an excessively short distance. Hence de/ds is very large ; and so it is said that the molecular forces are extremely power- ful, but that they are insensible at sensible distances. In confirmation of this we may firmly press together two leaden bullets at a part where their surfaces have been freshly cleaned. They will cling together so that the lower one may be lifted by means of the upper, the magnitude of the so-called molecular forces at the surface of contact being sufficient to overcome the gravitational attraction of the whole earth. If there is a film of oxide on the metal the experiment will not succeed, the thickness of the film being too great to allow of appreciable molecular attraction. (For further statements on this subject, see Chap. XII.) Many of the properties of matter may be said to depend on the nature of the molecular forces. Tenacity depends upon the extent to which these forces can overcome external forces which tend to draw the molecules apart. Malleability is a property essentially analogous to the preceding. It is the property in virtue of which a substance may be extended in two directions, while it is contracted in a direction perpendicular to these by the application of great pressure, its volume remaining practically unaltered. In testing tenacity the extension occurs in one direction only, and there is contraction in all directions perpendicular to that one. The brittle- ness of a body is due to the comparative ease with which the mole- cules can be separated beyond the range at which their mutual forces are appreciable. Eigidity depends upon the ability of the molecular forces to resist alteration of the relative positions of the molecules of a body. We therefore recognise two kinds of rigidity rigidity as regards bulk, and rigidity as regards form. Viscosity depends upon the ability of the forces to resist shearing motion. 84. In the immediately succeeding chapters, a more detailed examination of some of the most important of the properties alluded to above will be given. Others will be treated as occasion may arise subsequently. CHAPTEK VIII. GRAVITATION. 85. ALL bodies in the neighbourhood of the earth's surface possess potential energy, which, when circumstances permit, is invariably changed into kinetic energy of motion towards the earth. Hence we say that each body is ' attracted ' to the earth with a force which is termed its ' weight.' Bodies, made of the same material, may be roughly judged to be heav3 r in proportion to their bulk, i.e., in proportion to the quantity of matter contained in them. But we need not rest content with a roughly approximate rule, for the law that weight is proportional to mass is capable of as rigid proof as can be given by the most accurate physical methods. In the first place every body except in so far as the resistance of the air is concerned takes the same time to fall through a given distance. In other words, the acceleration of motion is the same in all cases. Hence, by Newton's Second Law, the force (weight) is proportional to the mass. The fact that the time of oscillation of a simple pendulum is inde- pendent of the mass of the bob furnishes a more readily obtainable 108 A MANUAL OF PHYSICS. proof of the proportionality of weight and mass. Let 9 be the angle through which the pendulum is deflected from the vertical. If w is the weight of the body the force acting in the direction of motion is w sin 9 ; which, for small angles, is practically wQ. This must be equal to the acceleration of momentum, which is ml9, if I is the length of the pendulum. Hence iuQ = mW\ and there- fore ( 51) a and a being constants. This shows that the motion is simple harmonic. And the time of a complete oscillation is 2, v^ since the value of 9 is unaltered if we increase t by this amount. Now experiment shows that this quantity is constant when I is constant. Hence w is proportional to m. Again, a body has the same weight whether its surface is large or small, and whether it is in a single lump or broken up into parts. This proves that the outer parts do not screen the interior parts from the action of gravitation. Indeed, perpetual motion could ensue if this were not so ; for, if matter were placed between one half of a vertically mounted wheel and the earth, the other half of the wheel would be permanently heavier. 86. The truth of Kepler's Laws regarding the motion of a planet would prove that the force of attraction between the planet and the (fixed) sun is in the direction of the line joining their centres, and is inversely proportional to the square of the distance between them. These laws are : I. Each planet moves in an ellipse of which the sun occupies one focus. (The path of a comet may be any conic section.) II. In that ellipse the radius-vector traces out equal areas in equal times. III. The square of the periodic time is proportional to the cube of the mean distance between the planet and the sun. If equal areas are traced out in equal times, the quantity is zero ( 50), which means that the attraction is central, and the magnitude of the central attraction is (43) r-r'0 2 . Now the GRAVITATION.. 109 equation of any conic section referred to a focus as pole is r = I/a (1 -f-e cos 0), a being the semi-axis major, and e being the eccentricity. Hence (writing r-Q = h, from which we a't once find we easily obtain by the methods of Chap. V., r rt? 2 = ah-fr 2 , which shows that the force is attractive and varies inversely as the square of the distance. Finally, since r 2 = h, we have a 2 (l + e cos 9)dO = hdt ; and the integral of this throughout a com- plete revolution, i.e., from 9 = o to 9 = 2?r, is lit = 2rf. But if p be the acceleration at the mean distance, a, so that h 2 = pa, we find 4^% 3 = /* a 3 . And so, from the third law of Kepler, we deduce the result that gravity depends only on the quantity, and not on the quality, of matter ; for any two bodies, having the same mean distance from the sun, would have the same periodic time, provided only that their masses were the same. 87. If, from the above evidence, we now assert Newton's great Law of Gravitation that Evert/ particle of matter in the universe attracts every other particle with a force whose direction is that of the line joining the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their dis- tance from each other, we see that Kepler's Laws cannot be strictly true. In the first place all the bodies in the solar system (including the sun) will revolve about the centre of inertia of the system, which is not necessarily nor actually situated at the sun's centre ; and again, their paths cannot be true ellipses because of mutual attraction. But we know that Kepler's Laws are not strictly true ; and, further, the deviations from them are precisely such as should result from mutual attraction amongst the various planets. Indeed, by assuming the truth of Newton's Laws of Motion and of the Law of Gravitation, Adams and Leverrier were able to predict in- dependently the existence and position of the previously unknown planet Uranus. The law of gravitation is supported by almost as strong proof as any theoretical statement can possibly have. 88. So far as we have gone we have looked upon the sun and the planets as mere material points. The justification of this is con- tained in the latter of two theorems due to Newton : 1, A spherical shell composed of uniform gravitating matter exerts no resultant attraction upon a particle in its interior ; and, 2, it 110 A MANUAL OF PHYSICS.' attracts an external particle as if its whole mass were condensed at its centre. Let A be a point external to the spherical shell PQQ'P', the centre of which is at 0, and the density of which per unit surface is p. Draw a cone, APQ, of infinitely small angle oj, having its vertex at A, and intercepting small surfaces of the sphere at P and Q. These surfaces are equally inclined to APQ, and so the masses which are cut off at P and Q, being equal to wpAP 2 sec OPQ and wpAQ 2 sec OPQ respectively, attract a particle at A equally. Now take an exactly similar cone at AP'Q', equally inclined to, but on FIG. 56. the opposite side of, AO. The elements of the shell at P and Q attract the particle at A equally. Let PQ' intersect AO in R. The position of R is obviously independent of that of P, and hence R is the vertex of a cone, of angle w', say, which intercepts the same elementary areas of the shell at P and Q' as the cones APQ and AP'Q' inter- cept. The masses are therefore /pPR 2 secOPR and w'pQ'R 3 secOPR respectively ; and, since OPR = OAP, their resultant attraction on unit mass at A is 'PR* , Q'] But PR/PA = BR/BA = OP/0 A = Q'R/Q'A, and so the resultant is Hence, summing all such quantities for each similar pair of ele- ments, the whole attraction of the shell is 4?rp OP 2 /OA 2 in the direction AO. And so the proposition is proved, since 4;rOP 2 is the whole surface of the shell. Since the proposition is true of a shell, it is also true of a solid GRAVITATION. Ill sphere, which may be regarded as built up of a number of such shells. It is, therefore, practically true of the planets and of the sun ; for the planets, even although in no case strictly composed of uniformly dense concentric layers, are yet at distances from the sun which are large in comparison with their own dimensions. [When A is inside the shell, P and Q are on opposite sides of it, and so the truth of the first theorem above is established.] The above case furnishes one example of that limited class of bodies which attract, and are attracted by, external bodies, as if their whole mass were condensed at a definite point, called their A Ef FIG. 57. centre of gravity. Another example is given in 317. The centre of gravity, -when it exists, always coincides with the centre of inertia. 89. The second theorem is made use of in Cavendish's method of de- termining the mass (and, consequently, the mean density) of the earth. Two small leaden balls, of mass m, are attached to a light rigid rod or tube, ab, which is attached at its middle point to a vertical wire. The couple required to twist the wire through a given angle is determined by observations upon the time of oscillation of the system. Two large leaden balls of mass M, which were originally in the positions A' B', are placed in the positions A, B. The mutual attractions of the balls A and a, B and b, deflect ab from its 112 A MANUAL OF PHYSICS. normal position, and it oscillates about a new position. The angle through which ab is deflected is determined by means of a beam of light reflected from a mirror which is fastened to the suspending wire. Similar observations are made with the large balls once more in the positions A', B', and finally in the positions A", B", at the same distance as before from ab, but on opposite sides of them, so as to deflect ab in the opposite sense. The mean of the deflections is taken, and the couple required to produce it is known. In this way the attraction between the two masses M and m, their centres being at a given distance r apart, is found ; and it may be compared with the attraction of the earth upon the small ball. The mass of the earth being p, and its radius E, we have &/t/B 3 =M/r 2 . This where p is the mean density of the earth, and Jf is a constant. Professor C. V. Boys has recently succeeded in drawing out extremely fine fibres of quartz, the torsional rigidity of which is so small that, by their means, the mutual attraction between two lead pellets can easily be made manifest. 90. The Schehallien experiment, undertaken with the object of determining the value of p, was of precisely the same nature. The deflection of the bob of a pendulum from the true vertical under the attraction of the mountain Schehallien was obtained by determining the apparent difference of latitude of two places one being situated to the north, and the other to the south of the mountain by means of a pendulum used as a plumb-line. If from this we subtract the true difference of latitude of the places, we get a measure of the attraction of the mountain upon the bob of the pendulum as compared with the attraction of the earth upon it ; for the attraction of the mountain pulls the pendulum from the vertical in such a way as to increase the apparent latitude of the northern point, and to diminish that of the southern. The great objection to this method lies in the fact that our knowledge of the .mass of the mountain is necessarily very imperfect. Yet another method consists in observing the times of oscillation of pendulums of the same length, one of which is situated at the top, and the other at the bottom, of the shaft of a coal-pit. In this way we get a comparison of the attractions of the whole earth, and of the earth minus a shell of thickness equal to the depth of the mine, upon a mass situated at known distances from the centre. Here, again, uncertainty arises, because of our limited knowledge of the density of the earth's crust, which must be assumed to be GRAVITATION. 113 equal to that in the neighbourhood of the shaft. This is called the Harton experiment, because it was performed at the Harton mines. The value of the mean density of the earth obtained by the Schehallien experiment is considerably less than, and the value obtained by the Harton method is considerably greater than, the mean of the (very closely accordant) results obtained by different experimenters who used the Cavendish method. Their results show that the mean density is about 5 '5 times the density of water. Hypotheses Framed to Explain Gravitation. 91. Various attempts at an explanation of gravitation have been made all more or less unsatisfactory. One of the most noted of these is Le Sage's hypothesis of ultra- mundane corpuscles. According to Le Sage, gravitation is due to the bombardment of bodies by numberless small material particles which are darting about in space with great speed in all direc- tions. A single material body placed in space would not be im- pelled in any one direction more than in another, for the corpuscles would batter it equally on all sides. But two bodies in space would be driven together provided, at least, that their distance apart were small in comparison with the free-path ( 148) of the corpuscles for those sides of the bodies which face each other would be shielded to a greater or less extent from the bombardment. If the dimen- sions of the bodies were very much smaller than their distance apart, the force of attraction (the incongruity of which term is evident from the fact that the force is here really one of impulsion) would vary in direct proportion to the cross-sectional area of each of the two bodies as seen from the other. But this is not the gravitational law. In order to obtain it, we must make the supposition that the molecules of matter are so far apart that the number of corpuscles which pass completely through (say) the earth, without striking any part of it, is enormously larger than the number of those which are stopped by it. In this case, every molecule in the interior will be bombarded equally with an exterior particle. The force also will be inversely as the square of the distance, and so we get Newton's Law. In order that the planets should not experience appreciable resistance to their motion around the sun, it is necessary to assume, further, that the speed of the planets is zero relatively to that of the corpuscles. And this indicates one great defect of the hypothesis ; for the energy of the corpuscles which must be spent 8 114 A MANUAL OF PHYSICS. in the maintenance of gravitation would be sufficient, by its transformation into heat, to completely volatilise any material substance of which we have knowledge. 92. The law of gravitation may be worked out into all its conse- quences (at least, so far as our methods avail) without any know- ledge of the mechanism by which it occurs. We require merely to assume that it acts directly at a distance. But, as Newton remarked, no one competent to think correctly on physical matters will be content with this assumption. He himself suggested the rarefaction of the ether in the neighbourhood of dense bodies as a possible explanation. Sir W. Thomson has pointed out a dynamical method of producing this diminution of pressure. An incompressible fluid, filling all space, which is brought into existence, or is annihi- lated, at the surface of every particle of matter at a rate propor- tioned to the mass of that particle, and which is annihilated, or produced, at an infinite distance, at the same total rate would supply the necessary means. .(See 74.) Waves traversing a medium would have the effect of making bodies immersed in it approach each other. The property of dilatancy (Chap. XXXIII.) in a medium com- posed of rigid particles in mutual contact would also account for a gravitational action on bodies placed in it. A certain stress in Maxwell's electro-magnetic medium (Chap. XXXII.) would account for it too. So, as we have seen, would Le Sage's ultra-mundane corpuscles. But, to every provisional hypothesis yet brought forward, some objection, more or less con- clusive, may be advanced. If gravitation be due to action propagated through a material medium, its propagation through finite distances must occupy a finite time. We can assert merely that the speed of propagation is large in comparison with planetary velocities ; for no such modifi- cation of the planetary motions, as would be entailed were it other- wise, is observable. The Nebular Hypothesis. 93. Various nebulae, when examined under sufficiently high mag- nifying power, are seen to be merely groups of stars like our own stellar system. But others cannot be so resolved ; and, among them, there is great variety of constitution. Some appear to be comparatively uniform throughout, while others seem to vary greatly in density at different parts. Others, again, have very dense nuclei with a faint nebulous surrounding. Such facts as these suggested to Laplace his famous Nebulae GRAVITATION. 115 Hypothesis. According to this hypothesis, we see in these nebulae solar systems, such as our own, in various stages of formation. The sun was formed by the mutual gravitation of its parts from a state of diffusion throughout space. And, to account for its rotation, these parts must have had the same amount of moment of momen- tum about an axis as that which the sun at present has. In all probability they had small translational velocity before they began to gravitate towards each other small, that is, relatively to the speed which they would acquire under the action of gravitation ; for, if this were not so the chance of their colliding would be vanishingly small. The collision of these parts would produce great heat, which would result in the production of a nebula, extending beyond the orbit of Neptune, and slowly revolving on its axis. As the nebula gradually shrank, its angular velocity would increase until, by ' centri- fugal force,' a ring of matter was left behind as the body of the nebula still farther shrank. The breaking up of this ring (which would usually occur, as dynamical principles show), and the subsequent agglomeration of its parts, would result in the formation of a planet. And so the development would proceed. Laplace's hypothesis cannot, as he originally stated it, give a, full account of all the phenomena of our solar system ; but this we do know, that some such hypothesis must be true if our sun's heat has had a physical origin such as our present knowledge is adequate to explain. We cannot conceive of any store of energy, sufficient to account for the immense radiation of heat from the sun, except the potential energy of separated masses of gravitating matter. Potential. 94. Since any force / is equal to the space-rate of variation of the kinetic energy of the material system upon which it acts, and since, by the principle of conservation, the change of kinetic energy is equal and opposite to the change of potential energy, we may write ,de dV where V represents the potential energy. The value of the quantity V depends upon the instantaneous position of the material system, and also upon the mutual configura- tion of its various parts. We may define the mutual potential energy of two material bodies in any given relative position, as the amount of work which may be obtained by allowing them to move, under their mutual repulsion, to an infinite distance apart. And, of course, this 82 116 A MANUAL OF PHYSICS. definition implies that we choose the configuration of infinite distance as the configuration of zero potential energy. The Potential at any point, due to any given distribution of matter, is the mutual potential energy between that matter and a unit of matter placed at the given point. To find its value we have only to multiply the force at any point of the path, along which the unit of matter is repelled, by the infinitesimal element, ds, of the path, and to sum all such quantities from the given posi- tion to an infinite distance. This quantity is represented by the symbol the meaning of which is that we are to find the general value of the integral oifds ; to replace in it, first, s by oo ; second, s by its actual value at the given point ; and, finally, to subtract the latter quan- tity, so found, from the former. This makes the work which is done depend only upon the initial and the final positions of the unit of matter a condition which must be satisfied if the forces are consistent with the principle of conservation. For, if the work depended upon the path along which the matter was repelled, we might cause it to return, by frictionless constraint, to its former position, by a path which necessitates an expenditure of less work than that which was done upon it by repelling forces. In such a case there would consequently be a continual gain of energy without any corresponding expenditure. Let us suppose, for example, that the mutual force is repulsive and inversely proportional to the square of the distance, s, between the two portions of matter. Let s increase from s to -s'. The work which is done is where m is the mass of the repelling matter, and we assume that the law of repulsion is similar to Newton's law of attraction, and define unit force as the force between two unit masses placed at unit distance apart. The general value of the integral is ( 35, example (3) ) and the work is therefore GRAVITATION. 117 If we now put s' = oo , the second term disappears and we see that the value of the potential is and the repulsive force is 95. Gravitational Potential. When the force is attractive, as in the case of gravitation, the potential denned as above becomes V=--- s The potential is therefore essentially negative ; that is to say, work is done against the forces when the distance between the particles is increased. But it must carefully be observed that the fact of the potential energy being negative, at distances less than infinite, is due entirely to our having, for convenience, chosen infinite distance apart as the configuration of zero potential. The increase of potential energy is positive as we pass towards infinity. Still, to avoid the inconvenience of defining the gravitational potential as a negative quantity at all finite distances, it is preferable to change the sign and write, in this case also, y_ m s dV where / is to be understood as signifying inward or attractive force. The potential V, therefore, does not represent the mutual potential energy. It is the exhaustion of potential energy which occurs when the unit of matter passes from infinity, under the attractive forces, to the position s. The potential at any point, due to a number of separate masses, is simply the sum of the separate potentials due to each mass. 96. Equipotential Surfaces. Lines and Tubes of Force. Any surface over which V is constant is called an Equipotential Surface ; and a line, the tangent to which at any point is always in the direction of the force at that point, is called a Line of Force. Each line of force is a possible path along which a material particle would move under the given forces. Since dV is zero as we pass from one point to another point of an equipotential surface, it follows that the force at any point of such a surface has no component along it. In other words, the lines of force are everywhere perpendicular to the equipotential surfaces. 118 A MANUAL OF PHYSICS., No two different equipotential surfaces can intersect with one another ; for this would imply that a finite change of potential would follow an infinitely small displacement of a material particle, i.e., the force would be infinite at any such intersection, and no examples of infinite forces occur in nature. If, through every point of an infinitely small closed curve drawn on an equipotential surface, we draw lines of force, a tube of infinitely small section will be formed. Such a tube is called a Tube of Force. Necessarily, the lines and tubes of force can originate only at a point where matter is situated ; for the force owes its origin to the presence of matter. And if, at any point of space, a force f exists, we may draw an infinitesimal tube of force, so as to con- tain that point and to enclose f lines of force per unit area of its normal section. The number of the lines of force per unit of sectional area therefore indicate the intensity of the force at the given point. If a given portion of a tube does not contain any matter, the number of lines which it contains remains constant, since no line can end in its interior, and none can pass out through its sides. This gives where f is the force at any point of the tube at which the sectional area is a, and c is a constant, numerically equal to the number of lines which the tube contains. 97. Special Applications. (1) Let the attracting body be sym- metrical about a point. In this case the tubes of force are cones, and the area of any normal section of each of them is proportional to the square of its distance from the point of symmetry. Hence the above equation shows that the force at any point is inversely pro- portional to the square of the distance of that point from the point of symmetry, say (2) Let the attracting matter be symmetrically arranged about an infinitely long straight axis. The equipotential surfaces are concentric cylinders whose common axis coincides with the axis of symmetry, and the tubes of force are wedges bounded by axial planes. The section is proportional to the distance from the axis, and therefore the force is inversely proportional to that distance, i.e., GRAVITATION. 119 (3) Let the matter be arranged homogeneously in infinite parallel planes. The equipotential surfaces are planes parallel to these, and the tubes of force are cylinders arranged perpendicularly to the planes. Consequently the force is constant at all distances, say 98. Total Force over a Closed Surface. Draw any closed surface S (Fig. 58), and, (1) Let m be a massive particle, outside the surface, to which the force is due. Draw any infinitesimal tube of force, mnp, cutting the surface at n and^?. (Of course, it may cut the surface in any even number of places.) The total force over the portion of the surface intercepted by the tube at n, is equal to that over the portion intercepted at p ; but, in the one case, the force is directed outwards over the surface, while, in the other, it is directed inwards. Hence, by consideration of an infinite number of such FIG. 58. tubes of force intersecting all parts of the closed surface, we see that the total inward force over the whole closed surface, due to a material particle situated at an external point, is zero. [The use of the word inward, of course, implies that the force is to be reckoned as negative when it is outwardly directed.] The same proof applies in the case of any number of material particles. (2) Let the particle, of mass m, be placed within the closed surface. Draw a sphere, with unit radius, from the point m as centre. The area of this spherical surface is 47r, and the force at any point of it, due to the attraction of the central particle, is equal to m. The total inward force is therefore 4-n-m. But, by the result of 96, the total inward force over this surface is equal to that over the given surface S. Hence, if m be the whole amount of matter contained within S, the total inward force over the whole closed surface, due to matter of amount m enclosed within it, is 120 A MANUAL OF PHYSICS. These results may be symbolised thus, /NdS = or 4tirm, where N rep esents the number of lines of force which cross the surface S per unit of area in the part where the element of surface dS is taken. 99. Special Applications. We may apply these results to the determination of the values of the constants a, b, and c, in 97. In example (1) of that section, the whole number of lines of force which cross any closed surface surrounding the spherical distribution of attracting matter is 47rm, where m is the whole amount of matter. Hence, if we suppose the enclosing surface to be a concentric sphere of radius s, we get as the total force Therefore Similarly, if, in example (2), we consider the force due to the matter m contained in unit of length of the cylindrical distribution, we get for 27TS is the area of unit of length of a concentric cylinder of radius s. Therefore b = 2m. Again, if (example (3) ) we draw a right circular cylinder of unit radius, perpendicular to the infinite planes, and close its ends by parallel planes on opposite sides of the given plane distribution of matter, the force exerted over the two ends is 27r/= 2?rc = so that c = 2w. In particular, if the given distribution consists of an infinitely thin plane layer, of infinite extent, and of finite surface density a, the force at any point is / = 2m = 2. [Such a distribution cannot occur in the case of gravitational matter, but the problem has a direct application in the theory of electricity.] It follows that the force at any point just outside a surface on which matter is distributed is normal to the surface and GRAVITATION. 121 equal to 2;r(r provided that the surface is of finite curvature ; for the point may be taken so close to the surface that, as seen from it, the surface is practically an infinite plane ; that is to say, any infinitely small portion of the surface is plane, and the point may be taken infinitely close to this portion in comparison with its dimensions, infinitely small though they be. . 100. In the case just considered, the normal force at the two sides of the surface are respectively /= 27r- = 3? + 7/2 + z i . whence drjdx = xjr, dr/dy = yjr, drjdz = z/r, for, since x, y, and , 122 A MANUAL OF PHYSICS. are independent variables ( 28), we must assume y and z to be constant when x varies, and so on. Therefore we get dV4 dr4 #V = 4 And, similarly, - 1- = ; whence 4 Around any point in space throughout which matter is distributed with density p, describe a sphere which is so small that the density of the matter which it' contains is sensibly constant. We may suppose all the attracting matter to consist of two portions that which is within the little sphere, and that which is external to it. We may divide the whole potential V into two parts, Vi and V 2 , of which the former is due to the sphere and the latter is due to the matter external to the sphere. Vi therefore satisfies the equation p being the density of the matter which produces the potential Vi at the given point. But the matter outside the sphere does not contribute to the density at the given point within. That is, at the given point, the density of the matter which produces the potential V 2 is zero, and therefore \dx' 2 + -VV = FIG. 59. Consequently, by addition, we get quite generally d*V + d^\_ + ~*~ GRAVITATION. 123 This shows us how to distribute matter throughout space, so as to produce a given continuous distribution of potential. The above result may be obtained in a totally different manner, which is even simpler, and which will make the meaning of the equation more evident. Take three rectangular axes at any point o (Fig. 59), and draw a little parallelepiped at this point with its edges parallel to the a?, y, and z axes ; and let the lengths of the edges be 8x, dy, Sz, respectively. Eesolve the forces in the neighbourhood of the parallelepiped into their components parallel to the axes. This will not alter the result of 98 ; and we may now draw the lines of force perpendicular to the various small surfaces. Let n f be the number of lines of force which cross unit area of the face of the parallelepiped which passes through the origin and is perpendicular to the x axis. The total number of lines which cross that face is therefore n^yd*, since Sydz is the area of the face. Similarly the number which cross the parallel face at the distance dx from the former is Now the lines which cross the former face are due entirely to matter outside the little volume, and they therefore cross the parallel face also. Hence the total number of lines which enter the little volume from without by the two faces is the difference of these two quantities. By similar reasoning for the other pairs of faces we find that the total number of lines which enter the little volume (that is, the excess of those which enter over those which leave) is (dn x + dn y + dn. \ dx dy dz But, by 98, this is equal to ^TrpdxSySz, where p is the density of the matter contained in the volume dttfyfy. Therefore dn r . dn v . dn x A -^ + -4 + -^=^- And the result is independent of the size of the little volume ; so that the meaning of the equation is that the volume density at any point of space is l/4w times the number of lines of force which originate, per unit of volume, at that point. 124 A MANUAL OF PHYSICS. , The following proposition is also of great importance. It is possible so to distribute matter over a given surf ace, ivhich encloses a given mass, as to produce, outside that surface, the same potential as that which the given mass produces. The mathematical proof of this proposition cannot be introduced here, but an experimental proof will be given in the chapter on electrostatics. 101. The calculation of the distribution of potential which is produced by a given distribution of matter is extremely difficult or even impossible in most cases ; but the beautiful method of electric images, due to Sir W. Thomson, enables us to deduce with great ease the solution of many unknown problems from the known solution of others. The further discussion of this subject may be left until we treat of the subject of electricity. CHAPTEB IX. PROPERTIES OF GA-SES. 102. Compressibility. Throughout this investigation we assume that the temperature of the gas remains constant. The effects which result from changes of temperature will be more conveniently treated in the chapter on the effects of heat. All gases are compressible ; that is, their volume can be diminished by the application of pressure. We shall see afterwards that sound could not pass at a finite rate through a gas which was not compres- sible. So that the mere fact that gases can convey sound constitutes a proof of their compressibility. 103. Boyle's Law. The law which very completely, though not with absolute accuracy, represents the relation between the pressure and the volume of air (and many other gases) was discovered experimentally by Boyle, who showed that the density of a gas is directly proportional to the pressure. In symbols, p being the density and p the pressure, this is p = cp t c being a constant. The density, that is the mass or quantity of matter in unit volume, is numerically equal to the reciprocal of the volume containing unit mass. Hence, v being this volume, we may write, instead of the above, pv = c, where the constant c is the reciprocal of the former one; or, in words, the volume is inversely proportional to the pressure. Boyle's apparatus consisted of a glass U-tube (Fig. 60) with a long and a short limb. The long limb was open to the atmosphere, while the short one was closed, and contained a quantity of air, which was separated, by means of mercury filling the bend of the tube, from the outside air. The level of the mercury was the same in both limbs of the tube, and so the enclosed air (the volume of which was carefully noted) was at atmospheric pressure (75). Mercury was then poured into the open limb, until a difference of level equal to 126 A MANUAL OF PHYSICS. a. the height of the mercury barometer was established. The air inside was therefore under a pressure of two atmospheres, and its volume was found to have been halved ; and so on with other values of the pressure. A slight modification of the. apparatus enables us to prove the law under diminished pressure. AB (Fig. 61) is a vessel containing mercury. The glass tube a&, which is closed at the end a, but is open at 6, is filled with mercury, and inverted in AB. Being shorter than the height of the mercury barometer, the tube ab remains filled. Air or any other gas may now be introduced into it until the mercury inside is at the same level as that outside. Under these conditions the gas in ab is under atmospheric pressure. If ab be raised, the mercury in it stands at a higher level than that outside, and the gas expands, since it is under diminished *^jj g pressure. The ideal gas which rigidly obeys Boyle's Law is called a perfect gas. FIG. 60. FIG. 61. 104. Compressibility of a Perfect Gas. One gas is more compres- sible than another in direct proportion to the alteration of volume produced by a given pressure, and in inverse proportion to the pressure required to change the volume to a given extent. Hence we measure the compressibility by the ratio of the percentage change of volume to the change of pressure which produces it. That is to say, if the volume V changes by the quantity v when the pressure alters by the amount p, the compressibility is measured by the ratio vfVp. By Boyle's Law we have and Therefore pV - ~Pv = 0, since we can neglect pi), which is the product of two small quantities. This gives v 1 that is, the compressibility of a perfect gas is inversely proportional to the pressure. 105. Deviations from Boyle's Laiv. Though no gas is perfect, yet many gases do not greatly deviate from Boyle's Law throughout a considerable range of pressure. Air is more compressed than it should be in accordance with the PROPERTIES OF GASES. 127 law until a pressure of nearly one ton's weight on the square inch is reached. After this point, the compression is less than the calculated value. A reason for this is simply that the volume of the gas is not capable of indefinite decrease, while Boyle's Law asserts that under infinite pressure the volume will become zero. Hydrogen, unlike air, is, at ordinary temperatures, always less compressible than the law indicates. Nitrogen, along with many other gases, resembles air. These results are exhibited graphically in Fig. 62. The actual [FiG. 62. volume of the gas is measured along the vertical axis, while the volume of a perfect gas under the same pressure is measured along the horizontal axis. The straight line passing upwards through the origin at an angle of 45 is obviously the graph for a perfect 128 A MANUAL OF PHYSICS. PROPERTIES OF GASES. 129 gas. "The curved line intersecting the perfect gas-line, and, like it, sloping upwards towards the right, represents the action of air ; and the other curve, sloping similar!}-, is the graph for hydrogen. 106. Compression of Vapours. A vapour, though it may obey Boyle's Law throughout a considerable range of pressure, ulti- mately deviates more and more from that law as the pressure rises. The direction of the deviation is similar to that of air at pressures less than 152 atmospheres, i.e., the vapour is more compressed than a perfect gas would be. When the pressure has become sufficiently great, the vapour begins to liquefy ; and the pressure then remains constant until the whole has become liquid. Further compression is comparatively a matter of extreme difficulty. The whole process above described must take place with extreme slowness in order that the condition ( 102) of constant temperature may be adhered to. We may now suppose the temperature to be increased to, and maintained at, a definite value higher than that which it formerly had. If the pressure has the same value as it had at the commencement of the former process, the volume of the vapour will be greater than before, for all vapours expand when heated under constant pressure. And, if the pressure be increased as in the previous case, a precisely similar series of phenomena will be presented; the volume of the substance, however, being always larger than formerly under the same conditions as regards pressure. But one important difference will be noted the change of volume during the process of liquefaction will be less than it was when the temperature was lower. Ultimately, when the temperature is sufficiently high, there is no sudden change of volume when the substance assumes the liquid condition. At still higher temperatures, the deviations from Boyle's Law become less and less marked. At all temperatures above the limiting one at which sudden lique- faction ceases, the substance is called a gas ; at lower temperatures it is termed a vapour. With this explanation, no difficulty will be experienced in under- standing the diagram on the opposite page. The volume, v, of a per- fect gas is measured along the horizontal axis from a point not shown in the diagram. The scale is such that 1,000 times the reciprocal of the abscissa represents the pressure in atmospheres. The actual volume, v', of carbonic acid gas is measured along the vertical axis. In this way a series of curves are shown which indicate the devia- tion of that gas from Boyle's Law at various temperatures. Portions of two of the nearly straight lines, which these curves would become if the gas were air, are drawn. The vertical portions of two of the 9 130 A MANUAL OF PHYSICS. curves indicate the stage during which liquefaction occurs the almost horizontal parts belong to the liquid carbonic acid. The substance is more or less compressible under given conditions of pressure and temperature than a perfect gas is, according as a line touching the curve (at the point satisfying these conditions) makes a greater or less angle with the horizontal axis than the angle whose tangent is v'jv times the tangent of the angle made by the corresponding perfect-gas line. Hence the substance, when in a condition resembling the liquid state, is less compressible than a perfect gas would be ; and we thus see that hydrogen is less com- pressible than a perfect gas, because, under ordinary conditions of temperature and pressure, it is in a state more analogous to that of liquids than to that of gases. Were it examined under conditions similar to those of carbonic acid in the upper right-hand region of the diagram opposite, i.e., under sufficiently diminished temperature and pressure, it would almost certainly be found to have a smaller volume than Boyle's Law shows. 107. Elasticity. All gases and vapours possess perfect elasticity of bulk. That is to say, they entirely recover their original bulk when allowed to do so by means of the removal of the distorting pressure. This may readily be proved by the simple experiment of inverting in water a glass vessel containing air or any gas which is not appreciably dissolved by the liquid. The gas may be subjected to, or relieved from, pressure by raising or lowering the glass vessel. The possibility of discharging a bullet from a gun, or of propelling a vessel or driving machinery by means of compressed gases furnishes another proof of their elasticity. And still another proof consists in the fact that they all convey sound, which would be impossible were they not elastic, just as it would be impossible if they were not capable of being compressed. 108. Viscosity. We have already given a general definition of this term as the property in virtue of which there is resistance to shearing motion. But it is convenient to use the word as referring to a specific property (one independent of the size of the body, 81). Hence we define viscosity as the tangential force per unit area of two indefinitely large parallel plane surfaces of the fluid which are at unit distance apart and move parallel to each other with unit relative speed. It follows that the tangential force per unit area of two such planes at a distance x apart, and moving with relative speed v, is rv/x, where T is the viscosity. But, in shearing motion v is always proportional to x, so that the tangential force is rdvldx. In making an actual determination of the value of r in any gas, PROPERTIES OF GASES. 131 various forms of experiment based upon the above definition might be used. Clerk - Maxwell used a circular disc which vibrated torsionally about a perpendicular axis through its centre. Two similar fixed discs were placed one on each side of the vibrating disc, and the gas occupied the intervening space. The disc would obviously oscillate more slowly in a viscous gas than in one which possessed small viscosity ; and the quantity r may be determined from the results of such experiments. The mathematical investiga- tion is somewhat more difficult than we can venture to introduce into an elementary work. This property varies very much from one gas to another. In hydrogen, carbonic acid, air, and oxygen, it increases from the first- mentioned to the last-mentioned, being about half as great in hydrogen as in air. It increases very markedly also with rise of temperature. The slow descent of clouds, or of fine suspended dust, in air is due to the viscosity of that gas. The weight of a drop of water, which causes its descent, is proportional to the cube of its diameter ; but the resistance which results from viscosity is proportional only to the first power of the diameter. Hence, if the diameter of a drop be reduced to one -tenth of its original value, the weight becomes one-thousandth of what it was before, while the resistance is merely reduced to one-tenth of its previous amount. That is to say, the re- sistance is relatively one hundred times more effective than formerly. 109. Diffusion. When two gases, which are not intimately mixed, occupy a certain volume, each gradually diffuses itself through- out the whole volume, so as to fill it just as it would have done had the other been absent. The only effect of the presence of the other gas (on the presumption that there is nothing of the nature of chemical action between them) is that the time taken by the first to uniformly fill the space is greatly increased. The above process is called diffusion, and the corresponding property is diffusivity. Experiment shows that the quantity of gas which passes in time t through -an area &, perpendicular to which the rate of variation of density per unit of length is r, is proportional conjointly to t, a, and r. Hence, if q be this quantity, we have q = Srta, where 8 is a constant (the diffusivity, or co-efficient of inter-diffu- sion) the magnitude of which depends upon the nature of the gases. If r, t, and a are each unity, we get q = t, and so we define the diffusivity as the quantity of tlic substance 92 132 A MANUAL OF PHYSICS. . which passes per unit of time through unit area across which the rate of variation of density of the substance per unit length is unity. The quantity r is generally called the ' concentration- gradient,' and may be written in the form dpjdx, where p is the density and x is measured along the line drawn in the direction in which the diffusion is taking place. We shall find afterwards that the kinetic theory of gases leads to the conclusion that the co-efficient of interdiffusion of gases should be approximately in inverse proportion to the geometrical mean of the densities of the two gases under one atmosphere of pressure. The figures in the first column of the accompanying table give relative values of $ for pairs of gases, the relative values of the reciprocals of the geometrical means of which are given in the second column : Carbonic Acid and Air ... ... 1 ... 1. Carbonic Acid and Carbonic Oxide ... 1 ... 1. Carbonic Acid and Hydrogen ... 3'9 ... 3'8. Carbonic Oxide and Hydrogen ... 4*6 ... 4*8. Oxygen and Hydrogen ... ... 5*2 ... 4*5. The greatest deviation occurs with oxygen and hydrogen. This is probably due to molecular action between these gases. 110. Effusion. The phenomena presented in the passage of gases through the pores of solids are of great interest, and have been elaborately investigated by Graham. The simplest results are obtained when the solid is practically of .infinite thinness and is non-porous, but has a small hole drilled through it. If a gas is kept under constant pressure at one side of the solid, while a vacuum is preserved at the other side, the process of passage of the gas is called effusion. The theoretical treatment of the question is extremely simple. The work done in the transference of unit volume of the gas is ( 62) numerically equal to p, the pressure ; for the total pressure on unit area then acts through unit distance. And the kinetic energy acquired is ^-pv 2 , where p is the density or mass per unit volume, and v is the speed of the escaping gas. Hence the speed of escape, and therefore the quantity of the substance which passes through in one unit of time, is inversely as the square root of the density. The observed and calculated quantities for four substances are given in the subjoined table : Observed. Calculated. Carbonic Acid 0'835 ... 0-809. Oxygen 0'952 ... 0-951. Air 1 ... 1. Hydrogen 3'623 ... 3'802. PROPERTIES OF GASES. 133 It appears that hydrogen and carbonic acid pass through more rapidly and more slowly respectively than the above law would indicate. The reason for this will be seen in next section. 111. Transpiration. When the non-porous septum, above referred to, is not thin, the small aperture becomes a tube of exceedingly fine bore, and the gas passes through by transpiration. Graham found that the rate of passage was altogether independent of the nature of the substance forming the walls of the tube. This suggests that a layer of the gas becomes deposited upon the interior of the tube, so that the gas has really to flow through a tube composed of its own substance in a highly condensed state. [It is well known indeed that most, and probably all, solids have a great power of condensing gases on their surfaces or within their pores.] Hence we would expect that transpiration is a process which depends upon the viscosity of the gas. This is borne out by the fact that the rates of transpiration of oxygen, air, carbonic acid, and hydrogen are, in increasing magnitude, in the order in which these gases are here named, being fully twice as great in hydrogen as in air an order which is the exact reverse of their order as regards viscosity. Hence we conclude that the abnormality in the rates of effusion of hydrogen and carbonic acid was due to viscosity, the hole in the thin plate acting to some extent as a short tube. 112. When the pores of a substance through which a gas passes are extremely fine (as in fine unglazed earthenware), the rate of passage follows the ordinary law of diffusion or effusion, i.e., gases pass through at rates which are inversely as the square roots of their densities. Hence we have a means known as the method of Atmolysis by which to separate a mixture of two gases of different densities. If the mixture be placed inside a porous earthenware vessel, the less dense gas passes through most quickly, so that, when the process has gone on for some time, we have two portions of gas, one containing in most part the less dense gas, the other composed mostly of the denser one. The process may be re-applied so as to separate the two constituents to any desired extent. It has been already mentioned ( 79) that carbonic oxide passes rapidly through red-hot iron ; and hydrogen passes through palla- dium, and even platinum, at ordinary temperatures. In some cases the gas combines chemically with the substance on one side, diffuses through it, and is given off on the other side. This occurs with india-rubber. CHAPTEK X. PROPERTIES OF LIQUIDS. 113. Compressibility . Liquids, like gases, convey sound, and are therefore compressible and elastic. But they differ from gases, in that their compressibility is usually extremely small. They differ, also, as widely in respect of the law of compression. An inspection of the diagram of 278 will show that a vapour such as carbonic acid becomes more and more compressible as it approaches the liquefy- ing stage, while, during liquefaction, the compressibility is infinite. The change is in the opposite direction when the whole substance has become liquid ; the compressibility is extremely small, and diminishes as the pressure increases. For example, the right-hand portion of the isothermal of 21'5 is practically a straight line, and therefore the quantity dvldp is constant. But the compressibility is dvjvdp ; and v diminishes as the liquefying stage is approached, so that the compressibility increases. Similar reasoning proves the above statement regarding the liquid condition. The earlier determinations of the compressibility of liquids were made by means of an apparatus called the piezometer, and the more perfect modern appliances all work on the same principle. This apparatus consists of a large glass bulb, having a narrow care- fully-graduated stem, which is open at the top. The internal volumes of the bulb and of the stem are accurately measured. The liquid, whose compressibility is to be determined, fills the bulb and part of the stem. A small column of mercury suffices to separate the liquid inside the bulb from water which fills a strong glass vessel, inside of which the bulb is placed. The outer vessel is closed, and pressure is applied by screwing in a plug so as to diminish the internal volume. The pressure is communicated to the liquid inside the bulb, since there is a complete liquid connection through the stem ; and its amount may be measured by means of the compression of air con- PROPERTIES OF LIQUIDS. 135 tained in a glass tube, which is closed at the upper end, and is also placed inside the outer vessel. If the glass were incompressible, the compression of the liquid would be at once found by means of the extent to which the mercury index descended in the stem. If the liquid were incom- pressible, while the glass was capable of compression, the index would rise. If the liquid and the glass were equally compressible, the position of the mercury would not alter. Hence we see that this experiment really gives the difference between the compressions of the glass and the liquid, so that we must first know by experi- ment the compressibility of the glass of which the bulb is formed. This, as we shall see in the next chapter, is a comparatively simple matter. Water is compressed by about one twenty-thousandth part of its bulk per atmosphere of pressure added. Unlike all other liquids hitherto observed, its compressibility diminishes when its tempera- ture is raised, a minimum being reached about 63 C. The compressibility of all liquids is lessened by increase of pressure. 114. Elasticity. All liquids, like gases, possess perfect elasticity of bulk, and, in common also with gases, have no elasticity of form. 115. Viscosity and Viscidity. Viscosity is very much more marked in liquids than in gases, and varies greatly from one liquid to another. The slowness of the descent of fine mud in water is due to the viscosity of that liquid, and the slowness of the fall of fine rain-drops is caused by the viscosity of air. Glycerine is one example of an extremely viscous liquid, while sulphuric ether has little viscosity in comparison with it. One extremely simple method of determining the viscosity of a liquid consists in forcing it under pressure through a cylindrical tube of very fine bore. The quantity which passes through per unit time is directly proportional to the difference of pressure per unit length of the tube and to the fourth power of the radius, and is inversely as the co-efficient of viscosity. Viscosity diminishes rapidly with increase of temperature. Viscidity is a related property in virtue of which a liquid can be drawn out into long threads. Other things being equal, a liquid is viscid in proportion to its viscosity ; but the molecular forces pro- duce another effect besides viscosity, which acts so as to prevent viscidity ( 125). 116. Diffusion. Under the diffusion of liquids we include the diffusion of solutions of solids. 136 A MANUAL OF PHYSICS. Diffusion of liquids is a very much slower process than diffusion of gases. If a solution of bichromate of potassium be carefully introduced at the foot of a vessel containing water, the process of interdiifusion may go on for months before appreciable uniformity is attained. Many methods (electrical, optical, etc.) exist, by means of which the co-efficient of interdiffusion (the definition of which is identical with that given in 110) may be determined. In one method, due to Graham, communication is established between two vessels, each of which contains a liquid capable of diffusing into that contained by the other. Special care is taken to avoid the production of currents whether in the act of establishing communication or because of difference of density of the liquids. The communication is closed after a definite time, and the extent to which diffusion has gone on is determined. A series of precisely similar experiments is made, each experiment of the series lasting for a different interval of time, and the diffusivity is determined from the results. 117. Osmose. Dialysis. Diffusion of liquids can take place through animal membrane, such as a piece of bladder. The less dense liquid passes through most quickly. If a vessel containing a strong solution of sugar be closed tightly by means of a membranous substance, and then be immersed in a vessel of water, the contents of the inner vessel will rapidly increase, and may finally cause the membrane to break. The process of such transference is called osmose. Liquids may be broadly divided into two classes with reference to the readiness with which they pass through animal membranes. Crystalloid substances, such as common salt, sugar, etc., pass easily through when in solution ; but solutions of colloid substances, such as glue, can scarcely pass at all. This is the basis of the process of dialysis, which is used for the separation of a mixture of colloid and crystalloid bodies. The mixture is separated from pure water by a portion of animal membrane, through which the substances pass in very disproportionate quantities. One or two repetitions of the process are sufficient to practically separate the two constituents of the mixture. The method is essentially analogous to the method of atmolysis, which was described in 112. 118. Cohesion. Cohesion is that property which, apart from the mere gravitation of the parts as a whole, results in the clinging together of portions of matter whether of the same or of unlike kinds. It may be regarded as a result of the so-called molecular PROPERTIES OF LIQUIDS. 137 forces. (See, again, 83.) When a body has been pounded down, so that its parts have been separated beyond the range of the mole- cular forces, cohesion may be brought about again by the application of pressure sufficient to place the molecules once more within the range of their mutual forces. In the case of a liquid, it is sufficient to merely place the separated parts in contact. (For further treat- ment, see under Properties of Solids.) 119. Capillarity. It is a well-known law of hydrostatics that the pressure ( 75) has the same value at all points of a fluid which are at the same level, so that we should expect that the level must be the same at all surfaces of a continuous fluid mass which are exposed to the atmosphere. Yet, in some cases, this is far from being the fact. If a fine capillary tube be inserted in some liquids, the leve is higher inside the tube than it is outside ; while in other liquids FIG. 63. the reverse is the case (Fig. 63). Thus water rises inside a glass tube, while mercury descends. These phenomena (called capillary phenomena) seem to be in direct violation of the above-mentioned law of hydrostatics, but in reality they are in strict accordance with it. 120. In proof of this we observe, first, that the surface of a liquid which rises in a capillary is always concave upwards, while the surface of one which descends is invariably convex upwards. Next, we observe that, if a surface, originally plane, is under tension and is curved, there must be more pressure on the concave than on the convex side. Otherwise, the surface would once more become plane, because of its tendency to shrink. Hence, if we can show that there is tension in the surface films of liquids, it follows that there is more pressure on the concave side than on the convex side of the curved surface of a liquid. 138 A MANUAL OP PHYSICS. But it is well known that the surfaces of liquids tend to become as small as possible. Many examples of this fact may be brought for- ward. A soap-bubble contracts of itself if the air inside it be in communication with that outside ; and the mere fact that the soap-bubble is naturally spherical constitutes another proof, for the sphere is the minimum surface which can enclose a given volume. For the same reason rain- drops are spherical which is proved by the perfect circularity and definiteness of the rainbow'. Again, if some alcohol be dropped on a thin layer of ink, the surface of the ink will decrease, while that of the alcohol is increased, because of the excess of the surface-tension of ink over that of alcohol. Let us suppose, now, that the surface of a liquid in a narrow tube becomes hollow upwards. Just underneath the outside plane surface there is atmospheric pressure, while just below the curved surface the pressure is less. At some point, p (Fig. 64), lower down in the tube FIG. 64. FIG. 65. the pressure (which is there increased because of the weight of the liquid) is equal to that of the atmosphere. Hence, in strict accord- ance with hydrostatic laws, the liquid must rise until the point p is at the level of the surface of the liquid outside. Similar considerations explain the depression, in a narrow tube, of a surface which is convex upwards. It only remains to explain why the surface becomes curved. We shall assume that the tube is of glass, that the liquid is water, and that the surrounding atmosphere is air. The surface of the liquid in contact with the glass is under tension, the amount of which per unit breadth of the film we may denote by K T g . There is also a film of condensed gas on the surface of the glass, the tension of which per- unit breadth we may similarly represent by T r The particles of the liquid at the edge will therefore be pulled upwards or PROPERTIES OF LIQUIDS. 139 downwards according us e T^ is greater than, or less than, .T,. In the case assumed, T^ is greater than ..T^, and so the water surface becomes concave upwards. The tension T W of the water surface which is in contact with the air which formerly acted straight outwards from the walls of the tube now acts downwards at an angle a with the side of the tube (Fig. 65). So we have now a total downward tension JS g + JT W cos a per unit breadth, and equilibrium will ensue when .Tj + .T. cos a=Jf g . 121. In the case of water this equation is not satisfied even when a vanishes, for .T, is greater than the sum of ,1, and .T, ; and so the surface of water in a narrow tube is hemispherical. It is very essential to keep the surface of the liquid free from impurity, and to ensure that the surface of the solid is chemically clean. The slightest trace of grease might entirely prevent the liquid from rising. The angle , which is called the angle of contact, may be found experimentally by the following method. Let AB (Fig. 66) be a plane plate of the glass (or other solid), and let it be dipped into the FIG. 66. liquid CD. If the liquid rises and wets the solid, making an acute angle a with it, it is evident that when AB is inclined at the angle a to CD, the level of the liquid is unaltered at that side of the plate which faces upwards. We may now find a by direct measurement. The figure, if turned upside down, corresponds to the case of a liquid which descends in a capillary tube. 122. We can now determine the height to which a liquid will rise in a tube of given bore. Let r be the radius of the tube, and let T be the tension per unit breadth of the surface separating the liquid from the air, while a is the angle of contact. T cos a is the 140 A MANUAL OF PHYSICS. upward pull per unit breadth, and hence 2;rr T cos a is the total upward pull. This is balanced by the weight of the raised liquid. Therefore, h being the mean height to which the liquid rises over the outside level, while p is the density of the liquid, and g is the value of gravity, we have 2?rr T cos a = 7rr 2 hpg. This gives h= 2T cos a pgr Hence the height is inversely proportional to the radius of the tube. When the liquid rises between two parallel plates of breadth b. placed at a distance d apart, the above equation becomes 26T cos a = bdhpg. So in this case h= 2T cos a - pgd that is, while the law is the same as formerly, the height will only be the same when the distance between the plates is equal to the radius of the circular tube. We can determine the value of T by either of the above methods, provided we know that of . The height to which the liquid rises is inversely proportional to d. And since, if the plates be not parallel, but be placed in contact along one of their vertical edges, d will be proportional to the distance from the common edge, the liquid, rising highest where d is smallest, will meet each plate in a curve which is a rectangular hyperbola. The axes of the hyperbolas will coincide respectively with the common edge, and with the lines in which the level surface of the liquid meets the plates. 123. The results of 120 enable us to explain the strong ' attrac- tion ' of two parallel glass plates between which a drop of water is placed. For, since the water becomes concave outwards, the pressure inside it is less than that of the atmosphere, and hence the plates are pressed, not attracted, together. The lifting of a stone by means of a leather ' sucker ' is similarly explained. The plates would be apparently repelled apart if the liquid did not wet them, i.e., if it became convex outwards. And it is easy to prove that this would result also if only one plate were wet. The liquid would rise to a greater height outside the one plate, and would descend farther outside the other, than it would rise or fall in the interior space. At the par^t ab (Fig. 67) there is less than PROPERTIES OF LIQUIDS. 141 atmospheric pressure outside (for the liquid is concave upwards), while, inside, there is atmospheric pressure, Also, at cd, there is more pressure on the inner, than on the outer, side. Thus, from both causes, the plates are pressed apart. 124. In the proof of 73 we may suppose that, instead of a cord stretched in a circular tube, we have a film of unit breadth stretched over a cylinder. The pressure per unit area will therefore be T/K. If the film be stretched over a spherical surface of the same radius K, the pressure would have the value 2 T/E, for there are now equal curvatures in two directions at right angles to each other. [The investigation of 122 furnishes a special proof of this. For if cos a 1, the surface of the liquid is hemispherical in the circular tube, and is cylindrical in the space between the parallel plates. But the formulae show that, if the radii of the d FIG. 67. i ,n h " H FIG. 68. cylinder and the sphere are equal, the liquid rises twice as high in the tube as it does between the plates. In other words, the pressure towards the centre of curvature of the film, which supports the weight of the elevated liquid, is twice as great in the first case as in the second.] In a soap-bubble, we must remember that the liquid film has two surfaces ; so that, when the bubble is spherical, the air in the interior is subjected to a pressure which is greater than that on the outside by the quantity 4T/E. The above considerations indicate a method, due to Sir W, Thomson, of measuring the value of T. A capillary tube is inserted in the bottom qf a vessel B, (Fig. 68) which is partly filled with the given liquid, and is connected by a syphon with a vessel A which also contained the given liquid. By raising or lowering A, the level 142 A MANUAL OF PHYSICS. , of the liquid in B may be altered at pleasure. The liquid will pass through the capillary tube, and will gather into a drop at the lower end of it ; but this drop will not fall away unless the difference of level, 7t, between its lowest point and the free surface of the liquid in B is too great. The inward pressure per unit surface of the drop is p + 2T/r, when p is the atmospheric pressure and r is the radius of the drop (measured by micrometric methods). The outward pressure per unit surface, due to the weight of the liquid and the pressure of the atmosphere, is j + Us, where s is the specific gravity of the liquid. Hence 2T = rhs. 125. We have hitherto regarded surface-tension as an observed fact merely, but it is easy to see that it is a necessary result of the mutual potential energy of molecules, or, as we may put it, of the molecular forces. Let p' (Fig. 69) be a molecule in the liquid, situated at a greater distance from its surface than the range through which the molecular forces are sensible. Draw a'sphere from^/ as FIG. 69. centre with the range of the molecular forces as radius. There is no mutual action between _p'and any molecule outside this sphere ; and it is equally attracted on all sides by the molecules inside the sphere. But any particle p, which is nearer the surface of the liquid than the given distance, is pulled inwards on the whole by the molecular attraction of the interior particles. And this inward pull on the surface particles produces the same effect as, and will obviously be manifested as, a surface-tension, tending to diminish the external periphery of the liquid. 126. The tension of a sheet of india-rubber increases in propor- tion to the augmentation of the surface, but the tension of a liquid film remains absolutely constant (at least through extremely wide limits) when the area of the surface is altered. If left to itself, the india-rubber will contract until the area of its surface once more attains its original value ; but the liquid will contract until its surface becomes as small as possible. Consider a film of breadth 6, the tension of which per unit PROPERTIES OF LIQUIDS. 143 breadth is T, so that the total tension is T6 in the direction of the length of the film, If the length of the film be increased by the amount I, the work done in the process is T6Z ( 62). But this is equal to TS, where S is the increase of surface. Thus the work is directly proportional to the increase of area, and we may look upon T as the amount of work done per unit increase of area instead of a tension per unit breadth, Taking this fact in conjunction with the result of last section, we can now obtain an expression for the exhaustion of potential energy of molecular separation (the work done by the molecular forces) when two separate masses of the same liquid are placed in contact over a given area. Let S be the area, so that 2S is the diminution of surface of the two masses. The work done is 2TS. Let T and T' be the surface-tensions of two different liquids, and let t be the tension of the surface separating the two when placed in contact. The work done by the molecular forces when they come into contact is obviously (T + T' - t} S. The above result is a particular case of this, for, when the two liquids are identical, we have t = o and T = T'. If in any case the work done is greater than (T -f-T')S, i,e. t if t be negative, the surface of separation of the two liquids must increase. This it may do by becoming puckered ; and, the smaller the scale of the puckering, the greater will be the increase of surface. Thomson regards this invisible replication of the separating surface as the commencement of the process of diffusion. 127. The surface-tension of a liquid diminishes rapidly with rise of temperature, and it vanishes entirely at the critical temperature ( 24, 278). The saturation pressure ( 275) of the vapour of a liquid depends upon the temperature. But, the temperature being fixed, it also varies with the curvature of the surface of the liquid, being greater the more convex outwards the surface is. Hence small drops of water in a cloud evaporate, the vapour being deposited upon the larger ones. From the fact that the surface-tension of liquids decreases with rise of temperature, we might deduce, by the principle of stable equilibrium ( 15), the result that sudden extension of a film will produce a fall in its temperature. For, since the system is in stable equilibrium, it follows that extension of the film will produce ar\ effect which results in an increase of the force resisting the exten- sion. It will, therefore, cause diminution of temperature, This is known to be the case, 144 A MANUAL OF PHYSICS. - The principle of dynamical similarity shows at once that the square of the fundamental period of vibration of a (weightless) liquid sphere is directly proportional to the density of the liquid and to the cube of the radius of the sphere, and is inversely proportional to the surface-tension. It also shows that the period of funda- mental vibration of a (weightless) soap-bubble is independent of its linear dimensions. CHAPTER XI. PROPERTIES OF SOLIDS. 128. Compressibility and Rigidity. The compressibility of a solid is defined in precisely the same way as that in which we have already denned the compressibility of a liquid or gas. It is the ratio of the percentage change of volume to the change of pressure which produces it. The reciprocal of this quantity is called the resistance to compression, and is usually denoted by the letter Jc. The compressibility is most readily determined by measurement of the alteration of length of a rod of the substance to which known hydrostatic pressure is applied. If p' be the percentage alteration of length, the percentage alteration of bulk is approximately p = 3p'. For Z, 6, and t, representing respectively the length, breadth, and thickness of a rod of the given substance, the new volume is lbt(l-p') 3 , which approximately is lbt(l-8p') = lbt(l-p). In all actual cases the value of p' is so small that any power higher than the first may be neglected. [It must be observed that we assume the substance to be isotropic, i.e., its properties are independent of direction. If this were not so, p' might have different values in different directions. This assumption will be adhered to throughout the chapter.] The rigidity of a solid is the measure of its resistance to change of shape. Let ABCD (Fig. 70) be a cube of a given solid, the edges of FIG. 70. which are of unit length. Let equal tangential forces, of magnitude T per unit area, be applied to the opposite faces AB and CD, and 10 146 A MANUAL OF PHYSICS. let them act in the directions indicated by the arrows. These forces will produce shearing of the cube, but they will also produce rotation in a direction opposite to that of the hands of a watch. To prevent this rotation, tangential forces equal to the former may be applied to the opposite faces AD and CB. The result of the application of this set of forces is that the square section shown in the figure becomes rhomboidal, the angles at D and B being made less than a right angle by the same amount that the angles at A and C are made larger than a right angle. If 9 be the change of angle, the rigidity, which is usually denoted by the letter n, is given by the formula n = T/0. 129. Let us denote the three pairs of parallel faces of a unit cube by the letters A, B, and C. Similarly we shall speak of the edges joining the A faces as the A edges, and so on. Let unit normal pressure per unit area be uniformly applied to the A faces. This will diminish the A edges by an amount Z, and will increase the B and C edges by a common amount I'. Now let unit normal tension per unit area be applied to the B faces. This will increase the B edges by the amount Z, and diminish the C and A edges by the amount I', small quantities of the second order of magnitude being neglected. The result is that the A edges and the B edges are respectively diminished and increased by the amount Z+Z', while the C edges are unaltered in length. Hence there is no alteration of volume. Now this result might have been produced by the method of last section. If any point in the face DC (Fig. 71) of the unit cube be slid forward relatively to a point in AB through the (very small) distance s, the increase of length of the diagonal DB is s cos 45 = s/>v/2. A B FIG. 71. Similarly the decrease of length of AC is / ^2. The given tangential forces are obviously equivalent to a pressure parallel to AC of magnitude 2Tcos45 = T A /2 per area ^2, i.e., T per unit area, and to a tension parallel to DB of the same magnitude. If, PROPERTIES OF SOLIDS. 147 now, we let T = 1, sj */2 is the alteration of length of the diagonal which contains \/2 units, and so s/2 is the percentage change of length in the direction of the diagonals due to unit tension parallel to BD and unit pressure parallel to AC. Hence, equating the results of the two methods, we get s = 2(Z+ 1'}. But s = 9, the change of angle of the unit cube. Hence 130. Unit pressure per unit area on the A faces shortens the A edges by the amount Z, and increases the B and C edges by the common amount I'. The quantities I and I' being extremely small, if unit pressure be now applied to the B and C faces, all the edges of the cube will be diminished by the amount I - 11' =p r . Hence (2). The quantity 1/Z (the reciprocal of the percentage change of length of a rod under unit tension or pressure per unit of its transverse sectional area) is called Young's Modulus. From (1) and (2) we find 91m This formula enables us to determine the value of either I, k, or n, provided that we know the values of the other two. The following table gives the value of I for a few substances, the unit of pressure per square inch being the weight of one pound : Steel ......... 30(10) -. Iron ......... 39(10)-9. Copper (hard) ...... 56(10) ~ 9 . (annealed) ... 64(10)- tJ . Glass (average) ... 141(10)-*. 131. The rigidity n is not found in practice directly by the process which was described in 128. It may be found by determining the moment of the couple which is required to twist a cylindrical rod of the substance through a given angle. Consider a circular ring of this rod, of radius r, of infmitesimally small breadth dr, and of thickness dh = dr. If the ring be divided into little parts by a series of planes passing through the axis OP of the rod, and making angles equal to dr/r with each other, each little part will be cubical in shape, and the number of cubes will be If, by the given twist of the rod, the upper side of any one 102 148 A MANUAL OF PHYSICS. of these little cubes is twisted forward relatively to the under one by the small angle d9, the consequent change of angle of the little cube is rdO/dh. And, as the same change of angle would be produced in FIG. 72. a unit cube by a tangential stress of the same magnitude per unit area as that which acts upon the elementary cube, we have ( 128) rd0_T dh ~n Hence the total tangential stress acting on the circular ring, which is the product of T into the area of the flat surface of the ring, has the value -yr dh The moment of the force on the ring is therefore where 9 is the twist per unit length of the rod, so that dQ = Odh. Taking the integral of this from the axis outwards, we find for the total moment of the force required to twist the rod through an angle 9 per unit of length the quantity r being the radius of the rod. The quantity 7rar 4 /2Z, where I is the length of the rod, is called the ' torsional rigidity ' of the rod. It is not, of course, a specific property. [It is easy to deduce the result (1) by elementary considerations. Let us imagine the rod to be divided into similar little cubes as above. The total tangential force (in a plane perpendicular to the ^ - _; IT ft. PROPERTIES OF SOLIDS. 149 axis of the rod) on one face of any such cube, at a distance r from the axis, is proportional to r- ; and therefore the moment of the force is proportional to r 3 . But the amount by which, with a given twist, the upper face of the cube slides forward relatively to the lower face is proportional to r. Hence the total couple, c, required to produce the twist is proportional to r 4 . But it is also proportional to 9 so long as Hooke's Law holds ; and we may therefore write where n is the rigidity, provided that we make a suitable definition of the various units involved.] Even a simpler experimental method consists in attaching to one end of the rod a body whose moment of inertia about the axis of the rod is very great. Let the rod be firmly clamped in a vertical position by its upper end, the body being attached to its lower end. The time of oscillation of the whole system about this axis depends upon the value of n. The moment of the couple about the axis is 10, I being the moment of inertia of the system ( 70). Hence, from (1), (2). This equation asserts that the angular acceleration is proportional to the angular displacement, and therefore the integral is ( 51) (3), a and 9 being constants. If we increase t by the constant quantity the value of 9 is unaltered. This means that the periodic time of vibration of the system is which furnishes a ready method of determining n. (It is here assumed that the rod is of unit length. If it be not so, the equation will still hold, provided that we divide the observed value of T by the square root of the length.) 132. The following table gives, in the same units, the values of n for the substances for which I was given in 130. From these numbers the values of k are calculated by means of (3) 130, and 150 A MANUAL OF PHYSICS. ' the observed values of & are given in the last column. There is con- siderable discrepancy between the calculated and the observed values, but this need not produce surprise, for there are many causes of variation in the experimental results. The value of r in the expres- sion for the rigidity is usually small, so that a large percentage error may occur in its measurement ; and, even if r were uniform through- out the rod, which is rarely the case four times this error will be produced in the calculated values of n, since r is involved to the fourth power. Again, the substances may not be really isotropic ; and the special physical treatment e.g., the drawing out of a wire to which an originally isotropic substance is subjected will fre- quently make it non-isotropic. n Steel ... 121(10)5 ...... 450(10)5 ... 284(10)5. Iron (wrought) ... 112(10) 5 ...... 120(10) 5 ... 213(10) 5 . Copper 64(10) 5 to 71(10)5 ... 122(10)5 to 288(10) 5 ... 227(10)5. Glass (average) ... 28'4(10) 5 ...... 47(10)5 ... 43(10)5. Amagat's observed values of ~k for steel, copper, and glass are respectively 220(10) 5 , 174(10) 5 , and 66(10) 5 . 133. The flexural rigidity of a bar in a given plane is measured by the moment of the couple which is required to produce unit cur- vature in that plane. In similar and equal bars of different substances it is directly proportional to Young's modulus ; and, in different bars of the same substance and of similar though unequal section, it is pro- portional to the square of the sectional area. However a bar be bent, the locus of the centres of inertia of all the transverse slices is un- altered in length. The length of all other lines is either increased or diminished. This shows why Young's modulus is involved. Consider a rectangular rod, of thickness d and of breadth 6. Let it be bent uniformly to unit curvature, and let us suppose that b and d are small in comparison with the radius of curvature. If we further imagine the rod to be composed of a very large number of rods, whose cross-sections, are similar to that of the given rod, and whose lengths are equal to the length of the given rod, it is easy to see from similarity that the couple which is required to produce the curvature must be proportional to 6, the breadth of the rod measured perpendicular to the plane of bending. Also the elementary rods which are further away from the centre of curva- ture than the central plane of the given rod are extended in pro- portion to their distance from that plane, while those nearer to the centre of curvature are shortened in the same proportion. Hence we see that the total force must be proportional (so far as PROPERTIES OF SOLIDS. 151 this effect goes) to cl and to w Young's Modulus ; and therefore the moment of the couple must be proportional to in and to d 2 . But the number of little rods, of a given size, is proportional to d ; and therefore, finally, the moment c must be proportional to &, to m, and to d? say where/ is a constant. Thus we .see that the flexural rigidity of a rectangular bar is proportional to its breadth and to the cube of its thickness in the plane of 'bending. 134. Elasticity. All solids possess elasticity, both of form and of bulk, to a greater or less extent. Within limits (which vary greatly in different substances) the elasticity is perfect, i.e., the original form or volume is entirely regained ; but if too great stress be applied, the body will either break or become temporarily or permanently distorted. Steel is a good example of the former class. Its limits of elasticity are very wide apart, and it breaks before much perma- nent distortion is apparent. On the other hand, lead can scarcely recover entirely from any distortion however slight. When perma- nent distortion occurs, the molecules have set themselves into new permanent groupings. When the distortion is only temporary, they can resume gradually their old positions. An elastic solid, if it be kept distorted for a considerable time and then be released so slowly that it does not vibrate, does not in general at once recover its original form, but gradually creeps back to it. If it be consecutively distorted, first in one sense and then in the opposite, it will slowly recover from the second distortion for a time, and then will undo the quasi-permanent part of the first. The limits of elasticity depend to a large extent upon the physical treatment to which the substance is subjected. The elasticity of a wire is greatly diminished if the wire be kept oscillating for a long period of time. That this is so is shown by the fact that its oscilla- tions die away much more rapidly in this case when it is left to itself after being set in oscillation. The elasticity is said to be ' fatigued ' by the process. 135. We may conveniently define elasticity as that property in virtue of which stress is required to maintain strain ( 68). Within the limits of elasticity, the necessary stress is proportional to the strain. This is known as Hooke's Law, and is usually stated in the form ' Distortion is proportional to the distorting force.' The constancy of the quantities n and k in equations (1), 129, and (2), 130, depends upon the truth of this law; for their con- stancy implies that, if the unit of pressure be multiplied in any 152 A MANUAL OF PHYSICS. ' proportion, the quantities on the left-hand side of the equations will also be increased in the same ratio. The constancy of the pitch of a note which is given out by a musical instrument proves that Hooke's Law is obeyed by the vibrating substance within the given limits of distortion. The rigidity and the resistance to compression will not remain unaltered if the distortion be too great ; but, between new limits, the law will again hold, n and & having new constant values. 136. Viscosity. Viscosity, or internal friction, is apparent in solids as well as in fluids. The vibrations of a tuning-fork die away at a geater rate than can be accounted for by the impartation of energy to the air in the production of sound. Internal friction transforms the original energy partly into the form of heat in the material of the instrument. The phenomenon of fatigue of elasticity is also due to this cause, the internal friction being increased by the process which induces ' fatigue.' 137. Cohesion. Tenacity. Cohesion is in general much more powerful in solids than in liquids. The parts of any body are kept together both by the force of cohesion and by mutual 'gravitation. In small bodies, such as stones, the part played by gravitation is totally iiegligable in com- parison with that due to cohesion. But, in large masses, such as the earth, gravitation has much the larger effect. The force of cohesion has generally been regarded as a molecular attribute distinct from gravitation. But Sir W. Thomson has pointed out that cohesion may be explained by means of the gravi- tational law. If a bar of lead be cut into two parts, such that the freshly-cut surfaces accurately fit each other, the parts will readily reunite by cohesion when the surfaces are brought sufficiently close to each other. Such a phenomenon as this could not occur if matter were continuous and of uniform density throughout. For the range of the molecular forces ( 145) is so small that only an extremely small amount of matter at one surface of the 'bar could sensibly attract, according to the gravitation law, a given particle at the opposed surface. But, in order to get comparatively a very large mass at one surface within ' molecular range ' of the given particle at the other, we have only to suppose that, as regards density, matter is intensely heterogeneous on an invisibly small scale ; and thus the molecular gravitational force might be sufficiently great to account for cohesion. 138. The property of tenacity, in virtue of which there is resist- ance to the drawing asunder of the parts of a body, is obviously PROPERTIES OF SOLIDS. 158 cohesion regarded from a different point of view. We may measure the tenacity of a substance by the tension which is required to rupture a rod or wire of that substance whose cross-sectional area is unity. The following table gives its values, for sudden rupture, in a few substances. Eupture will be slowly produced by some- what smaller (frequently considerably smaller) tensions. These numbers, however, can only be used for purposes of rough calcu- lation, as the property varies with the physical treatment and chemical purity of the substance. They represent the number of kilogrammes whose weight will produce sudden rupture of a rod which has a sectional area of one square millimetre. Lead 2-4 ... 2. Tin 2-9 ... 3-6. Gold ... ... 28 ... '11. Silver 30 ... 16. ' Copper 41 ... 32. Iron 65 ... 50. Steel 99 ... 54. Oak 7 Ash 12 The numbers in the first and second columns refer to the drawn and annealed states of the metals respectively. The tenacity of wood is of course very much smaller when the length of the rod is taken across the grain than when it is taken in the direction of the grain. CHAPTEE XII. THE CONSTITUTION OF MATTER. 139. EARLY in the history of science, discussion arose regarding the possibility of the infinite divisibility of matter. A drop of water may be subdivided into smaller drops of water ; but this process cannot be indefinitely continued, for a point is ultimately arrived at beyond which subdivision cannot be carried without alteration of the chemical nature of the substance. But the problem with which we are now concerned goes deeper than this : we wish to know whether or not we would ultimately reach an indivisible part, or atom, of matter. No answer can be given yet to this question, though various hypotheses have been framed regarding the ultimate constitution, or structure, of matter. One of the most famous of these hypotheses is known as the Lucretian Hypothesis of hard atoms. According to Lucretius, hard atoms exist ; and they are indivisible because they are infinitely hard. The reason which he gave for their existence was that there must be a limit to the 'decay of matter which he asserted to be a more rapid process than the agglomeration, or building up, of matter. If there were no such limit, all matter would, he said, have disappeared in the course of infinite past ages. His hypothesis may be true, but his assumption, upon which it was based, is not true ; for we know that the building up of matter into larger parts is a more rapid process than its disintegration. Lucretius further asserted that there must be void spaces between the atoms, otherwise motion of solids in fluids, such as that of a fish in water, could not occur. Here, again, his reason is not con- clusive, although his conclusion is correct. Motion of solids in fluids could occur even if the hard atoms were closely packed together. This we know by the principle of fluid circulation in THE CONSTITUTION OF MATTER. 155 re-entrant paths. But the fact that all matter is compressible shows that there must be intervals between the hard atoms. 140. The atomic hypothesis of Boscovich is a mere mathematical device, which enables us to avoid the physical difficulties of the problem. Boscovich looked upon the atoms as mathematical points endowed with inertia and with attractive or repulsive forces varying' with the distance. These forces are always attractive when the distance exceeds a certain superior limit, and are always re- pulsive when the distance falls short of a certain lower limit. They become infinite when the distance vanishes, so that no two atoms can occupy the same position at the same time. This thepry may be worked out so as to determine the properties of a continuous medium so constituted. It has recently been developed by Sir W. Thomson. 141. The most recent atomic hypothesis and the one possessed of the greatest interest at the present day is the vortex-atom hypothesis of Sir W. Thomson. According to Thomson, matter consists of the rotating parts of a perfect ( 74), inert fluid, which fills all space. Since the fluid is perfect, any part of it, being once set in motion, will remain in motion for ever, and will be completely distinguished from all other parts. Thus the principle of conservation of matter is an essential part of the hypothesis. The ultimate parts or vortices are indivisible, not from being perfectly hard but, because it is impossible to get at them in order to divide them. The properties of such vortices may be illustrated by means of smoke-rings, such as those which are occasionally seen issuing from the funnel of a locomotive when the door of the furnace is suddenly closed, or those which are at times blown from the mouth of a cannon. These smoke-rings may readily be produced by means of a box one end of which is flexible and the other end of which is per- forated by a circular opening three or four inches in diameter. If a strong solution of ammonia be sprinkled on the floor of the box, and if a vessel containing a strong solution of hydrochloric acid gas (or, preferably, common salt on which strong sulphuric acid is poured) be also introduced into it, dense white fumes of ammonium-chloride are formed. A slight blow on the flexible end of the box is sufficient to drive out a portion of the air from the interior. This part revolves round and round, in the manner which is indicated in section in Fig. 73, and forms a complete circular vortex, the course of which may be traced for some distance through the surrounding still air. As the ring advances, its speed diminishes, and its 156 A MANUAL OF PHYSICS. ' diameter increases; and at last it disappears, its motion being stopped by friction. If a second ring be projected after the former, with slightly greater speed and nearly in the same line, actual collision will not ensue, but each will move aside vibrating as an elastic ring would after direct impact. If it be projected exactly in the same straight line as the first, the second one will grow smaller as they approach, and, its speed increasing, will pass through it. If the relative speed of approach be not too great, the two rings will not separate, but FIG. 73. will continue to revolve round each other in the manner indicated. This corresponds to molecular or chemical combination of vortex atoms. Currents are produced in the surrounding fluid which flow for- wards in the direction of motion through the interior of the ring and pass back round the outside of it. It is the action of these currents which prevents actual contact between two impinging rings, and which makes it impossible to divide a ring formed in a perfect fluid. The circular atom is the simplest which can exist, but a vortex - atom may be built up of any number of simple, or knotted, rings, linked, or locked, together in any manner whatsoever. The mathematical difficulties which beset the investigation of the motion, and the mutual action, of vortices are so great that they have been overcome as yet only to a very slight extent. But, in so far as they have been overcome, the results are not unfavour- able to the hypothesis. The great recommendation of the hypothesis is that it postulates nothing but the inert vertically moving fluid. All the known properties of matter are to be deduced from that one postulate. Other hypotheses assume the existence of special inter-atomic forces, and, if one assumed set fails to produce required results, a new set can be conjured up to suit the case. Further adaptations of the vortex theory will be given in next chapter, and in Chap. XXXIII. 142. Standing in distinct contrast with these atomic hypotheses, we have the hypothesis that matter is continuous but intensely heterogeneous. The atomic hypotheses may be very roughly illus- THE CONSTITUTION OF MATTER. 157 trated by a brick wall built without mortar. The separate bricks represent the various atoms, and there are gaps between them. The hypothesis which we now consider is similarly illustrated by. a wall in which the gaps are filled up by cement. Viewed from a distance, the whole seems homogeneous (as matter does even when we inspect it with the most powerful microscopes) but, if suffi- ciently closely inspected, it is seen to be heterogeneous. Heterogeneity is a necessity whether atoms exist or not. This is indicated by many physical phenomena, notably by the disper- sion of light, which, if the undulatory theory be true, could not occur were matter homogeneous- and continuous. 148. If we assume the existence of molecules which exert action in all directions throughout a small range, we can explain the various phenomena of crystalline structure. Let us suppose first that the molecules are in .stable equilibrium under their mutual action when their centres are at a definite distance apart. We can obviously make a model of such an arrangement by means of equal spherical balls or marbles, a molecule being supposed to be situated at the centre of each. FIG. 74. We may lay down a plane triangular arrangement of marbles (Fig. 74), and then build up a solid by placing a second set of marbles above the first, so that each marble in the second set fits into the hollow between three in the first set, and so on. In this way we form a regular tetrahedron. This tetrahedron has six edges, and, if these edges be pared off by planes equally inclined to the adjacent faces, we ultimately get a cube. The cube has eight vertices, and, if these vertices be pared off by planes equally inclined to the faces which meet at these vertices, a regular octahedron is obtained. If the twelve edges of the cube be bevelled by planes equally inclined to the adjacent faces, a rhombic dodecahedron is ulti- mately produced. It is by no means difficult to account in this way for all the 158 A MANUAL OF PHYSICS. various symmetrical forms of crystals belonging to the cubic system ; and, if we replace the component spheres by ellipsoids of revolution, we can account for all known crystalline forms. We are thus led to believe that crystalline bodies are built up of particles which set themselves, under the action of their mutual forces, in the position of least potential energy, i.e., in the position of stable equilibrium. The strongest proof of this which we can. have is contained in the fact that the lengths of the edges of any given natural crystal are expressible as multiples of small whole numbers. 144. The square order of arrangement (indicated in the above figure) in which any one sphere touches four spheres in the layer immediately below, is in no way different from the triangular arrangement just considered; for, if we remove an edge row of spheres from the triangular pyramid, it. is at once evident that the particles are arranged in square order in planes which bevel an edge symmetrically. In both the triangular and the square order of arrangement, any one particle touches twelve others. In the triangular order, a particle touches six others in its own layer, and three in each of the adjacent layers. In the square order a particle touches four in each of these layers. It is true indeed that all the crystalline forms of the cubic system may be explained by two other methods of arrangement, but these are of no interest to us physically, as they do not correspond to that position of most stable equilibrium which free, mutually attractive, particles would naturally assume. Thus we might build up a cube first of all on the square order of arrangement a oo oo FIG. 75. particle in any one layer resting on the top of one in the subjacent layer ; and, from this cubic form, we might then, as above, produce the other crystalline forms. Again, we might start with an open square arrangement (indicated in Fig. 75) the particles in any one layer being so far apart that another particle which is set in the hollow between four will just touch a particle which is set in the corresponding hollow on the other side of these four. In this way, THE CONSTITUTION OF MATTER. 159 also, all the crystalline forms of the regular cubic system may be produced. In the former method, any particle touches four particles in its own layer, and one particle in each of the adjacent layers six in all ; in the latter method, a particle touches none in its own layer, four in each of the layers immediately preceding and succeed- ing its own, and one in each of the layers outside of these ten altogether. In neither of these cases is the equilibrium so stable as in the case which was considered in last section. 145. Many other physical phenomena, besides those which are exhibited by crystalline bodies, make it a practical certainty that matter is molecular in its structure ; and these phenomena also enable us to obtain approximate estimates of the range of the molecular forces, and of the average distance between contiguous molecules. One such class of phenomena is that exhibited by liquid films. We have seen ( 125) that the existence of molecular forces would account for surface-tension. And we have seen, also ( 126), that the surface-tension remains practically constant until the thickness of the film is very largely reduced. There are optical methods ( 218-221) by which the thickness can be very accurately ascer- tained. Keinold and Riicker have shown by these methods that the surface-tension of a soap-bubble begins to diminish when the thick- ness is between 96 and 45 micro-millimetres (millionths of a millimetre, one inch being about equal to 25'4 millimetres). It diminishes to a minimum when the thickness is 12 micro-milli- metres, and then increases again to a maximum. Plateau had previously shown that the tension is unaltered when the thickness is reduced to 118 micro -millimetres, and he concluded that the range of molecular forces is less than 59 micro-millimetres. (Maxwell, however, has given theoretical reasons for the belief that the tension may not alter until the total thickness of the film is equal to the range of the forces, which would make Plateau's superior limit 118 micro-millimetres.) By measurements of the height to which liquids rise, because of the so-called capillary forces, between parallel glass plates which were coated with very thin wedge-shaped metallic films, Quincke was led to the conclusion that the forces between the glass and the liquid became evident when the thickness of the metallic film was 50 micro-millimetres. Wiener has found that the phase of the vibrations of light ( 243) which is reflected from a thin film of silver deposited on mica begins to alter when the thickness is 12 micro-millimetres, and that it was possible to detect the presence of a film of silver not exceed- ing 0'2 micro -millimetre in thickness. 160 A MANUAL OP PHYSICS. Estimates of the magnitude of the molecular range have also been based upon the thickness of the films of gas which are con- densed upon solids, but these are open to great objection. From all these results we may conclude that the order of magni- tude of the range of molecular forces is about 50 micro-millimetres, or one five -hundred-thousandth part of an inch. 146. It is possible, also, to obtain an estimate of the coarse- grainedness of matter; that is, of the average distance between molecules. If (see 324) .two plates of different metals, say copper and zinc, be placed in contact, each becomes electrified oppositely, and so they attract each other. If the plates have an area of one square cen- timetre and be at a distance of one-hundred-thousaiidth of a centi- metre apart, they will, when joined by a metallic connection, attract each other with a force of two grammes weight. Hence the work done in bringing them into this position by means of the electric attraction alone would be 2/100,OOOths of a centimetre-gramme. If we now build up a cube of such pieces of metal, alternately zinc and copper, the thickness of each being l/100,000th of a centi- metre, and the distance apart of each pair beihg l/100,000th of a centimetre, the work done by the electric attraction is 2 centimetre- grammes. If this work were spent in heating the mass of metal, the temperature would rise by less than the sixteen-thousandth part of a degree centigrade. But if the thickness of the plates and their distance apart were l/100,000,000th of a centimetre, the work would be sufficient to raise the temperature of the mass by nearly 62 C. And, if the plates and the spaces between them were made yet four times thinner, the work would produce more heat than is given out by the molecular combination of zinc and copper. Hence the magnitude of the contact-electrification of zinc and copper must sensibly diminish before the substances are so finely divided as we have supposed. But this suggests that there cannot be many molecules in a thickness of the l/100,000,000th of a centimetre possibly the coarse-grainedness may even be on a larger scale. We have already seen ( 126) that the work required to increase the area of a water film by one square centimetre is numerically equal to the tension of the film per linear centimetre. But the magnitude of the tension is about 16 centigrammes weight per linear centi- metre. Hence, if we increase the surface of a water film by n square centimetres, we have to do n times 16 centimetre -centi- grammes of work. But a water film cools when it is stretched ; and Sir W. Thomson has shown that, if the temperature is to be kept constant which is necessary if the tension is to remain THE CONSTITUTION OF MATTER. 161 constant we must supply heat to the film, to an extent which would require the expenditure of about half as much work again. So, finally, if the area of a water film be increased, at constant temperature, by n square centimetres, work must be expended to the extent of 24ti centimetre-centigrammes. If we start with a cubic centimetre of water, and increase its area (and therefore diminish its breadth) one-hundred-million-and-one-fold, we must expend 2,400,000,000 centimetre-centigrammes of work. But this work, if spent in heating the liquid, would be more than sufficient to completely volatilize it. Hence, if the thickness can be diminished to this extent, the surface-tension must greatly decrease in magni- tude. And so we conclude that there cannot be many molecules of water in a thickness of l/100,000,000th of a centimetre. The fact of the dispersion of light in its passage through dense transparent media proves, as has been already stated; the hetero- geneity of these media. Starting from certain assumptions regard- ing the constitution of such media, Cauchy deduced results which indicate that there are only about ten molecules to the wave-length of violet light (about 4'10- 5 cm.) in ordinary glass. This cannot be admitted for many reasons. But Thomson has recently shown that it is possible to modify Cauchy's theory in such a way as to widen the limit which it sets. The kinetic theory of gases gives ( 153) another, and even more certain, indication of molecular magnitude. From these four courses of reasoning, Sir W. Thomson concludes that there are not more than 10 9 , nor less than 5(10) 6 , molecules per linear centimetre in ordinary liquids or solids. Another method consists in measuring the thickness of the dielectric layer separating an electrolyte from the electrodes. This thickness is presumably the distance between the molecules of the liquid and the contiguous molecules of the solid electrode. The method gives values of the number of molecules per linear centi- metre which vary from 10 7 to 5(10) 8 . Again, if we suppose a cubic centimetre of any liquid to be divided up by three sets of n planes parallel to the three pairs of faces of the cube, the surface of the liquid is increased by 6n square centimetres. The work which is required to produce this increase of surface is (neglecting the equivalent of the heat required to keep the temperature constant) 6wT centimetre-grammes, where T is the surface-tension expressed in grammes weight per linear centimetre. [It is assumed, of course, that T is practically constant during the process. We have seen that Plateau found it to be constant in a water film down to a thickness of 118 micro-millimetres. And 11 162 A MANUAL OF PHYSICS. Eeinold and Kiicker have shown that the value of the second maximum, which is reached when the thickness is 12 micro-milli- metres, only differs from the former value by about 0*5 per cent.] If n be the number of molecules per linear centimetre, the quantity 6nT is approximately equal to the work required to break up the cube of the liquid into its constituent molecules. It is therefore equal to the work-equivalent of the latent heat ( 276, 289) of the liquid. If L be this quantity, we have, therefore, as an approxi- mation, n = L/6T. The liquids, water, alcohol, ether, chloroform, carbon bisulphide, turpentine, petroleum, and wood spirit, have, according to this method, 50, 52, 30, 15, 19, 30, 40, and 70 millions, respectively, of particles per linear centimetre. These numbers all lie well within the extreme limits given by Thomson. Of course, no stress is to be laid upon the relative, or even the absolute, values of the figures; the point of interest is the close agreement as to the order of the unknown quantity. CHAPTEK XIII. THE KINETIC THEORY OF MATTER. 147. THE first glimmerings of the idea that the observed properties of matter may be due to motion. occurred as far back as the time of Democritus and Lucretius. But the idea did not develop into an actual physical hypothesis until Hooke, and, later, Daniell Bernoulli suggested that gaseous pressure may be due to the impact of the molecules of the gas upon the sides of the vessel which contains it. Somewhat later, Le Sage, as we have already seen, applied the same principle to the explanation of gravitation, and various developments were made by Prevost and Herapath. In 1848, Joule calculated the speed which the particles of a given gas must have in order to pro- duce a given pressure. But the full mathematical development of the Kinetic Theory of Gases is due mainly to Clausius and Maxwell. 148. In the kinetic theory it is supposed that the particles of a gas are darting about in all directions with great average speed. Some of the particles may, for a short lime, have very much smaller speed than this average may indeed be at rest for a moment ; and others may be moving with much greater speed than the average. This average is the square root of the mean of the squares of the individual speeds of the various molecules, and is called the mean square speed. Collisions are supposed to occur amongst these particles. In the interval between any two successive collisions of a particle there is a certain average distance which the particle describes, and this distance is called the mean free path. The mean free path, under ordinary conditions, is large in comparison with the diameters of the molecules, which are regarded as being smooth hard spheres with unit co-efficient of restitution. The collisional force between two molecules is assumed to be repulsive, just as it would be in the case of elastic solids. But, actually, in. many, or even most, of the so-called 'collisions,' true contact may not occur. The real forces may be attractive, and 112 164 A MANUAL OF PHYSICS. two rapidly-moving molecules coming within range of mutual attraction may whirl round each other in sharply-curved paths, as a comet dashes round the sun, the result being the same as if actual contact had taken place. Some contacts must occur unless the molecules are infinitely small, which we cannot admit. Experiments made by Joule and Thomson on certain gases show that, if the density of each be varied while its total energy remains constant, the temperature is somewhat higher when the density is greater. Hence, if ( 150) equality of temperature in two portions of gas means equality of average kinetic energy per molecule, it follows that the potential energy is somewhat less in the denser condition. But this indicates molecular attraction at the average distance of the molecules experimented with. 149. Gaseous Pressure. Boyle's Law. Let n be the number of molecules per unit volume which are moving in a given direction with speed which differs extremely little from a certain quantity u. The number of such molecules which pass per unit time across unit area drawn perpendicular to the direction of motion is nu ; and, if m be the mass of each molecule, the momentum which these particles carry with them is mnn?. By Newton's Second Law of Motion, this must be equal to the pressure produced by such mole- cules on the side of the vessel. Hence, the square of tlie speed being involved, we see that, so far as the pressure is concerned, we may assume that each particle is moving with the mean square speed. The total pressure per unit of area in the direction considered, which we may assume to be that of the #-axis, is therefore 7?zN% 2 , where N is the total number of particles per unit volume, and u 2 is the mean value of u 2 for all the molecules. Similarly the pressures per unit area in the direction of the y and z axes may be written wN^and mNw 2 respective^'. But, in a gas which is at rest as a whole, all these quantities are equal ; and so we have for the pressure p the expression p = wN(. 2 + y 2 + w 2 ), which we may put in the form p = imNV 2 = ipV 2 ...... (1) where V is the mean square speed independent of direction and is the density of the gas. By means of this result, Joule calculated the value of V in various gases. In hydrogen it is somewhat over 6,000 feet per second. _ We have seen that V is constant when the temperature is steady, so that this equation asserts that the density of a gas is directly proportional to the pressure which is Boyle's Law. THE KINETIC THEORY OF MATTER. 165 150. Avogadro's and Charles' Laws. The equation (1) may be written pv = T (2) where v is the reciprocal of p, i.e., it is the volume of unit quantity of the gas. If we compare this with the expression ( 266) py = RT, we see that the mean square of the speed of molecular motion is proportional to the absolute temperature as measured by a gas thermometer ( 266, 267) filled with the particular gas under consideration. It follows from the principles of the kinetic theory that, in a mixture of two gases in equilibrium, the average kinetic energy per molecule of each gas is the same. And, if we assume the truth of Avogadro's law that there is the same number of molecules in unit volume of each of two gases at given temperature and pressure, (1) shows that in two such gases the average kinetic energy per molecule is identical. Conversely, if we assume as is done in the kinetic theory that two gases at the same temperature have the same average kinetic energy per molecule, we can deduce Avogadro's law as a consequence. But it must be carefully observed that the truth of Avogadro's law does not establish the truth of this assump- tion ; it merely proves on this theory that two gases at the same temperature and pressure have equal average kinetic energy per molecule. If, however, the gases rigidly obey both this law and Boyle's Law, the truth of the assumption follows : for then, in any one gas, mV' 2 remains constant, no matter how much p may vary. Equation (2) shows that any gas which obeys these laws will expand equally for equal increments of temperature, provided that the temperature be measured by the average kinetic energy per molecule, which is a generalisation of the above assumption. Andj further, with this proviso, the equation also shows that any two such gases will expand proportionately as their common temperature rises. These two results constitute Charles' Law ( 265). The deviations from Boyle's and Charles' Laws can be explained on the kinetic theory if we take into account molecular action between the particles at greater than collisional distances. 151. Diffusion. Thermal Conductivity. Viscosity. The ques- tion of gaseous diffusion has been already discussed in Chap. VIII. The kinetic theory asserts that each particle is darting about with great speed, and is only prevented from moving rapidly away from the neighbourhood of its position at any given instant by means of the great number of collisions to which it is subjected by other 166 A MANUAL OF PHYSICS. particles. These collisions change the direction of motion of the molecule an immense number of times per second, and thus the diffusion of particles throughout a mass of gas is a very slow process. The law o&inter-diffusion given by the theory is identical with that deduced from observation. The diffusing molecules carry their kinetic energy with them, and there is interchange of energy during collision. This trans- ference of energy constitutes (Chap. XXIV.) the process of thermal conduction. And this process is slightly more rapid than the transference of the molecules themselves ; for, though a molecule may be turned back by a collision, its energy is handed on by means of the molecule which collides with it. If two portions of gas are moving relatively to each other, the molecules of each inter-diffuse, and consequently there is inter- change of momentum. But, on the average, the momenta of the particles of each gas are in opposite directions, and so the relative motion is gradually stopped. This explains gaseous viscosity. An illustration may be taken from two railway trains running on parallel lines past each other. If luggage wer.e thrown constantly from each train into the other, the interchange of momentum resulting from the impacts might soon reduce the trains to relative rest. 152. Evaporation, Dissociation, etc. As a gas becomes more and more condensed the mean free path of its molecules becomes smaller and smaller, until, -in the liquid state, its magnitude is excessively small in comparison with the magnitude of the mean free path of a gaseous particle. Still, in the liquid, the molecular action is precisely of the same character as that of a gas. But, as a result of the comparative closeness of the molecules, the trans- ference of energy by impact (that is, the conduction of heat) is a much more rapid process than the transference of the molecules themselves ; and the rate of diffusion of the molecules of a liquid is very small in comparison with the rate of diffusion of gaseous molecules. In a liquid some of the quickly -moving particles may escape from the attraction of neighbouring molecules and become particles of vapour. At the same time some vapour particles may become entangled in the liquid. When these two processes take place at the same rate, the liquid and its vapour are said to be in equilibrium. When the former process occurs most rapidly, the liquid is said to be evaporating; and, when the latter process preponderates, the vapour is said to condense. Dissociation occurs when an impact is so violent as to break up a compound molecule into its constituent parts. Probably dissocia- THE KINETIC THEORY OF MATTER. 167 tion occurs in all fluids to a slight extent, even when their tempera- ture is far below the ordinary temperature of dissociation, but, in this case, recombination occurs as rapidly. As the. temperature rises, the impacts become more violent and dissociation goes on at a greater rate, but not at so great a rate that recombination cannot occur as quickly. When, finally, the so-called temperature of dissociation is reached, recombination is unable to balance dissocia- tion, and the substance is resolved into its components. If the temperature is again allowed to fall, recombination may, or may not, occur, depending on the condition whether or not energy will be degraded ( 11) in the process. If more energy will be degraded by the occurrence of recombination than by its non-occurrence, recombination will ensue. [See 280.] An important result of the kinetic theory is that, in a vessel filled with a mixture of different gases, the final distribution of each gas under the action of gravity is the same as if no other gas had been present. Another very important result (which, together with the former, is due to Clerk- Maxwell) is that a vertical column of gas, when in equilibrium under the action of gravity, has the same temperature throughout, or, in other words, gravity has no effect upon the conditions of thermal equilibrium. 153. An expression has been deduced from this theory by which we can calculate the length of the mean free path of a molecule in an ordinary gas, such as air, in terms of observed values of the viscosity, and of the material and thermal diffusivities. According to Maxwell the value of this quantity in the case of hydrogen is 3'8(10)~ 6 of an inch (roughly four millionths). The theory also shows that the number of particles per cubic inch of ordinary gases is about 3(10) 20 (that is, 300 million million millions), and that the diameter of a (supposed hard and spherical) molecule is about 2-3(10) -8 inch. The following results are given by Maxwell : Hydrogen. Oxygen. Carb. Oxide. Carb. Acid. Mean square speed at C. expressed in feet per second 6190 ... 1550 ... 1656 ... 1320 Mean free path in thousand- millionths of an inch ... 3860 ... 2240 ... 1930 ... 1510 Number of collisions per millionth of a second ... 17750 ... 7646 ... 9489 ... 9720 Diameter per inch x(10)-n ... 2300 ... 3000 ... 3500 ... 3700. The length of the free path increases as the density of the gas diminishes. Tait and Dewar first found that, in a good vacuum, it 168 A MANUAL OF PHYSICS. , may amount to several inches. The action of Crooke's radiometer depends upon the great length of the free path. [This instrument consists of four vanes of mica mounted at the ends of two light rods, which are fastened by their centres to a vertical axis which is free to rotate. Each disc of mica lies in the vertical plane which passes through the rod to which it is attached, and the two rods are placed at right angles to each other. Also each disc is blackened on one side and is bright on the other, the blackened sides being all similarly situated with regard to rotation around the vertical axis. The whole is mounted inside a glass vessel from which the air is largely extracted. Eadiant heat (or light) falling upon the vanes is absorbed more freely by the black surfaces than by the bright surfaces, and so the former become hotter than the latter. Hence the particles of air which impinge upon them are driven off again more violently than are the particles which impinge upon the bright sides, and so, by the third law of motion, the necessary reaction results in rotation. The exhaustion of the air is carried to such an extent that a particle which is repelled from a disc rarely encounters another particle before it strikes the side of the vessel. If the exhaustion is carried too far the effect is diminished, because the number of particles which strike a disc in a given time is lessened.] Gaseous matter whose molecules have very large free paths is sometimes called ' radiant ' matter. 154. The statement that the diameter of one of these hypothetical (hard, smooth, spherical) molecules is 2*3 hundred-millionths of an inch is not intended to imply that this is the size of an actual material molecule. It merely asserts that, on the average, this is about the least distance to which the centres of two molecules can approach during collision. But, even if they be smooth and spherical, the molecules of matter cannot be hard ;" for, in this case, the time of describing the mean free path must vary inversely as the average speed of the particles. But, on the contrary, Maxwell's experiments on gaseous viscosity show that the time is independent of the speed. This would be possible with soft elastic particles. There are other, even more cogent, reasons why the molecules cannot be smooth, hard, and spherical. Thus the great complexity of the spectra (Chap. XVII.) of many gases and vapours shows that the molecule is a very complex structure with a great many degrees of vibrational freedom. But a hard, smooth, spherical molecule has, in effect, only three degrees of freedom all translational. And again, the theory asserts that each additional degree of freedom THE KINETIC THEORY OF MATTER. 169 which the molecule possesses requires that the ratio of the specific heat ( 271) of the gas at constant pressure to that at constant volume shall be made larger ; and the actually observed values of this ratio are far smaller than that indicated by the theory. Further, it seems certain that, in a gas whose molecules are small, elastic, solid bodies, the energy of translation of the molecules must gradually be converted into energy of vibration of smaller and smaller period, so that finally the particles would come to rest. 155. But these difficulties with which the theory is beset do not lead us to discard it. Its results, briefly indicated in part in the preceding sections, place it upon far too firm a basis; and we seek, rather, to inquire more closely into the truth or probability of the fundamental assumptions of the theory, and into the correctness of our deductions from them. The difficulty regarding the specific heats seems to be due largely to a too rapid theoretical generalisa 1 tion. And the difficulty regarding the transformation of the transla- tional energy of elastic solids into vibrational energy does not, Sir W. Thomson remarks, seem to apply to the case of fluid vortices. 156. We are not, however, to rest content with a kinetic theory of gases only. We wish, if possible, to recognise all the properties of matter as the results of motion of the parts of a medium in which we postulate nothing but incompressibility and inertia. We can make, from rigid portions of matter, a complex which possesses elasticity. A gyrostat (essentially a heavy fly-wheel, of great moment of inertia, free to rotate about its axis) mounted on gymbals, but not set in rotation, serves very well as a model of a plastic body. It may be turned about into any position which we please, and has no tendency to re-assume its original position. But, if the fly-wheel be set in rotation about its axis, the system at once acts as if it possessed rigidity and elasticity. If it is sharply struck, so as to suddenly turn its axis to a slight extent from its original direction, it will oscillate rapidly about, and finally come to rest in, its first position. Again, by means of rigid rods on which revolving fly-wheels are pivoted, we can construct a framework, which acts like an elastic spring, and may represent an elastic molecule. By joining together millions of such arrangements, it is possible to construct a model of an elastic solid through which distortional waves will pass, and which, under proper conditions, will exhibit, with regard to these waves, the same phenomena as those which are shown when light passes through a medium placed in a field of great magnetic force. 170 A MANUAL OF PHYSICS. ' The vortex theory shows that all that can be done by such a model might be done by a model in which vortices in a perfect fluid take the place of the revolving fly-wheels. And further, with such vortex molecules we can construct what cannot be done by the rigid fly-wheel molecules a model of a gas, whose molecules exert mutual force. Such considerations lead us to believe that we may ultimately be able to explain apparently statical properties of matter in terms of motion. CHAPTEE XIV. SOUND. 157. THE word sound is generally used with reference to the physiological effect which results from excitation of the hearing organ. In physical science the term is applied to the external cause of this subjective impression. We are accustomed to say that sound travels through a given medium, whether solid, liquid, or gas ; and by this we imply that the particles of the medium do not move forward from the source of the sound to the place at which it is heard. Intermittent impact of such particles would account for the vibration of the tympanum which is necessary to the production of the mental impression of sound. But it is quite obvious that the particles of a solid cannot move for- ward in the manner indicated, and it would be unscientific to assume that the method of transference of sound in a gas differs totally from the method of transference in a solid body, especially when we consider that sound, after travelling through a solid, may be communicated to, and travel through, a gaseous medium, such.as air. We do not, however, need to rely upon any such semi-meta- physical argument, for the speed of sound in air is so great that the forward motion of the particles did it occur would correspond to a hurricane of much greater violence than any ever observed. In short, we know that the passage of sound through air is not accompanied by such motion witness the well-known fact that sound is usually best heard in still air. But the passage of sound through air may communicate vibratory motion to objects immersed in it e.g., the tympanum and it therefore involves transference of energ} r , which implies motion of matter. This motion can be nothing else than vibratory motion of the particles of the medium, the state of motion being handed on from particle to particle in the form of a wave which may cause vibratory motion of any solid object which it reaches, just as a wave sent along a stretched cord may cause motion of any object to which the cord is attached. 172 A MANUAL OF PHYSICS. The vibrations of the particles of a medium may be transverse to the direction in which a wave travels, or they may take place in that direction. A little consideration will show that, in a sound-wave, it is the latter which must take place. Let us suppose that a sound is started by an explosion at a certain point. We know that the sound travels outwards in. all directions from this point as centre ; and we know also that the particles at the point are driven outwards by the explosion, so as to produce a state of great con- densation in the immediate neighbourhood. But, the medium being elastic, the compressed portions expand, and so cause com- pression of the adjacent parts, and thus the state of compression travels outwards from the centre. Since a state of rarefaction is produced at the very centre of the explosion, the particles in the compressed part just outside rush back so as to fill the partial vacancy, and so the state of compression is succeeded by a state of rarefaction, which travels outwards at the same rate. Thus we see that sound consists in the propagation of a condensational-rarefac- tional wave, the particles of the medium vibrating to and fro in the direction in which the wave travels. [It is easy to construct a model, which will roughly illustrate this process, by means of a row of equidistant balls which are attached by strings of equal lengths to a horizontal straight rod.] The distance, measured in the direction of propagation, from any particle to the next similarly -moving particle, is called the wave- length. It may be measured, for example, between points of maximum condensation, or between points of maximum rarefaction. The actual motion of any particle is a simple harmonic motion, and the terms used in the discussion of such motion ( 51) amplitude, phase, period, etc. are therefore used here with similar meanings. It is customary to speak of a single disturbance propagated through the air as a noise. The term sound is employed when a periodic disturbance, or series of disturbances, is propagated the sound being musical, or non-musical, according as the disturbances are of regular or of irregular period. When a musical sound is produced, it is usual to speak of the note or tone which is given out, but it is better to limit the application of the latter term to a simple sound of one definite period only. We shall see subsequently that no sound usually given out by a musical instrument consists of a simple tone only. The term note may conveniently be applied to the composite sound which is actually given out. All sounds differ from each other in three points only intensity, pitch, and quality. These will be treated in detail afterwards. SOUND 173 158. We shall now proceed to investigate the propagation of sound through any gaseous medium. For the sake of simplicity, we shall suppose that a disturbance is propagated in the form of a plane wave ; that is to say, we assume that any continuous set of particles, the motion of which is in the same phase, lie in a plane. Let u be the speed with which the sound-wave travels. The actual speed, v, of any particle is the resultant of this speed u and the speed due to the simple harmonic motion of the particle. Hence an ideal plane, which is perpendicular to the direction in which the sound is going, and which moves in that direction with speed u, possesses the characteristic that the instantaneous speed of all particles which cross it is constant. (The value of this speed depends, of course, upon the initial position of the plane.) A statement equivalent to this is that the plane moves so as to be always in a position of constant density. If p be this density, we may write pv = c, .......... (1), where c is a constant ; for this equation expresses the fact that the total momentum of the particles which cross unit area of the plane in unit time is invariable. But any two such planes, which remain equidistant, obviously contain between them a constant amount of matter which moves with constant total momentum. Hence c is absolutely constant, and so, by the methods of Chap. IV., we get from(1) If the pressure per unit area of the plane under consideration be jp, the pressure per unit area of another plane at a distance dx from it is (p -\- dp/dx , dx). Hence the difference of pressure per unit area of two such planes is - dp/dx . dx. But, by the second law of motion, this must be equal to the instantaneous rate of increase of the momentum of a column of the gas, of unit section, contained at the given instant between these planes. Hence, the amount of matter in this little volume being pdx, we get pdv/dt = - dp/dx. That is, pvdv/dx = - dp/dx, and so dp~ pv ' From (2) and (3) we have at once d (4). 174 A MANUAL OF PHYSICS. In a gas which obeys Boyle's Law and Charles' Law, the relation p = Rfy holds E being a constant and t being the absolute tempera- ture. If the temperature be constant, this gives and (4) takes the form i,2 = P = IM .......... (6). P But the compressions and rarefactions which take place when sound passes through a gas take place so rapidly that the equation p = Tltp does not represent the actual conditions. Instead of it we must write ( 302) where y is the ratio of the specific heat of air at constant pressure to its specific heat at constant volume ; so that, instead of (5), we get and (6) becomes ^= r ^ = r R* ........ (6'). 159. The general equation (4) gives the value of v under all con- ditions of the substance, and therefore (6') gives the value of v under ordinary conditions of the gaseous medium, provided only that we put the normal values of p and p in the expression on the right-hand side of that equation. Hence we see that, in cases in which (6') is sufficiently nearly true, the speed of sound is independent of the pressure ; and that, in all such cases, it is proportional to the square root of the absolute temperature. Also, in different gases under equal pressure and at the same temperature, the speed is inversely proportional to the square root of the density provided that (6) is true. The following table shows how nearly the theoretical relative speeds correspond to the actual relative speeds as observed in a few of the more common gases : Observed. Calculated. Air ... ......... 1-000 ...... 1-000. Carbonic Acid ...... 0-786 ...... 0-811. Oxygen ......... 0-953 ...... 0-952. Hydrogen ......... 3-810 ...... 3'770. SOUND. 175 It must be remembered that these results are calculated on the supposition that the gases are perfect. [The following considerations enable us to deduce, in a much less strict manner, the general conclusions arrived at by means of the investigation of 158. The speed of propagation of a given state of compression depends upon the readiness with which the compressed gas recovers its original condition ; and this, in a perfect gas, is proportional to the resistance to compression, which, 104, is measured by the pressure. But the readiness of recovery depends also upon the mass which has to be moved before the substance can expand, i.e., it depends upon the density, being greater the smaller the density is. Hence we see that, in a perfect gas, the speed of sound depends only on the temperature ; for (the pressure remaining the same) the density diminishes as the temperature increases, while, if the temperature be constant, the pressure and density vary proportionately.] 160. Equation (4) gives, not merely the speed of sound in a gas but, the speed of plane sound-waves in any substance. The resistance to compression, &, is by definition, 104, the ratio of the increase of pressure to the percentage decrease of volume which it produces. But the percentage decrease of volume is equal to the percentage increase of density. Hence we have & = (> . dp /dp, and so (4) becomes In the case of a perfect gas, Jc = p, and (7) becomes identical with (6). The speed of sound in water is nearly four times as great as it is in air. This was found experimentally by means of observations made at the Lake of Geneva, the waters of which are compara- tively free from currents. A bell was sounded under water, and the mechanism which rang the bell simultaneously fired a gun placed at the surface of the water at the same spot. A large receiver filled with air was placed below the surface at a known distance from the bell. An observer, who listened at the extremity of a tube con- nected with the receiver, knew when the sound which travelled through the water reached the receiver. He also observed the instant at which the sound of the report of the gun reached him through the air; and so, knowing by the flash of the gun the instant at which both sounds were produced, he compared the speeds in air and water. In solids the speed is still greater. 176 A MANUAL OF PHYSICS. The speed in air may be roughly found by observing the interval of time which elapses between the instant of seeing the flash of a distant gun and the instant at which the report is heard. .If the air is not still, two simultaneous observations must be made, in one of which the sound travels with the wind, and in the other of which it goes against the wind. The mean of the two results must be taken. A more accurate method of determining the speed in gases will be given afterwards ( 172). 161. The loUctness, or intensity, of a sound is due to the kinetid energy per unit volume of the medium, and so it depends upon the amplitude of vibration of the particles, being proportional to the square of that quantity. For the same reason it depends also, other things being equal, upon the density of the medium, being greater as the density is greater, and vanishing when the density becomes zero. Thus no sound is heard when a bell is struck inside a receiver from which the air has been exhausted as completely as possible. [For the displacement of a particle which executes a simple harmonic vibration is 2ff , a sin 7p, where T is the periodic time, and a is the amplitude. The speed of motion, which is the time-rate of variation of the displacement, is therefore 27T 27T and the energy of the particle is 27T 2 where m is the mass. In an interval of time r, which is very large in comparison with T, the energy may be regarded as constant and, equal to its average value during the period T. But this is SOUND. 177 The first term vanishes since T is very large in comparison with T, and hence the kinetic energy is mTr-or/T 2 per particle, or p?r 2 a 2 /T 2 per unit volume. The maximum value of the kinetic energy per particle is 2w7r-/yT 2 , and so the largest possible value of the kinetic energy per unit volume is 2p7ra 2 /T 2 . Hence the total energy per unit volume of a system of particles in simple harmonic vibration is (in an interval of time which is large in comparison with the period of a complete vibration) one-half kinetic and one-half potential. This statement will obviously still be true, without the above restriction regarding the interval of time, provided that the number of particles per unit volume is very great, and that the wave-length of the dis- turbance is very small.] In still homogeneous air sound spreads outwards from the source uniformly in all directions ; and therefore particles, whose vibrations are in a given phase, lie on the surface of a sphere, the radius of which grows uniformly. Since the energy of vibration is distributed over the surface of this growing sphere, it follows that the intensity diminishes proportionately as the surface increases. It is therefore inversely proportional to the square of the distance from the source. Sound travels faster than usual when the air through which it moves is blowing in the direction in which the sound moves, for the two speeds are simply superposed. Similarly, when the wind is blowing in a direction opposite to that in which the sound moves, ct a,' a," 3C FIG. 76. the speed is distinctly diminished. And, In addition to this, motion of the medium affects the distance at which the sound may be audible. For, if ab represent the (vertical) front of a plane-wave of sound travelling with the wind in the direction ax, it is evident, since the motion of the upper strata of air is less retarded by 12 178 A MANUAL OF PHYSICS. friction than that of the lower strata, that the wave-front gradually becomes more and more inclined to the vertical as it moves forward. Successive new positions are indicated by the lines a'b', etc. And the sound, instead of travelling straight forwards, is thrown down towards the ground in the manner indicated by the curved lines in Fig. 76. When the wind blows in a direction opposite to the direction of motion of the sound, the sound is thrown up from the earth's surface so as to be inaudible at comparatively short distances. 162. Reflection of Sound. When sound strikes an obstacle, it is reflected in such a way that the reflected ray (the word is used by analogy from the phenomena of light, which, we shall see subse- quently, are due also to wave-propagation) and the incident ray. make equal angles with, and lie in a 'plane passing through, the normal to the surface. Thus, if ab (Fig. 77) represent a plane surface from which the ray ec is reflected in the direction cf, the line cf lies in a plane passing through ec and the normal cd, and the angles ecd and fed are equal. This law is identical with that which obtains in the reflection of light, so that all the results which are deduced in Chap. XVI. regard- ing the reflection of light from surfaces will at once apply to the case of reflection of sound. The law can be deduced as a result of the fact that sound consists in wave-motion, in precisely the same way as that in which the corresponding law is deduced in 186 as a result of the wave-theory of light. Echoes are due to the reflection of sound from buildings, rocks, trees, clouds, etc. They occur even when there is no visible object to account for their existence. In this case the reflection must occur at the common surface of two large masses of air of different densities, or containing very different amounts of moisture per unit volume. If the reflecting surface be curved, the reflected sound may be conveyed to a focus so as to be much more distinctly heard than the direct sound. SOUND. 179 163. Refraction of Sound. Since the speed of sound has different values in media of different densities, and since, in any one homogeneous medium, the sound spreads out uniformly in all direc- tions from a centre of disturbance, it follows that the direction in which a ray is travelling is, in general, suddenly changed when the sound passes from one such medium into another. If it pass from a less dense into a more dense medium, the direction of propagation is inclined to the normal at a smaller angle after the interface is passed than before ; and the opposite statement holds when the first medium is denser than the second. This phenomenon is known as the refraction of sound. The law is that the incident and the refracted rays lie in one plane with the normal to the refracting surf ace, and make with it angles whose sines bear a con- stant ratio to each other. If ab (Fig. 78) represent the intersection of the common surface of the two media by the plane of the paper, while cd is the normal, and if eo be the direction in which the sound impinges upon the surface, while of is the direction which it takes after entering the second medium, the angles i eoc and r=fod are connected by the relation sin i=n sin r, where /* is a constant. These results are identical with those which are observed in the refraction of light .( 187), and the reasoning of 200 may be applied directly to the present case. A lenticular bag, filled with carbonic acid gas, has been found to convey sound to a focus in precisely the same manner that a glass lens conveys light to a focus. 164. Diffraction of Sound. When sound enters a room by an aperture such as a window, it is equally well heard at all parts inside the room. It bends round so as to penetrate every portion, and does not cast a sharp ' shadow ' of an obstacle as light does. This bending 122 180 A MANUAL OF PHYSICS. of sound into the region behind an obstacle is termed diffraction. We shall see afterwards that the same phenomenon also occurs, under suitable conditions, in the case of light. But it is possible, by proper means, to produce sound shadows. The necessary condition is that the aperture through which the sound passes, or the obstacle by which it is intercepted, shall be large in comparison with the wave-length of the sound. Thiis a sound made at one side of a steep high bank may be totally unheard at the other. So, also, a sound may be clearly heard through a hollow between two mountains, while it is inaudible in the region behind either mountain. And, indeed, by using sound of sumciently short wave-length, we can produce comparatively sharp sound shadows of obstacles which are only a few inches in diameter. (For' an explanation of the phenomenon, see the discussion of the diffraction of light, Chap. XVIII.) 165. Interference of Sound. As sound consists physically of waves of condensation and rarefaction, it follows that two sounds may ' interfere ' with each other, according to the usual laws of inter- ference of waves. Thus, if two sounds are travelling in the same direction through a given medium, and if the intensity and wave- length of these sounds are identical, there will be no resultant dis- turbance of the medium, i.e., no sound will be heard, provided that the maximum condensation due to one of the disturbances occurs simultaneously with the maximum rarefaction due to the other. If both condensations, and therefore both rarefactions, take place together, a sound of four times the intensity will be heard, for the resultant amplitude of vibration of the particles of the medium is doubled. And, if the wave-lengths of the separate disturbances are not precisely identical, the resultant sound will periodically vary in intensity from the former (zero) to the latter (quadruple) value. 166. Pitch. The resultant vibration of a particle of air is, in general, extremely complex ; but, when a pure tone is transmitted through air, the vibration of each particle is simple harmonic. There can, therefore, be no difference between one tone and another except such as is due to a difference in the amplitude of vibration or to a difference in the period of vibration. As the amplitude of vibration determines the intensity of the sound, we infer that the pitch of a note depends upon the frequency of vibra- tion, i.e., upon the number of vibrations which are executed per second. That this is actually the case may be roughly ascertained by means of very simple apparatus. If a piece of cardboard be pressed against t*he edge of a toothed wheel, which revolves at a SOUND. 181 definite rate, a sound is emitted which is of a fairly definite pitch. As the speed of the wheel increases, i.e., as the number of impacts per second between the teeth of the wheel and the cardboard increases, the pitch of the note rises ; and, to a note of given pitch, there corresponds a definite rate of rotation of the wheel, and consequently a definite rate of vibration of the cardboard. A much more accurate proof is obtained by means of the syren. This instrument consists essentially of a perforated disc of metal, the perforations being arranged in a circle round the centre, as in the figure. A tube, through which air is driven, is placed behind the disc ; and, as the disc revolves, each successive opening in it FIG. 79. comes opposite the end of the tube. As each blast of air passes through, a state of condensation is produced, which is succeeded by a rarefaction in the interval between tw T o blasts. Hence a sound-wave of constant period is set up when the disc revolves at a uniform rate. The rate of rotation, and the number of perforations in the complete circle, being known, we get at once the number of vibrations per second corresponding to any note of given pitch. When the disc revolves very slowly, no musical sound is heard, but each separate air-blast can be distinctly heard. As the speed increases, the ear ceases to distinguish the separate pulses, and a note of very low pitch becomes audible. The speed still increasing, the pitch of the note produced becomes higher and higher, and at last it becomes so high that the note is no longer audible. The limits of audibility vary considerably in different observers, but, roughly speaking, a sound becomes inaudible as a note when the rate of vibration falls short of 20 times or exceeds 70,000 times per second. The fact that a melody is perfectly heard at different distances from the source shows that the speed of sound does not depend upon the wave-length. / "The apparent pitch of a note depends tipon the relative motion of I ' 182 A MANUAL OF PHYSICS. , the hearer and the instrument upon which the note is sounded. If the hearer approaches the instrument, the pitch of the note seems to be heightened, because more than the normal number of vibra- tions reach his ear in a given time ; and, conversely, if he recede from the instrument, the pitch of the note will appear to be lowered. As a particular case, if the rate of recession were greater than the speed of sound, no sound could be heard. 167. Musical Intervals. If any particular note be taken as a keynote, it is found that there are certain other notes, definitely related to it as regards pitch, which produce specially pleasing effects upon the ear when sounded with each other or with the key- note. These are, therefore, the notes which are employed in the ordinary major and minor scales, and the difference in pitch of any two notes is called the interval between them. When two notes have the same pitch, they are said to be in unison. The chief interval is the octave, and it is found that, in order to produce the octave of any given note, the rate of vibration must be exactly doubled. The intervals into which the octave is subdivided are not equal. The following table indicates their values. The first column gives the names of the intervals which separate each note from the keynote, and the relative rates of vibration in the two notes forming each interval are given in the second column. For example, nine vibrations are performed in the higher of two notes separated by the interval of a second in the time in which eight vibrations are performed in the lower : Unison ]-. Second g. Minor Third ... ... ... . Major Third |. Fourth \. Fifth for 2 (f). Minor Sixth f or a(). Major Sixth f or a($). Minor Seventh ... ... ... ^ or 2(|). Major Seventh */. Octave f. A very little consideration will show that in order to find the sum of two intervals we must multiply together the fractions given above for each interval. Thus a fifth is the sum of a minor third and a major third. Also, to find the difference between two intervals we must divide the fraction corresponding to the larger by SOUND. 188 the fraction corresponding to the smaller. The second method of writing the fractions corresponding to the minor seventh, the major and minor sixths, and the fifth, shows that the interval between the notes so indicated and the octave are respectively a second, a minor third, a major third, and a fourth. 168. Vibrations of Bods. We shall assume that the extent of the vibrations is such that Hooke's Law ( 135) is followed. In this case, since the period of vibration is independent of the extent of vibration, a musical note of constant pitch will be heard, provided that the rate of vibration is sufficiently rapid. Two kinds of vibra- tions have to be considered. Transverse Vibrations. We see from 51, 63 that the time of simple harmonic vibration of any material system varies directly as the square root of the mass to be moved and inversely as the square root of the stress called into play by a given displacement. It therefore, in the case under consideration, varies inversely as the square root of the flexural rigidity. Now both the flexural rigidity and the mass of a rod are ( 133) proportional to its breadth, and so the period of transverse vibration is independent of the breadth of the rod. Again, the mass of the rod is proportional to the length, while the rigidity varies inversely as the cube of the length. Hence the period is proportional to the square of the length. Similarly, it varies inversely as the thickness, since the rigidity is directly pro- portional to the cube of the thickness. Hence we conclude that the time of transverse vibration of similar rectangular rods is in direct proportion to their linear dimen- sions. And considerations of a like kind enable us at once to extend this statement to the case of similar rods of any form ; for the masses of such rods are in proportion to the cubes of their linear dimensions, while their rigidities vary as the first power of the dimensions. Longitudinal Vibrations. If a rod be fixed at one end and receive a smart blow on the other end in a direction parallel to its length, a wave of compression will travel along it to the fixed end, from which, by reflection, it will return to the free end. The rod will now extend, because of its elasticity, to a length greater than its normal length, and a wave of rarefaction will travel along it to the fixed end at which it too will suffer reflection. The free end is obviously a loop, or place of maximum motion, while the fixed end is a node. Hence the length of a wave is four times the length of the rod, and the time occupied by the disturbance in travelling from end to end of the rod is therefore one-quarter of the periodic time of longitudinal vibration. 184 A MANUAL OF PHYSICS. . If the rod be fixed at both ends, the wave-length is equal to twice the length of the rod, since each end is now a node. In each case the periodic time is obtained by dividing the length of the wave by the speed of the disturbance. The speed is given by equation (7), 160, ~k being in this case the constant called Young's modulus. From this we see that the period is proportional to the length of the rod and the square root of the density, and is inversely proportional to the value of Young's modulus, while it is independent of the sectional area. The rod which is fixed at both ends has twice the rate of vibra- tion of a similar rod fixed at one end only. A rod which is free at both ends, and has its node at the centre, obviously vibrates at the same rate as does a like rod with both ends fixed. 169.. A rod which is fixed at one or both ends may have more than one mode of transverse or longitudinal vibration. In Fig. 80 a represents the fundamental mode of vibration of a rod fixed at one end, while b and c represent the modes which stand next to it in order of simplicity. This is the case of a vibrating tuning-fork. When the fork is feebly excited, the fundamental mode of vibration is the chief one which occurs ; but, under violent excitation, the forms 6, c, and other higher forms, are superposed upon it. The extra node in b is distant from the fixed end two -thirds of the whole \/ i c d e FIG. 80. length of the rod, since the free part and the looped part must have a common period of vibration. The length of the free part in b is therefore one-third of the length of the free part in a, and conse- quently the period of vibration in the case b is one -ninth of the fundamental period. A reference to the table of 167 will therefore show that the note given out by a fork vibrating in the mode b differs in pitch from the fundamental note by the interval of three octaves and a second. In the figure, d, e, and /, show the three simplest modes of trans- verse vibration of a rod which is fixed at both ends. The interval between the notes given out by similar rods vibrating as in d and e SOUND. 185 is two octaves, while the interval in the cases indicated by d and f is the same as between those indicated by a and b. The modes of longitudinal vibration of rods are precisely. identical with those of air - columns, which will be discussed shortly. 170. Vibration of Plates. A plate may be set in vibration by drawing a bow across its edge. The modes of vibration depend on the form of the plate, and may be greatly varied by forcing certain points to lie upon nodal lines which may be done by keeping the fingers in contact with the desired points. As in 168, we can without difficulty deduce the result that the periods of vibration of similar plates (similar, of course, as regards thickness as well as form) of the same material are in proportion to their linear dimensions. Also, as in the section referred to, we may show that the period is inversely as the thickness of a plate of given form and area. By combining these two results we see that the periods of vibration of similar plates of the same thickness are pro- portional to the squares of their linear dimension, that is, to their areas. 171. Vibrations of Strings. The speed, v, with which a distur- bance runs along a stretched cord is (^ 73) proportional to the square root of the tension, T, and is inversely proportional to the square root of p, the mass per unit length of the string. Hence, if X be the wave-length, the time of a complete vibration is where p is the density of the material of the string and s is its area of cross-section. The figure below shows the three simplest forms of vibration of a stretched string which is fixecl at its ends. Any disturbance of the FIG. 81. string is propagated with equal speed in both directions along the string from the point of disturbance, each part being reflected when it arrives at a fixed end. Hence, ( 53), the string is thrown into 186 A MANUAL OF PHYSICS. segments, the wave-length in each case being equal to twice the length of a loop. Thus the above equation enables us to assert that the funda- mental period of vibration of a string 1 is directly jpropor/icmfl/ to its length, to the square root of its density, and to the square root of its sectional area, and is inversely proportional to the square root of its tension. If a string of length I vibrates n times per second (determined by means of the syren), the speed of the wave is lln. The higher vibrations correspond to tones which are respectively one, one-and-a-half, two, two-and-a-quarter, etc., octaves above the fundamental tone< It is interesting to compare this result with the corresponding result in the case of a vibrating rod which is fixed at both ejids. In the latter case, the transversely propagating force is the flexural stress ; in the present case, it is the tension of the cord, in which flexural stress has no existence. 172. Vibration of Air -Columns. When a condensation reaches the closed end of a pipe, or tube, containing air through which sound is travelling, the increase of pressure which takes place can only be relieved by expansion of the air backwards along the pipe. Similarly, when a rarefaction reaches the closed end, the consequent diminution of pressure can only be changed by a flow of the air contained in the pipe towards that end. In other words, the wave is simply, reflected at the closed end, the result being that two sound-waves are travelling simultaneously along the pipe, in opposite directions, with equal speed. And the state of pressure due to each is always in the same phase at the closed end of the pipe. But, at a certain distance from the closed end, an outgoing condensation meets an incoming rarefaction. Hence, on the assumption that no energy has been lost in the process of reflection, the pressure has here its normal value. And this normal condition is maintained constantly so long as the wave-length of the disturbance is un- altered, for the two oppositely - travelling waves are always in opposite phases as regards pressure at this place, which is evidently distant by one-quarter of a wave-length from the closed end of the pipe. The conditions of motion of the particles of air are simply reversed when the wave is reflected, and, consequently, the incident and the reflected disturbances are in opposite phases as regards vibration when they are in the same phase as regards pressure or density. Hence the places of maximum variation of pressure are places of no vibration, and the places of uniformly normal pressure are those of maximum vibration. Thus a node occurs at the closed SOUND. 187 end of the pipe, and other nodes appear at equal intervals of half a wave-length. It is obvious that a loop, or place of greatest motion, must occur at the open end of a pipe. For, when a condensation reaches it, the air is free to expand outwards much more free to do so, in fact, than to expand inwards and consequently the condensation is succeeded by a rarefaction which is propagated back through the pipe. Similarly, an incident rarefaction is filled up by the influx of the surrounding air, and so a condensation is returned. Thus we see that there is great vibration at the open end of the pipe, which therefore corresponds to a loop. [This might otherwise have been determined by means of the consideration that, in the immediate neighbourhood of the open end, the air-pressure is uniform and equal to the normal atmospheric pressure. Consequently the open end is a place of maximum vibration.] Stopped and Open Pipes. A pipe which is closed at one em is called a ' stopped ' pipe, while one which is open at both ends is called an ' open ' pipe. In a stopped pipe the nodes evidently occur at the same positions as do the nodes of a transversely-vibrating rod which is fixed at one end. Thig is indicated in the figures below, in which the vertical distance between the waved lines at any part of the length of the pipe may be supposed to indicate the extent of the vibrations at that part. The first figure shows the fundamental mode of vibration. The wave-lengths, and consequently the periods of vibration, in all the possible modes are evidently inversely as the odd numbers 1, 3, HC X X FIG. 82. 5, etc. ; and, in the fundamental mode of vibration, the length of the wave is four times the length of the pipe ; for a condensation which leaves the open end has its phase reversed when it again reaches that end and is reflected. Two reversals are therefore necessary in order that the original phase may be attained. 188 A MANUAL OF PHYSICS. , In an open pipe the nodes occur at the same positions as do the nodes of a transversely-vibrating rod which is not fixed at either end. The positions are indicated below. The possible periods of vibration are inversely as the natural numbers 1, 2, 3, etc. The FIG. 83. wave-length in the fundamental mode of vibration is twice the length of the pipe ; for a condensation passing from one end of the pipe is reflected from the other end as a rarefaction, and leaves the original end once more as a condensation. Hence an open organ- pipe which has the same fundamental tone as a given closed organ- pipe must be twice as long as the closed one is*. Speed of Sound. In order to determine very accurately the speed of sound in any given gas we merely require to sound an organ- pipe of given length which is filled with that gas. By means of the syren the number of vibrations which are made per second is found. And if I be the length of the pipe, while n is the number of vibra- tions made per second, the speed is 4nl, or %nl, according as the pipe is closed or open. Under ordinary atmospheric conditions the speed of sound in air is about 1,120 feet per second. 173. Partial Tones. Resonance. From the results of the last few sections it is now evident that no note given out by a musical instrument is, strictly speaking, a pure tone ; in most cases it very obviously is not so. Thus the ' tongue ' of a reed is simply a rod or strip of metal which is fastened at one end, and is caused to vibrate by means of a current of air which is rendered intermittent by the vibrations which it produces in the tongue ; and these air-pulses produce a note in which occur tones corresponding to the various forms of vibration which the tongue assumes. These constituent tones of a note are called the partial-tones ; and it is usual to distinguish between the fundamental tones (corre- sponding to the fundamental mode of vibration of the sounding body) and the higher partial-tones, which are termed the overtones. SOUND. 189 Much (greater force is required to set a body in rapid vibration than to set it in slow vibration, and consequently the overtones of a note are feebler than the fundamental tone each overtone, taken in order of pitch, being weaker than the preceding one. In this way a note is said to have the same pitch as its fundamental tone. The unaided ear is able to detect in great measure the various overtones which are present in a given note ; but, in this analysis, it may be greatly aided by means of instruments, the action of which depends upon the principle of resonance, according to which any sounding body can readily absorb, and give out again of itself > a sound wliicli is emitted by another sounding body which has a period of vibration identical ivitli its own. To understand this principle, we need only refer to a well-known dynamical analogy : A pendulum of given length has a definite period of vibration, and oscillations of great magnitude may be induced in it by the application to the bob of the feeblest impulses, provided only that these impulses are communicated regularly at instants the interval between which is equal to the natural period of oscillation of the pendulum. The effects of all the impulses are in the same direction, and so the total effect may be very large although each single effect is excessively small. And, similarly, the feeble periodic impulses which are communicated through the medium of the air to a body which is capable of giving out sound may, by being properly timed, set that body in such a state of vibration as to give out the sound of itself after the original note has ceased. The resonator of v. Helmholtz consists of a hollow brass ball with two apertures at opposite ends of a diameter. (This is indi- cated in Fig. 84.) Sound is communicated to the air in the ball through the large aperture, and the small aperture is applied to the FIG. 84. ear. The air inside the ball has a definite fundamental period of vibration, and therefore that tone (should such exist) which corre- sponds to this period is very distinctly heard all others being entirely, or almost entirely, suppressed. 190 A MANUAL OF PHYSICS. In many cases it is desirable to intensify the fundamental tone of a note. The principle of resonance shows that this may be done by associating with the sounding body a resonating column of air which has the same fundamental period of vibration. Thus a reed is fitted at one end of an open pipe of the proper length ; and a tuning-fork is attached to a ' sounding-box,' which is simply a closed pipe of such a length that the contained air-column has the same fundamental period of vibration as the tuning-fork has. The box itself is set in vibration by the fork, and so the motion is communi- cated to the enclosed air. The cavity of a violin acts similarly as a resonator. The sounding-board of a pianoforte is thrown into vibration by the wires which are attached to it, and so increases the mass of air which is made to vibrate when a note is struck on the instrument. Thus, by its resonance, it intensifies the sound. 174. Quality. There is between two pure tones of the same pitch no difference of quality such as that which distinguishes the same note when played on two different instruments. We therefore conclude that difference of quality between two notes is due to the existence of partial-tones. Experiment shows that this is actually the case, and it indicates further that the quality of a note depends upon the number of partial-tones which are present in it ; that, the number being the same, it depends upon the particular set of tones which go to make up that number ; and, lastly, that it depends upon the relative intensities of the partial-tones. [It is independent of the particular phase in which any one of the constituent vibrations may be when the phases of the others are given. Now the nature of the resultant vibration essentially depends upon this condition ( 52, (1) ), and so the above result appears somewhat startling. Its truth is due simply to the fact that the human ear is an instrument which resolves a compound vibration into its separate constituents, and thus mere alteration of phase of a constituent has no effect upon the nature of the sound which is heard.] As a special example, we may instance the difference of quality between two notes of the same pitch sounded respectively on a closed and an open organ-pipe. In a closed pipe the odd partial- tones alone occur, while in the open pipe both the odd and the even tones are present. 175. Beats. Consonance and Dissonance. It has already been pointed out ( 165) that the intensity of the resultant sound due to two component vibrations of slightly different period varies SOUN&. Idi periodically from a minimum to a maximum. These regularly- occurring maxima constitute beats. One beat occurs in the time in which one vibration gains a complete period upon the other ; and, consequently, the number of beats which occur per second is equal to the difference of the number of vibrations which take place per second in the two component tones. When the beats succeed each other with too great rapidity, the ear becomes unable to distinguish them from one another, but it still recognises a cKstinct discontinuity in the sound, which produces a harsh effect known as dissonance. The number of beats which occur per second when two given pure tones are sounded depends, of course, upon the absolute, as well as upon the relative, pitch of these tones. When the tones are in the neighbourhood of the middle C, maximum dissonance is pro- duced when they differ in pitch by about half a tone ; and there is almost no dissonance when the interval is a minor third. On the other hand, beating may occur between the partial-tones of two notes, and the result is that there may be considerable dis- sonance between the two, even although their fundamental tones differ by more than a minor third. Indeed, from this cause, none of the intervals below the octave form a perfect concord. Still, it is only in the cases of the second, and the major and minor sevenths, that the result is actually classed as a discord. In some cases the greatest dissonance, as above denned, does not produce the most disagreeable effect upon the ear. 176. Combination Tones. Dissonance of Pure Tones. When the rapidity of the beats is sufficiently great, the effect upon the ear is that of a musical note the pitch of which is the same as that of a tone in which the number of vibrations per second is equal to the difference (or sum) of the number of vibrations per second in the two tones which are producing the beats. Such tones are called combination tones. Combination tones of a higher order may be produced between the first combination tone and either of the primary tones, and so on. Beating may occur between a combination tone and a primary, or between two combination tones. The result is that dissonance may occur in the case of two pure tones where the interval is greater than a minor third. Thus, when the interval is a major seventh, the higher tone will make 450 vibrations per second if the lower makes 240 per second. The combination tone has therefore 210 vibrations per second, which produces 30 beats per second with the lower primary. CHAPTEE XV. LIGHT : INTENSITY, SPEED, THEORIES. 177. Rectilinear Propagation. Intensity. In a homogeneous medium, light, in general, moves in straight lines. The fact that the shadow of an object which is cast on the ground by sunlight is not defined with mathematical accuracy, but has a more or less blurred edge, does not disprove the statement. . The indistinctness of the boundaries of the shadow is due to the finite size of the sun's disc, the light proceeding from each point of which produces a separate shadow. The cases in which 'the above rule is departed from will be discussed in Chap. XVIII. The total intensity of a luminous source is measured by the amount of light which it emits per unit time, and the intensity of the light at any given point of the medium is measured by the quantity which falls per unit time upon unit area taken perpen- dicular to the direction in which the light moves at that point. And, since light moves out uniformly in all directions from a point - source in a homogeneous medium, it follows that its intensity varies inversely as the square of the distance from the source ; for the same total quantity is, at different instants, spread over the surfaces of different concentric spheres, the areas of which vary directly as the squares of their radii. (We assume, of course, that the medium is one which does not absorb the light in its passage through it. If any absorption did occur, the quantities of light passing through the different concentric spheres could not be equal.) Instruments used for the purpose of comparing the intensities of different sources of light are called photometers. The simplest form of photometer consists of a sheet of paper upon which there is a grease spot. If the paper be illuminated from behind, the spot appears bright ; if it be illuminated from the front, the spot seems dark ; if it be illuminated to an equal extent on both sides, the spot vanishes. Under the latter condition the intensities of the sources are inversely as the squares of their distances from the spot. LIGHT : INTENSITY, SPEED, THEORIES. 198 In another form of the instrument two grease spots are employed, each being illuminated from behind by one source alone. The distances of the sources are varied until the two spots appear equally bright. 178. Speed. In last section we have spoken .of the motion of light. The use of the term is justified by the fact that a flash of light is not simultaneously seen by two observers who are situated at different distances from the source. When the distance between the two points of observation is not very large a few miles, say the interval of time which is occupied by the light in passing from one point to the other is so small that it cannot be measured except by very special means. But there are two astronomical methods of determining the speed of light which do not involve the measure- ment of a small interval of time. The first of these is due to Homer, who observed that the eclipses of Jupiter's satellites do not appear to recur at equal intervals of time, and pointed out that this would be a necessary consequence of the finite speed of light. In order to understand more clearly how this may be, we may take an illustration from the phenomena of sound. If an observer be situated at a fixed distance from a point at which a gun is fired off at equal intervals of one minute, he will hear the report at equal intervals of one minute. But if, between two successive discharges of the gun, he move nearer to it, he will hear the next report at a shorter interval of time than one minute ; while, if he move farther from it, the interval will necessarily be greater than one minute. (When applied to light ( 204), this principle is usually called Doppler's principle.) The speed of sound may be determined from the results of two such observations. Let r be the time which elapses between two successive discharges, and let t be the interval noted by the observer between two successive reports when he has meanwhile increased his distance from the gun by the amount d. Then, t' being the (unknown) time taken by the sound to pass over the distance d, we get = r-H'> and therefore the speed of sound is given by the quotient of d by t - r. Now the eclipses of Jupiter's satellites occur at instants which are very accurately calculable from known astronomical data. But the observed instants at which the eclipses apparently take place as seen from the earth do not coincide with the calculated instants ; and the errors at different times of the year are (assuming the finite speed of light) due to the variation of the distance between Jupiter and the earth. The greatest difference of apparent errors is the time which light takes to pass over the greatest difference of dis- 13 194 A MANUAL OF PHYSICS. tance. But the greatest difference of distance between Jupiter and the earth is the diameter of the earth's orbit ; and so we obtain the time taken by light to pass over this known distance. The speed of light as deduced by this method is about 186,000 miles per second. The other astronomical method is due to Bradley. He observed that the fixed stars appear to describe small ellipses on the surface of the heavens in the course of a revolution round the sun, each star being displaced from the centre of its elliptic path in the direction of the earth's motion in its orbit, and each to the same amount. He concluded that this was due to the finiteness of the speed of light as compared with the speed of the earth in its orbit. A simple illustration may make this clear. On a still day, rain- drops fall vertically downwards. But, if one moves forward with considerable speed, they do not seem to fall vertically ; they apparently fall in a slanting line, which is inclined forwards from the vertical in the direction of the observer's motion. And it is evident that the apparent velocity of the drops is the resultant of their actual velocity and a velocity equal and opposite to that of the observer. The light which comes from a star appears to come in a direction which depends in the same way upon the velocity of light and the velocity of the earth in its orbit. This latter velocity and the maximum angular displacement of a star from its true position being known, we can calculate the speed of light. The value which is obtained by this method agrees very closely with that obtained by Eomer's method. Fizeau was the first (1849) to determine the speed of light by direct experiment. He caused a beam of light to pass out through the gap between two of the teeth of a toothed wheel, the teeth and gaps of which were all of one size. This beam was reflected from a mirror, placed at a distance of a few miles from the wheel, in such a way that it passed back again through the same gap between the teeth of the wheel. The wheel was then caused to rotate, and, at a certain rate of rotation, it was found the light ceased to pass back between the teeth ; the reason being that an adjacent tooth had moved into the place of the gap in the time that the light took to travel twice over the distance between the wheel and the mirror. The rate of rotation of the wheel, and the number of teeth which it contained, being known, the time which was taken by the light to pass over the given distance can be readily found. If N is the number of revolutions which it made per unit of time, while n is the total number of gaps and teeth in its circumference, LIGHT : INTENSITY, SPEED, THEORIES. 195 the speed of light is 2dNn, d being the distance between the wheel and the mirror. Fizeau's experiments were repeated some time afterwards by Cormi with improved apparatus ; and, still more recently, a further improvement of the same method has been effected by Professor G. Forbes and Dr. J. Young. In Foucault's method (recently improved by Michelson), a beam of light, after passing through a slit, falls on a mirror which can be made to rotate about an axis parallel to the slit. After reflection from the mirror, the light passes through a lens, which brings it to a focus on a fixed mirror. This mirror being so placed as to exactly reverse the course of the beam, the light once more falls upon the first mirror, and is reflected from it. If the latter is rotating, and has turned through a sensible angle in the time taken by the light to pass twice over the distance between it and the other, the beam will not pass back through the slit, but will be deflected from it through a measurable distance. The speed of light may be found in terms of the two distances just mentioned, together with the rate of rotation of the revolving mirror and the distance between it and the slit. These two experimental methods give values of the speed of light which agree very closely with the values obtained by the two astronomical methods. Foucault's method is so sensitive that it may be used successfully when the distance between the mirrors is only a few feet, and it lends itself readily to the determination of the speed of light in different media, such as glass, water, etc. The speed is found to be less in dense, than in rare, media. 179. Theories. The transference of light involves motion of matter ; for when light is absorbed by any body, increased motion J* of the particles of that body is generally produced. Indeed, in the radiometer ( 153), visible motion of a considerable mass of matter may follow the absorption of light. Another marked example will be found in 376. Hence we see that transference of light implies transference of energy ; and it is in this sense that we speak of light as a form of energy. We are therefore limited to two suppositions regarding the physical nature of light. It may consist in the actual propagation of material particles, or corpuscles, from the luminous object ; or, it may consist in the propagation of wave-motion through a material medium which fills space, 132 196 A MANUAL OF PHYSICS. The former theory is known as the Corpuscular Theory, the latter as the Wave Theory, or Undulatory Theory, of light. If the corpuscular theory were true, the mass of a corpuscle must be excessively small. For vision, according to this theory, is due to the impact of the corpuscles upon the retina; and the speed of these corpuscles is so great that, unless their individual mass were almost vanishingly small, the structure of the eye would be completely destroyed by the impact. The theory is met by a number of difficulties at the very outset. Thus it is somewhat difficult to account for the fact that the corpuscles have the same speed whatever be the temperature of the object from which they are projected. Again, the mass of a luminous body must be appreciably affected by the emission of particles ; but there is no evidence of any such effect. Still, if we boldly overlook any such preliminary difficulties, we shall find that the theory enables us to account readily for many of the phenomena of light, although ultimately it fails us altogether. On the wave-theory, vision is due to the communication of the vibrations of the assumed luminiferous medium (called the ether) to the nerve-ends of the retina. The molecules of a luminous body are ( 202) in rapid vibratory motion, and this motion is communicated to the particles of the ether, and is propagated through it from par- ticle to particle giving rise to a series of waves which travel with the speed of light. The investigation of 161 has a direct applica- tion to the present case, and shows that the intensity of light is pro- portional to the square of the amplitude of vibration of the particles of the medium, and that the energy of the medium, when light passes through it, is one-half kinetic, one-half potential. (On the corpuscular theory the intensity must be proportional to the space - density of the corpuscles.) We shall find subsequently (Chap. XIX.) that the direction of vibration in the medium must be perpendicular to the direction of propagation of the waves. The wave-length is the distance, measured in the direction of propagation, from any point to the next point where the motion is similar. (Compare 157.) The wave-theory is not without its difficulties many of them, indeed, are of a most formidable nature. But, as will appear, the evidence in favour of it is of such an overwhelming nature that we now practically regard its truth as definitely established. Newton rejected it because he was unable to explain by it the rectilinear propagation of light. We now know that the existence of rays is a necessary consequence of the fundamental principles of the theory. LIGHT : INTENSITY, SPEED, THEORIES. 197 180. Colour. Many different kinds of light are recognised. We speak of red light, blue light, etc. On the corpuscular theory the difference must be inherent in the corpuscles. On the wave-theory the difference is a mere difference of wave-length, or of vibrational period which is only another way of stating the same thing, since the speed of propagation of light of all colours has, in free space, one definite value only. CHAPTER XVI. LIGHT : REFLECTION, REFRACTION, DISPERSION. 181. Laws of Reflection. When a ray of light reaches the bound- ing surface of a homogeneous medium through which it is passing, it is, in part at least, bent back or reflected, and pursues a different, though still rectilinear, path. The reflected and incident rays lie in one plane with, and make equal angles with, the normal to the surface. FIG. 85. Let EBF (Fig. 85) represent a section of the bounding surface by the plane of the paper, and let BD be the normal to the surface at the point B whereon the incident ray AB falls. Then, BC being the reflected ray, the angles i and r, which AB and BC make with BD, are equal, and AB, BC, and BD all lie in one plane, which is normal to the reflecting surface. The angles i and r are called, respectively, the angle of incidence and the angle of refraction. Many surfaces, such as those of chalk or of rough white paper, scatter the incident light in all directions. But this is merely a special case of reflection. At every point of such a surface rays are reflected in accordance with the above law ; but the whole surface is practically made up of an excessively great number of very small planes, which are indiscriminately inclined in all possible ways. The intensity of the reflected ray depends upon the angle of inci- dence, being greatest when the angle is large, and having its least LIGHT : REFLECTION, REFRACTION, DISPERSION. 199 value when the incidence is perpendicular. It varies (the intensity of the incident ray being fixed) with the nature and the state of surface-polish of the reflecting substance, and it depends also upon the nature of the medium through which the light is travelling. 182. Reflection from Plane Surfaces. If a ray of light be emitted from a point B (Fig. 86) and reach a point A, after reflection D FIG. 86. from a plane surface, CD, the actual length of the path APB is the shortest possible consistent with the condition of reflection at the given surface. For an eye placed at the point A will see the light in the direction AP as if it came from a point B', which is situated on the normal drawn from B to the surface. [This is so since the eye sees an object by means of a cone of rays : and the angle of the cone is un- altered by reflection since (Fig. 87) the angles which a^pi and a. 2 p. 2 FIG. 87. make with the reflecting plane are respectively equal to the angles which Pid'i and^ 2 a' 2 , the continuations of the lines bpi and bp. 2 , make with that plane. And so the continuations of a^ and a. 2 p 2 meet at a point &', which is such that bpib' and bp. 2 b' are both bisected by the surface ; and therefore b' and 6 lie on the same normal to the surface, and are equally distant from it.] And any other path, AP'B, being equal to AP'B', is greater than APB, which is equal to APB'. The point B' is called the image of the point B. If B were a body of finite size, each point of it would give rise to an image ; and the whole congeries of these point-images constitutes the image of the body B. 200 A MANUAL OF PHYSICS. 183. Reflection from Curved Surfaces. Let Q (Fig. 88) be the section of a spherical mirror by the plane of the paper. Let be the centre of the sphere, and let a ray, UP, emitted by a luminous U FIG. 88. object at'the point U, be reflected to the point V. We have PVQ = QPV+ PUQ = 2UPO + PUQ. Therefore PVQ + PUQ = 2(UPO + PUQ) = 2POQ. When P and Q are nearly coincident this becomes approximately PQ/PV+PQ/PU = 2PQ/PO or 1/PV+1/PU = 2/PO. If we denote by u, v, and r, the lengths of the lines PU, PV, and PO respectively, this gives This equation enables us to calculate the position of the image V when the position of the object U is given. If U be situated at infinity towards the left-hand side of the diagram, V is half-way between and Q. This point is called the principal focus of the FIG. 89. mirror. As U moves in from infinity V moves out to meet it, and the two points coincide at O, the centre of the sphere. The positions of U and V are now interchanged, and finally, when U is at the principal focus, V is at infinity towards the left. Whenever U comes nearer Q than the distance of a semi-radius, the quantity v becomes negative, that is, the image passes away (Fig. 89) behind the mirror, and gradually approaches it from infinity in this direction until both object and image coincide at Q. LIGHT : REFLECTION, REFRACTION, DISPERSION. 201 In all cases V and U are interchangeable, that is to say, V may be the object and U will then be the image. A slight inspection of the two diagrams will make this clear. The image, when on the same side of the mirror as the object is on, is called a real image ; when on the opposite side, it is called a virtual image. It is evident that an object on the convex side of the sphere can have a virtual image only. In this case the above formula becomes l_l = 2 f v u r which is the modification of the formula necessary to make it apply to the case of reflection from a convex spherical mirror. FIG. 90. The law stated in last section with reference'to a plane mirror that light takes the shortest possible path between two points con- sistent with the condition of reflection at the given surface, still holds in the case of any surface provided that we limit the statement to other paths which do not finitely differ from the actual path of the light. The necessity for this limitation will be evident if we consider FIG. 91. that light diverging from a focus of a reflecting ellipsoid may take the longest possible, as well as the shortest possible, path. Figs. 90 and 91 show positions of real and virtual images : a and a' are mutually real images ; b and b f are mutually virtual images ; b being the image of b' in a convex mirror, while a, a', and b' are 202 A MANUAL OF PHYSICS. the images of a', a, and 6 in a concave one. When formed by one reflection, or by an odd number of reflections, a real image is inverted, but a virtual image is not inverted. The positions of the various points of the image corresponding to given points of the object are found by means of the above formulae. 184. Caustics : Focal Lines. Let CBQ (Fig. 92) represent the section of a spherical mirror by the plane of the paper, and let PC, FIG. 92. PB represent two rays which, diverging from the point P, fall upon the mirror : let also the reflected rays, Cpf and Bp/, intersect in the point p. Since the vertical angles at the intersection of Cp and OB (a radius) are equal, we have OCp+COB = OBp+CpB, or OCP+COB This gives COQ - CPO+COB=BOQ -BPO+CpB, whence CPB+C#B [The results of last section follow as a particular case of this.] Let CB be an infinitesimally small arc of constant length, and let T be the point at which the tangent from P meets the circle CBQ. CPB always diminishes, and therefore (by the above equation) CpB constantly increases as C moves from Q towards T. Hence the length of Cp always diminishes as its inclination to CO increases, until finally p coincides with T. The locus of p is called the caustic curve, and is indicated by the dotted curve in the figure. It touches the circle at the point T and the line PQ at a point m, the position of which may be found by the formula of last section. LIGHT : REFLECTION, REFRACTION, DISPERSION. 203 If the whole figure be rotated about the line PQ, the circle CBQ traces out the spherical reflecting surface, and the caustic curve traces a continuous surface which is called the caustic surface. All points on this surface are more intensely illuminated by the reflected light than any point which does not lie upon it, and the cusp (at ra) is the place of most intense illumination. All rays which are reflected from the surface pass through the line PQ. Let us suppose that a small, but finite, circular cone of rays falls upon the reflecting surface in the neighbourhood of the points B, C. All rays from points on the small circle, the pole of which is Q, and which passes through B, intersect PQ in the point /; and all rays from points on the similar circle through C pass through/'; and so on. It is evident, therefore, that a plane which is perpendicular to the axial line of the reflected cone, and which passes through the point in which the axial line intersects the line PQ, will cut the cone in an elongated figure-of-eight-shaped area, which may be regarded as a straight line, and is called the secondary focal line. Again, a plane drawn perpendicular to the axial line through the point in which that line touches the caustic surface, cuts this surface in a circle, and all the reflected rays will pass through the plane in the immediate neighbourhood of a small, practically straight, portion of the circle ; so that there is another, nearly linear, normal section of the reflected cone. This is called the primary focal line. The two focal lines are mutually perpendicular. An approximately circular section exists between the two linear sections. This is called the circle of least confusion, and is the place where the reflected light most nearly converges to a point. 185. The law of reflection follows readily from the principles of the corpuscular theory. Let pq (Fig. 93) represent the path of a FIG. 93. corpuscle, and let AB be the surface from which the corpuscle is reflected. When the corpuscle comes within a certain small distance from the surface, indicated by the line ab, it experiences the attraction of the medium which is bounded by the surface AB, and so is bent from its rectilinear path. The mutual action of the par- 204 A MANUAL OF PHYSICS. ticle and the medium may alternate from attraction to repulsion many times according to an unknown law, but it must ultimately be a repulsion which (at r) stops the motion of the corpuscle towards the surface. The mutual forces still acting as before, the particle must now describe a path rq' precisely similar to rq, until, at q r , being freed from the action of the reflecting medium, it describes a rectilinear path, q'p', which is inclined to AB at the same angle as pq is. And, since the action of the medium is everywhere in lines perpendicular to the surface AB, the particle retains its velocity parallel to AB unaltered during the process of reflection, and the lines pq and p'q' are in one plane with the normal to the surface. 186. The wave-theory also affords a ready explanation of the phenomena of reflection. But, before dealing with this point, it is necessary to consider the explanation, on this theory, of the rectilinear propagation of light. Newton did not see how to account for it, and so supported the cor- puscular theory. Huyghens was the first to show that the wave- theory furnishes a ready explanation of the phenomenon. FIG. 94. Let AB (Fig. 94) represent a portion of a spherical wave-front diverging from the point 0. All points, such as a, 6, c, on this sur- face become centres of disturbance from which secondary spherical waves diverge. With radius AA', or BB', equal to the distance which light will travel in a certain time, t, describe circles from a, b, c as centres. These circles will all touch another spherical surface, A'B', concentric with AB. This constitutes the new wave- front. At all points of this surface secondary wavelets are super- posed, and so a strong resultant effect may be produced. At no other points, besides those on A'B', are the effects of the separate wavelets superposed, and the isolated wavelets are too feeble to pro- duce light. Hence the rays included in the region AOB diverge outwards in straight lines. The above explanation is due to Huyghens, and will suffice for LIGHT : REFLECTION, REFRACTION, DISPERSION. 205 our present purpose. But, as Fresnel pointed out, Young's principle of interference is essential, in addition to the above, in order to make the demonstration rigorous. See 226. We shall assume for the sake of simplicity that we are dealing with a plane wave propagated through the luminiferous medium. Let ADF (Fig. 95) represent the reflecting surface, and let the given disturbance have reached the position ABC ; so that ABC represents a portion of the plane wave-front, to which the rays (of which three are indicated in the figure) are everywhere perpendicular, the medium being assumed to be homogeneous and isotropic. When the wave reaches the point A, the particles of the ether at that point are set in vibration and give rise to a spherical wave which spreads out from A as centre. If, from A, we draw a sphere with radius AP = CF, we get the position of this spherical wave when the disturbance originally at C has reached the reflecting sur- face. Similarly, DE being parallel to ABC, if we draw from D as centre a sphere with radius DQ = EF we get the corresponding posi- tion of the spherical wave which is originated at the point D when the wave-front reaches it. All such spheres touch a plane surface, PQF, which is, therefore, the wave-front after reflection. Thus we see that a plane wave remains a plane wave after reflection. But, further, AP = CF, and AF is common to the two right-angled triangles ACF and APF. Therefore the angles CAF and PFA are equal, that is, the reflected wave-front has the same angle of inclina- tion to the reflecting surface as the incident wave-front has. And this since the rays are perpendicular to the wave-fronts gives the known law of reflection. 187. Laws of Refraction. A ray of light, on passing from one medium into another of different density, is in general bent from its original direction, and is said to be refracted. The angle which the refracted ray makes with the normal is called the angle of refraction. The refracted and incident rays lie in one plane with the 206 A MANUAL OF PHYSICS. normal to the surface and the sine of the angle of incidence bears a constant ratio to the sine of the angle of refraction, The angles of incidence and refraction (Fig. 96) being denoted by the letters i and r respectively, the above law may be written The constant, /i, is called the index of refraction. When the refraction takes place from a less dense into a more dense medium, H is generally greater than unity ; and, conversely, /* is usually less than unity when the light passes from a more dense into a less dense medium. FIG. 96. FIG. 97. The intensity of the refracted ray depends upon the angle of inci- dence. It is greatest when the incident ray is perpendicular to the refracting surface, and diminishes as the angle increases. In the case of refraction into a denser medium, minimum intensity is attained at grazing incidence : in the case of refraction into a rarer medium, the minimum is reached when the angle of incidence is less than 90 ; and at higher angles of incidence no refraction occurs. 188. Refraction through a Plane Surface. Let a ray, aO (Fig. 97) fall upon the plane surface, AB, of a medium, the refractive index of which is ju that of the first medium being taken as unity. If we assume p to be greater than unity, the path of the ray in the second medium will be a line 06, such that sin aOc = p sin bQd, CD being perpendicular to AB at the point of incidence 0. Also light travelling in the direction 60 will emerge into the first medium in the direction Oa, which is such that sin 60d = sin aOc. /' Now suppose ^00 = 90. The light will enter the second medium LIGHT: REFLECTION, REFRACTION, DISPERSION. 207 in a direction Oc, such that sin cOD=//; and, conversely, light travelling in the direction cO will emerge into the first medium so as just to graze along the surface. Light passing in the direction eO, where eOD > cOD, cannot enter the first medium at all, but will suffer total reflection in the direction oe', in accordance with the ordinary law. (This proves the concluding remark of last section.) The ray eo is said to suffer Total Reflection, and the limiting angle cOT> is called the Critical Angle. The refractive index of water, relatively to that of air, is about 4/3 for ordinary yellow light. Hence an eye placed underneath the surface of water will see objects above the surface by means of rays which are crowded into a cone, the sine of the semi-vertical angle of which is 3/4. The angle of deviation of a, ray from its original direction by a single refraction is i-r,i and r being respectively the angles of incidence and refraction. FIG. 98. From (Fig. 98) as centre describe two circles APB and CQD, with radii equal to unity and // respectively. Let COQ = r, and draw QPN parallel to CO. Then ON = sin OPN =// sin OQN =/* sin r, and so OPN = i. Now OP and OQ are fixed in length, and PQ is always parallel to OC ; so that OPQ and OQP become more nearly right angles, and PQ becomes greater at an increasing rate as i increases up to 90. Hence i - r increases faster and faster as either i or r increase uniformly. Next suppose that PQ moves out from OC through an infinitesimal distance into a position P'Q', and let OQ' be then turned back to coincide with OQ. P' will then take a position p such that AJO is 208 A MANUAL OF PHYSICS. greater than AP. By this process i-r has increased by the amount Pp, and PQO = r has increased by the amount >QP = Pp cos t/PQ. Hence r (i r) 2r~i has increased by the angle Pp (cos -i/PQ 1). This vanishes when cos i = PQ. But cos i=NP, and hence the condition is NP = PQ. This implies AC < OA, that is, fi < 2. Again, the increase of 2r -i is positive or negative according as NP > or < PQ. Hence it is always positive until the limiting angle is reached, after which PQ still increases, while NP diminishes, so that 2r-i diminishes continuously. In other words the con- dition NP = PO or 2 cos i = p cos r (since NQ = /* cos r) indicates a maximum value of 2r i. This condition, combined with sin i=n sin r, gives 3 sin 2 i = 4 - ^ 2 . Similarly, the angle 3r - i increases under the same conditions by the amount Pp (2 cos i/PQ - 1) ; and the condition for a stationary- value is 2NP = PQ or 3 cos i=(J> cos r, which implies /* < 3. As in the former case the condition indicates a maximum. From this along with sin i ^ sin r we get 8 sin 2 i 9 - /* 2 . These results are of importance in the discussion of the primary and secondary rainbows. FIG. 99. 189. Focal Lines: Caustics. Let OPo' (Fig. 99) represent an axial section of an infinitesimal circular cone of rays diverging from a point P placed in a dense medium. The section, by the plane of the paper, of the cone a'pa, by which the object P is actually seen by an eye situated in the rare medium, has its vertex at a different point, p, which is nearer the surface than P is. Let i' t r', and i, r, represent respectively the angles of incidence and refraction (regarded from the rarer medium) of the rays a'o'P LIGHT : REFLECTION, REFRACTION, DISPERSION. 209 and aoP respectively. The angle of the cone, whose vertex is at P. is r' -r, and the angle of that whose vertex is at^> is i'i. To find the relation between these angles we may write the equation sin i' = p, sin r' in the form sin (i' i+i} = n sin (r' r-\-r). This gives sin (i'i) cos i-f-cos (i'i) smi = p [sin (r'r) cos r + cos (r' r) sin r] , and so, remembering that i' -i and r'r are vanishingly small, we get i' i __ cos 'r_tan i r' r cos i tan r But i' -i=0o r cos i[0p, and r'-r = Oo' cos r/OP. Hence OP _ sin i . cos 2 r. Op ~~ sin r cos 2 * Now let Op be prolonged to meet PQ (the line drawn through P perpendicular to the surface) mp f , and we get Op' sin z = OP sin r; so that Op = 0p' * cos - r This formula, which will be of use when we deal with spherica lenses ( 194), shows at once (cf. 184) that no more than two finitely distinct rays in any given vertical plane section can inter- sect in one point, p ; but, when the cone is very small, all the rays in a given plane section pass approximately through one point. We have next to consider the lateral divergence of the beam of light. This is obviously unaltered by refraction ; for, after refraction, a ray remains in the same plane, passing through the normal, which contained it before refraction. Hence, laterally, the rays diverge from points on PQ. And thus, since the cone has some lateral thickness at the pointy, we see that the emergent light appears to pass, at >, approximately through a small line, the length of which is perpendicular to PQ. This is the primary focal line. The secondary focal line is the intersection of the refracted cone with a plane drawn perpendicular to its axial line and passing through the point in which the axial line cuts the line PQ. At perpendicular incidence, Op = Op' OP//z ; so that the light appears to diverge from a point which is closer to the surface than the actual point in the ratio of 1 to fi. For example, the depth of water appears to be only three-fourths of its true depth. The existence of the primary focal line implies the existence of a caustic surface. We may map out (Fig. 100) successive points, 14 210 A MANUAL OF PHYSICS. jPi> A> G * C M corresponding to successive positions of the point o, by means of the last equation above together with the condition sin i^n sin r. We thus obtain the caustic curve popiP*o f , which FIG. 100. touches QP at p$ and Qo' at o r . The line Po' indicates the limiting position beyond which total reflection takes place. If the caustic curve be rotated around the line PQ, the caustic surface will be described. 190. Refraction through Parallel Layers. A ray of light, which is bent out of its original direction on entering a refracting layer (Fig. 101) with parallel plane sides, is, on re-entering the original \ FIG. 101. medium on the far side of the layer, bent back again into its first direction. This follows at once from the symmetry of the arrange- ment. If an object is looked at perpendicularly through such a layer of thickness, t, its distance from the eye is apparently diminished by the amount (/* !)//. For, if d be the distance of the object from the layer, an eye situated in the layer would see it as if it were at a distance dp from the layer. Therefore, when the eye is just at the far side of the layer, but still inside, the apparent distance of the object is t-\-djji. And, if the eye be now just outside the layer, this LIGHT : REFLECTION, REFRACTION, DISPERSION. 211 distance is decreased in the ratio of p to unity, which proves the statement. If a beam of light passes through three such layers the total deviation of the beam from its original direction is the same as if FIG. 102. the intermediate layer were absent. For, /t x and ju 2 being the refrac- tive indices of the first and second media with reference to the second and third respectively we have (see Fig. 102) sm ?! _ sm sin ro sin sin r, sm sin r sin r. sm ^ 3 sin r 2 But, since /<., is the index of the second medium expressed in terms of that of the third as the unit, while ju x is the index of the first expressed in terms of that of the second as the unit, ^3 is the index of the first medium expressed in terms of that of the third. It follows that no effect is produced by any number of intermediate layers. For this reason it is possible to calculate the refraction error in the apparent altitude of stars, etc. 191. Mirage. When a beam of light passes iion-perpendicularly through a medium composed of parallel layers of continuously B FIG. 103. varying density, the direction of the beam constantly alters. Let us suppose that, over the surface of the ground AB (Fig. 103) there 142 212 A MANUAL OF PHYSICS. exists a stratum of air the density of which continually increases with its distance from the surface. We may suppose also that, above the plane ab, the density remains uniform. An object, nm, will be seen by an eye situated at p by means of rays which enter the non-homogeneous medium and are then bent upwards until, re- entering the uniform medium, they pass straight to p. The object appears to be in the direction of m'n', and is obviously inverted since the rays from the upper and lower extremities cross each other before reaching the eye. But the object may also be seen directly through the medium above ab, and so the impression of an object with an inverted reflection is produced. This is the case of the ordinary mirage of the desert. In Fig. 104, AB again represents the surface of the ground, and FIG. 104. the density of the air is suppose to decrease from below upwards. A direct mirage is thus produced. In Fig. 105 the air is supposed to be of uniform density between AB and ab, but is supposed to diminish continuously in density a FIG. 105. above ab. This condition obviously gives rise to an inverted image. 192. Prisms. When the two plane bounding surfaces of a medium are not parallel the emergent ray is no longer parallel to the incident ray. Such an arrangement constitutes a prism. Let ABC (Fig. 106) represent the section, by the plane of the paper, of a prism of some dense medium, such as glass. Let the edge, B, of the prism be perpendicular to the plane of the paper, and let abed be a ray which makes an angle i with the (inward) normal LIGHT : REFLECTION, REFRACTION, DISPERSION. 213 to the face AB and an angle i' with the (outward) normal to the face BC. Let r and r' be the corresponding angles of refraction. The deviation of the ray from its original direction by the first refraction is i r; and at the second surface this deviation is FIG. 106. farther increased by the amount i' -r r . But ( 188) the alteration of i-r, consequent on a given alteration of i, is greater and greater the larger i or r is ; and any increase or decrease of r involves an equal decrease or increase of r'. Hence, if i be greater than i', a given diminution of i produces a diminution of i r, which is greater than the simultaneous increase of i' r'. Hence the ray has mini- mum deviation when i=i', that is, when the ray passes through the prism so as to make B6 = Be. When the angle of the prism is less than sin -1 1//* the deviation i' - r' may be in the opposite direction to the deviation i r, but it is easy to see that the above result is true in all cases. The total deviation, d, is, in the standard case above, i'+i- (r'-fr), which, in the minimum position, becomes 2(i r)=%i-a, being the angle of the prism. If /* be the refractive index of the substance of which the prism is composed, this gives sin sin /2 This affords a ready means of determining the refractive index of the substance of the prism ; and, if the prism be hollow and have its sides made of glass plates of uniform thickness, the refractive index of any liquid placed in the hollow may be found. 193. Refraction through Spherical Surfaces : Lenses. Let QA.B (Fig. 107) represent a part of a spherical surface the centre of 214 A MANUAL OF PHYSICS. which is at O ; and let the refractive index of the substance on the convex side of the surface be ju. Let rays PA, PB, diverging from P, meet the surface in the points A and B respectively. Suppose that Ap, Bp are the backward prolongations of the refracted rays /i being assumed to be greater than unity. Join PO and continue the line to meet the surface in Q. If we denote the angles PAO, PBO, pA.0, pRO, BPA, BpA, BOA, by the letters i', i, r f , r, $, $, 9, respectively, the diagram at once gives i'i = (j>-0, and r ' - r = ty B. Hence, when the angles are so small that we may write sin i=i, cos i 1, etc., we have ( 188) i' -i=p (r' r), and therefore Gu-1) 0=^-0 (1). When A coincides with Q and AB is very small this becomes ?-l = /* _ 1 . OQ jpQ PQ When /i=-l, this gives the first formula of 183. Now let the rays (Fig. 108), apparently diverging from the point p, fall upon the second spherical surface of the dense medium at the O' FIG. 108. points A', B'. Let 0' be the centre of the new surface, and let O'p meet it at the point Q'. Denote the angles p'B'O', _pA'O', p'B'O', LIGHT : REFLECTION, REFRACTION, DISPERSION. 215 yA'O' A'/B', A'O'B', by the letters i f , i, r', r, ij/', 6', respectively. Since A'^B' = ;//, the figure -gives i'-i = ^ 9' and r 1 -r = V -0'. Hence, when A'B' is very small (i// 0') = /*(4 / ""^') or Oi-l)0' = ,4-*' ........ (2) [Of course (2) might have been deduced from (1) by the substitution of 1/u for ft.] From (1) and (2) we get (/-I) (0-00 = *'-* ...... (8) When A (Fig. 107) coincides with Q, A r (Fig. 108) coincides with Q' ; and so, when the angles are sufficiently small, we may write (3) in the form r and s being the radii of the two spherical surfaces, while v and u are respectively the distances of p' from Q', and of P from Q. The quantity on the left-hand side of equation (4) is constant. We may therefore write (4) in the form UI-I ........... (5) / v u where / is a constant, called the principal focal distance. When u is infinite, that is, when the incident rays are parallel, we get f=v; in other words, / is the distance from Q' of the point from which originally parallel rays seem to diverge after passing through the medium. This point is called the principal focus. P and p' are called conjugate foci. A portion of a medium bounded by two spherical surfaces is termed a lens. In the case just discussed we have taken s>r, and have assumed that the concave sides of the spherical surfaces are directed towards P. All lines drawn in the direction PQ are regarded as positive. Lines drawn in the direction of QP are therefore negative. Equation (4) applies to all lenses. The quantity /is negative (1), when s is positive but less than r, (2) when r is infinite and * is positive, (3) when r is negative and s is positive. In such cases the principal focus and the source of the parallel rays are on opposite sides of the lens, which is necessarily thickest at the middle (Fig. 109). But / is positive (1) when r and s are both positive and s > r, (2) when s is infinite and r is positive, (3) when s is negative and r is positive. 216 A MANUAL OF PHYSICS. In the three latter cases the principal focus and the source of the parallel rays are on the same side of the lens, which is now thinnest (Fig. 110) at the middle. Such a lens diminishes the convergence or increases the divergence of the incident rays, while a lens of the FIG. 109. previous type diminishes the divergence or increases the convergence of the incident rays. Equation (3) makes this evident at once. The formula (4) is true of a given small cone of rays which falls perpendicularly upon the lens at its central part, and it shows that, FIG. 110. after passing through the lens, all such rays appear to diverge from, or are converged to, another perfectly definite point. We shall see in next section that the same result holds in the case of a small obliquely-incident pencil of rays. 194. Lenses : Oblique Refraction. Let DCBA (Fig. Ill) repre- sent a ray, which, entering the lens CBQQ' at C, emerges at B and cuts the line OQ in A. (The letters O, 0', Q, Q' have the same meanings as in last section.) Let the ray be such that OB and O'C are parallel. Since BC makes equal angles with OB and O'C, it follows that AB and CD make equal angles with them. Therefore AB and CD are parallel. Let CB be continue^ to meet OQ in K. We have OB/OB = O'B/O'C, and so the lengths of OE and O'K bear a constant ratio to each other. Therefore K is a fixed point, and is called the centre of the lens. Thus no ray which passes through the centre of the lens is deviated from its original direction by its passage through the lens. LIGHT I REFLECTION, REFRACTION, DISPERSION. 217 It is easy to find an expression for the distance BQ in terms of the thickness of the lens and its radii of curvature. When the lens is very thin K practically coincides with either Q or Q'. We shall hereafter assume that we are dealing with thin lenses only. The final equation of 189 may be written in the form opjop' = I*? (1 sin 2 i}/(i^~ sin 2 i), or, approximately, ^ (1 i 2 )/(/* 2 ft). This shows that, when i is so small that its square may be neglected, p coincides with p'. And it is easy to prove that, when the reflecting FIG. 111. surface is spherical instead of plane, p and p 1 still practically coin- cide when i 2 is small. (The formulae of 193 enable us to calculate the positions of the focal lines.) Hence we conclude that a given, sufficiently small, pencil of rays will, after refraction through a thin lens, be brought to a focus at, or appear to diverge from, one definite point ; and this point must be situated on the line drawn from the vertex of the incident pencil through the centre of the lens. We are now in a position to investigate the production of images by means of lenses. 195. Formation of Images by Lenses. Let AB (Fig. 112) be a thin lens, of which C is the centre, and CF is the principal focal length. If MN be an object which is situated at a distance from the lens greater than CF, rays diverging from N will be brought to a focus at a point n such that l/CF = l/CN + l/Cra. Similarly 1/CF = I/CM + I/Cm, and so on. Thus an inverted real image of MN is formed at mn, and the rays diverging from that image may be examined by an eye which is 218 A MANUAL OF PHYSICS. situated at the distance of about ten inches from it that being the usual distance for most distinct vision. FIG. 112. On the other hand, if MN be slightly nearer (Fig. 113) to the lens than the principal focus, the lens is only able to diminish the diver- gence of the incident rays, and the erect and virtual image of MN FIG. 113. is situated at a greater distance from the image than the object is. For the purpose of correct vision as regards an eye placed close behind the lens, this distance must be about ten inches. The magnifying power of the lens is the ratio of inn to MN, and is therefore approximately equal to 10//, where f is the principal focal length expressed in inches. The object-glass of the ordinary astronomical telescope acts in the manner first described above. The practically parallel rays from a very distant object, MN (Fig. 114) are converged to the principal focus of the object-glass, and so an inverted image, mn, is formed. The eye-glass is placed at a distance from mn which is slightly less LIGHT I REFLECTION, REFRACTION, DISPERSION. 219 than its principal focal length, and so forms a magnified image, m'n r t still inverted, at a distance of ten inches from the eye. N The magnifying power of the telescope is the ratio of the angle which mm, subtends at the eye-glass to the angle which it subtends at the object-glass. It is, therefore, approximately equal to the ratio of the focal length of the object-glass to that of the eye-glass. The arrangement of lenses in the compound microscope is essen- tially the same. The object is placed at a distance from the object- glass which is slightly greater than its principal focal length, and so a magnified inverted image is formed, and is further magnified by the eye-glass. 196. Dispersion : Aberration. In all the preceding sections it was assumed that we were dealing with light of one definite kind or colour alone. But rays of light of different colours are differently refracted by any given substance ; and we must, therefore, with this in view, reconsider briefly the action of prisms and lenses. Let a ray of white light, ab (Fig. 115), fall upon a prism, ABC, in a direction perpendicular to its edge. The single ray of white FIG. 115. light will, on entering the prism, be broken up into a series of coloured rays, fee, be', etc. The red rays are least deviated, and the 220 A MANUAL OF PHYSICS. blue or violet rays are most deviated, from their original direction. The rays intermediate between the red and the violet are usually broadly characterised in order as orange, yellow, green, blue, and indigo. Let be and be' represent respectively a violet, and a red, ray ; and let these lines meet the perpendicular from a on AB in the points r and v. So long as the square of the angle of incidence is negligable, an eye placed in the substance of the prism will see, no white point a but, a coloured line, rv , red at the end nearest to, and violet at the end farthest from, the prism. After emergence from the face BC, the two rays considered will take the directions cd, c'd'. Drop the perpendiculars rr' and vv' on BC, and let dc meet vv' at the point v', while d'c' meets rr' at the point r f . The angles which the emergent rays make with BC being small (which necessitates ABC being small), we see that an eye placed in the same medium as the point a, but on the opposite side of the prism from it, perceives, instead of a, a coloured line, r'v'. Let us suppose now that a represents the section, by the plane of the paper, of a luminous white line, which is parallel to the edge of the prism. In this case r'v' represents a coloured band, which is called a spectrum. This separation of the constituents of white light by refraction is termed dispersion. The measure of the dis- persion produced by any given substance is the difference of the refractive indices of that substance for the extreme rays of the visible spectrum. Now let P (Fig. 116) be a luminous white point from which diverging rays fall upon a lens AB. The refractive index of the substance of the lens for violet rays being greater than its refractive index for red rays, the violet rays will be brought to a focus at a FIG. 116. point, P) which is nearer the lens than the point p', to which the red rays converge. The light in the regions BpA and B'jp'A' is nearly colourless, but, on the whole, it is somewhat violet in the former and reddish in the latter. In the region immediately outside BjjA the light is red, while, in the region immediately outside B'p'A', it is violet. LIGHT : REFLECTION, REFRACTION, DISPERSION. 221 A lens is said to produce aberration when it fails to bring all rays diverging from one point to a focus at another point. The aberra- tion is called chromatic aberration when it is due to dispersion : it is called spherical aberration when it is due to the spherical form of the faces of the lens. The former depends on the first power of i, the angle of incidence, while the latter ( 194) depends on i 2 . So long as the lens is sufficiently thin, and the incidence is sufficiently direct, both are negligable. 197. Dispersion : Achromatism. It is possible to get nearly rid of dispersion while refraction remains, and thus we can obtain a practically achromatic lens. This result can be obtained, since some highly refracting substances produce comparatively small dis- persion, while some substances of low refracting power produce comparatively large dispersion comparatively, that is, to their refraction. Let us suppose that two prisms, of equal angles, but of different substances, produce dispersions d, d 9j respectively, and let their refractive indices for the extreme red light of the spectrum be p lt /* 3 respectively. If now we alter the angle of the prism, the dispersion of which is d v in the ratio d. 2 /d^ and form a compound prism of the two with their edges turned in opposite directions, we shall have a prism which will not produce dispersion of the red and violet rays ; but refraction will still take place unless dj^^de,/^. Similarly, we may construct a compound lens for the purpose of avoiding chromatic aberration. By no pair of substances yet found, however, can we produce complete achromatism. If a lens is completely achromatic for two definite kinds of light it will not be so for any other pair. For if given dispersion is produced by one prism, between a series of pairs of definite kinds of light, equal dispersion will not be produced, by any other prism, between more than one of these pairs. This is known as the Irrationality of Dispersion. Compound lenses, made up of three constituent lenses, can produce closer approximation to achromatism than can a compound lens made up of only two. 193. Rainbows : Halos. We are now in a position to discuss the formation of rainbows and halos. Let AB (Fig. 117) represent a ray of sunlight which, falling on a drop of water at B, is refracted to C. After reflection at C the ray emerges at D. The whole figure is symmetrical about the line drawn from C through the centre of the drop. The semi-angle between AB and ED is obviously 2r - i, i and r being respectively the angles of incidence and refraction. 222 A MANUAL OF PHYSICS. Now ( 188) this is a maximum when 3_sin 2 'i = 4-/* 2 ; and, in the case of water and yellow light, /* = 1-336.' This makes the semi- angle, corresponding to the maximum value of 2r-t, equal to 21 1' nearly. But the existence of a maximum means that the rays are crowded closely together in the immediate neighbourhood of the maximum angle, and so an eye situated at E will see com- paratively strong yellow light in the direction ED. Now consider AB to be a ray of white light. This becomes dis- persed on refraction, and a blue ray (say) being more refracted, will suffer reflection at a point C' such that the angle %r i is smaller than before. Hence the maximum value of the angle 2r - i becomes smaller as /* increases. From E draw EP parallel to AB, and let a very large number of drops be situated symmetrically around EP in the direction of the sun's rays. (Of course, in the figure, the size of the drop is immensely exaggerated relatively to other magnitudes.) The eye at E will see a circle of yellow light, of radius 4a; the centre of which is situated on the line EP. Inside this a circle of blue light will be seen, and, outside it, a red circle will appear the colours of the spectrum succeeding each other, in order of decreasing refrangi- bility, from within outwards. This constitutes the explanation, by geometrical optics, of the primary rainbow. In the actual bow the colours are, of course, impure. For the light proceeding from each point of the sun's disc gives rise to a separate bow, and the bow which we see results from the superposition of all these distinct bows. If the ray AB (Fig. 118) suffers two reflections in the interior of the drop, the emergent ray EP makes an angle with it which is LIGHT : REFLECTION, REFRACTION, DISPERSION. 223 double of 7T/2 (3r i). But, 188, 3r i reaches a maximum when 8 sin- i = 9- /A Hence the angle BPE is a minimum when i has the value indicated by this formula. For yellow light, falling on a drop of water, the vertical angle is about 50 58'. A ray of higher refrangibility will be reflected at points C f , D', such that the perpendicular from on C'D' will intersect AB at an FIG. 118. angle which is larger than BPO. Hence the minimum value of the angle 7r/2 - (3r - i) increases as // increases. And so, in the secondary rainbow, which is due to two such internal reflections, the colours succeed each other, in order of increasing refrangibility, from within outwards. Also, since the primary bow corresponds to a maximum value of the angle which the emergent ray makes with the incident ray, while the secondary bow corresponds to a minimum value of this angle, we see that the space between the bows is devoid of light due to rays which have suffered one or two reflections inside the drops, while there is some such illumination in the region inside the first bow and also in that outside the second. The bows which are due to more than two internal reflections are too feeble to be visible. Halos are due to the refraction of light through ice-crystals. The rays are most densely crowded in the directions of minimum deviation. The red rays are least deviated, and therefore appear always on the interior portions of halos. The size of a halo depends upon the effective angle of the ice -crystal to refraction through which it is due. Parhelia and paraselenes are simply exceptionally bright portions of halos. Colourless halos are produced by the reflection of light from the plane surfaces of the crystals. 199. The general phenomena of reflection and refraction receive a ready explanation whether on the corpuscular or on the undulatory theory of light. Let AB (Fig. 119) represent the bounding surface between two refracting media. Let p q r s t represent the path of a corpuscle. 224 A MANUAL OF PHYSICS. During the rectilinear motion from p to q the corpuscle experiences no resultant attraction in any direction. When it reaches the distance from AB indicated by the line ab, the greater (say) attraction of the medium on the other side of AB preponderates and the path becomes concave towards the surface of separation. This continues until a point, s, at which the resultant attraction again becomes zero, is reached. The rest of the path st is therefore straight, and is inclined at a less angle to the normal to the surface AB than is the part pq. a A a FIG. 119. Even if the refracting surface is not actually plane, it is yet prac- tically plane, in all cases of finite curvature, so far as the present reasoning is concerned, for the portion qs of the path of the corpuscle is excessively small. Now (cf. 42) the square of the resolved part of the speed of the corpuscle along the normal increases by a constant amount in pass- ing from ab to a'b', while the resolved part along the surface remains constant, and hence the total speed, v f , of the particle in the second medium bears a constant ratio to its total speed, v, in the first. Let, as usual, i and r denote respectively the angles which pq and st make with the normal. We have v sin i = v' sin r, which is identical with sin i~p sin ?*, where /* is equal to v'/v. But this is the known law of refraction. A It follows necessarily that the speed of a corpuscle is greater in a dense medium than in a rare one. 200. Let ABC (Fig. 120) represent a plane wave-front, which travelling through the air in the direction indicated by the arrows, reaches the surface, ADF, of a refracting medium. From A, as centre, describe a sphere of radius, AF, such that light will travel over the distance AF in the refracting medium in the same time that it will describe the distance CF in the air. Similarly, from D LIGHT : REFLECTION, REFRACTION, DISPERSION. 225 describe a sphere of radius, DQ, such that DQ bears to EF (DE is parallel to ABC) the same ratio as the speed of light in the refracting medium bears to its speed in air. It is obvious from the construction that all such spheres touch a plane, PQF, which is therefore the wave-front after refraction. FIG. 120. Let us denote the speeds of light in the medium, and in air, respectively, by the letters v', v. Let i and r be respectively the angles of incidence and refraction. Then CAF = i, AFP = r, and CF = AF shit, AP = AF sin r. But CF/AP = v/v r . Hence sin i= vjv' . sin r = ^ sin r, if \i = v\v* ; and so the known law of refraction is a consequence of the wave-theory of light. Observe that, in a dense medium in which /* is larger than unity, v' is necessarily less than v. Hence, on this theory, the speed of light must be less in a dense medium than it is in a rare one. This conclusion is in direct opposition to that derived from the principles of the corpuscular theory ; and so we are furnished with a crucial test between the two theories. The result of experiment is ( 178) entirely in favour of the undulatory theory. Consequently, we shall hereafter deal with the results of this theory alone. It is easy to deduce from the above theory the fact that the time taken by light to travel from a given point in one medium to a given A . FIG. 12L point in another is a minimum. Let PAQ (Fig. 121) be the actual path of light proceeding from P to Q, and let PBQ be a very near path. Draw AC and BD perpendicular to PB and AQ respectively* 15 226 A MANUAL OF PHYSICS. Then CB/AD = sin i/sin r = v/v'. That is, CB/v = AD/^', or AD and CB are described in equal times. Therefore, and since PA = PC, DQ = BQ, the paths PAQ and PBQ are described in equal times. But, as B moves away uniformly from A, QA QB increases at a diminishing rate, while PB PA increases at an increasing rate. Hence the time of description of PAQ is a minimum. This law of least time was first given by Fermat. It is true in the case of reflection also (see 182). The corpuscular theory gives PAv+AQ-y' = a minimum. This sum is termed the action. 201. Hamilton's Characteristic Function. In the earlier part of the present century Sir W. R. Hamilton introduced a general method by which all optical problems may be solved by a single process. The following extracts from an ' Account of a Theory of Systems of Kays,' written by Hamilton himself, and published in his Life will indicate the nature of his work. ' A Bay in optics is to be considered here as a straight or bent or curved line, along which light is propagated, and a System of Rays as a collection or aggregate of such lines, connected by some common bond, some similarity of origin or production, in short, some optical unity. Thus the rays which diverge from a luminous point compose one optical system, and, after they have been reflected at a mirror, they compose another. To investigate the geometrical relations of the rays of a system of which we know (as in these simple cases) the optical origin and history, to inquire how they are disposed among themselves, how they diverge or converge, or are parallel, what surfaces or curves they touch or cut, and at what angles of section, how they can be combined in partial pencils, and how each ray in particular can be determined and distinguished from every other, is to study that System of Rays. And to generalize this study of one system so as to become able to pass, without change of plan, to the study o other systems, to assign general rules and a general method whereby these separate optical arrangements may be connected and harmonized together is to form a Theory of Systems of Rays. Finally, to do this in such a manner as to make available the powers of the modern mathesis, replacing figures by functions and diagrams by formulae, is to construct an Algebraical Theory of such Systems, or an Application of Algebra to Optics ' The method employed in that treatise (Malus's Traite d'Optique) may be thus described: The direction of a straight ray of any final optical system being considered as dependent on the position of some assigned point upon that ray, according to some law which characterizes the particular system and distinguishes it LIGHT: REFLECTION, REFRACTION, DISPERSION. 227 from others ; this law may be algebraically expressed by assigning three expressions for the three co-ordinates of some other point of the ray, as functions of the three co-ordinates of the point proposed. Malus accordingly introduces general symbols denoting three such functions (or, at least, three functions equivalent to these), and pro- ceeds to draw several important general conclusions, by very com- plicated but yet symmetrical calculations, many of which conclusions, along with many others, were also obtained afterwards by myself, when, by a method nearly similar, without knowing what Malus had done, I began my own attempts to apply algebra to optics. But my researches soon conducted me to substitute, for this method of Malus, a very different, and (as I conceive I have proved) a much more appropriate one, for the study of optical systems ; by which, instead of employing the three functions above mentioned, or at least their two ratios, it becomes sufficient to employ one function, which I call characteristic or principal. And thus, whereas he made his deductions by setting out with the two equations of a ray, I, on the other hand, establish and employ the one equation of a system. 'The function which I have introduced for this purpose, and made the basis of my method of deduction in mathematical optics, had, in another connection, presented itself to former writers as expressing the result of a very high and extensive induction in that science. This known result is usually called the law of least action, but sometimes also the principle of least time, and includes all that has hitherto been discovered respecting the rules which determine the forms and positions of the lines along which light is propagated, and the changes of direction of those lines produced by reflection or refraction, ordinary or extraordinary. A certain quantity which in one physical theory is the action, and in another the time, expended by light in going from any first to any second point, is found to be less than if the light had gone in any other than its actual path, or at least to have what is technically called its variation null, the extremities of the path being unvaried. The mathematical novelty of my method consists in considering this quantity as a function of the co-ordinates of these extremities, which varies when they vary, according to a law which I have called the law of varying action ; and in reducing all researches respecting optical systems of rays to the study of this single function : a reduction which presents mathematical optics under an entirely novel view, and one ana- logous (as it appears to me) to the aspect under which Descartes presented the application of algebra to geometry.' 152 CHAPTER XVII. RADIATION AND ABSORPTION : SPECTRUM ANALYSIS. ANOMALOUS DIS- PERSION. FLUORESCENCE. 202. WE have already discussed the reflection (including scattering) and refraction of light at the common surface of two media. We have now to consider the absorption of light hi its passage through material media, together with other associated phenomena. According to the undulatory theory (which we now assume to be true, and of the truth of which we shall receive additional evidence as we proceed) light consists of waves propagated through a medium (called the ether) which fills space. The particles of a body which is emitting radiation must therefore be hi rapid vibratory motion, and must communicate their motion to the ether. The parts of the body, the vibrations of which are communicated to the ether, may be the molecules, or the constituent parts of the molecules, or even the atoms. When a bell is struck violently and frequently the resulting sound is extremely complex and consists of notes, of various pitches, which may differ greatly from each other in intensity. The more violent the blows upon the bell become, and the more rapidly they are made, the more complex will the clang be. New forced vibrations appear, and the intensity of each of those previously existing is increased. It is only when the blows are excessively feeble and unfrequent that the fundamental tone is heard by itself. Now the molecules of a solid body, of high temperature, are con- stantly colliding ; and the vibrations induced in a molecule by one collision do not die out before another is sustained. Hence the radiation given off by such a body consists of vibrations of many periods ; and, as the temperature of the body becomes higher, vibra- tions of shorter and shorter period make their appearance, and the intensity of all previously existing vibrations becomes greater. Thus, if we examine the spectrum of a body the temperature of which is gradually raised, we may at first perceive no luminous RADIATION AND ABSORPTION. 229 radiation at all ; but, as the temperature rises, first red light will appear, then yellow, and so on, until a complete continuous spectrum is seen, and the luminosity of every part gradually increases as the temperature rises. In the case of an ordinary gas, however, the molecules usually are sufficiently free from collisions to allow of them vibrating in their own proper modes. Hence radiations of definite periods only will be emitted ; and thus the spectrum of a gas is discontinuous,! and consists of bright lines. It varies with the temperature and( pressure. As the pressure is increased the lines broaden out, and the spectrum gradually becomes continuous, like that of a liquid body or of a solid. We already know ( 173) that a body which has a definite period of vibration, and which is at rest, may be set in vibration by the communication to it, through an intervening medium, of vibrations, of its own proper period, which are emitted by another body. Hence, if radiation travelling through the ether enters a material medium (solid, liquid, or gas), and if the natural period of oscillation of the molecules coincides with the period of some of the ethereal vibrations, the molecules will be set in motion, and the energy of the radiation will be diminished. This is the process which is termed absorption. In many cases (if not really in most cases) the period of the induced vibration is longer than that of the ethereal vibration which induces it ( 208). In almost all cases the absorbed energy is manifested by a rise of temperature. 203. Equality of Emissivity and Absorptive Power. The Absorptive Power of a body, under given conditions, for any definite radiation, is the fraction of the whole incident radiation of that kind which it absorbs. Now a black body is one which absorbs all the incident radiation ; so we might define the absorptive power for the given radiation as the ratio of the amount of it which the body absorbs to the amount of it which a black body would absorb. The Emissivity of a body, at a given temperature, for any given radiation, is the ratio of the quantity of that radiation which it emits to the quantity of it which is emitted by a black body under the same conditions. An extremely simple relation connects these quantities: The Emissivity and Absorptive Power of a body, at a, given tem- perature, for any radiation, are equal. The proof of this law was given by Stewart in 1858. The law is (see 255) an extension of the statement, made by Prevost about a century ago, that the radiation emitted by a body depends solely 230 A MANUAL OF PHYSICS. ' upon the nature of the body, and upon its temperature ; and various experimental illustrations of it were known long before Stewart's proof was given. Brewster had shown that definite portions of the sun's light are absorbed in its passage through the earth's atmo- sphere. Foucault had pointed out that, while the electric arc emits (more freely than its other radiations) yellow light of two definite refrangibilities, the light from one carbon pole is robbed of these two kinds of radiation when it passes through the arc. Stokes also had explained this by the analogous properties of sounding bodies ( 173). It is known, as an experimental result, that a number of bodies, at different temperatures, placed inside an enclosure which neither allows radiation to pass outwards from within it nor inwards from without it, will ultimately arrive at, and maintain, one common temperature. But this could not result unless each body emitted radiation at precisely the same rate as that at which it absorbed the radiation. This proves the law so far as radiation as a whole is concerned. [No such enclosure as has been postulated exists in nature ; but a polished reflecting surface of silver would form a sufficiently close experimental approximation ; and, the more nearly the condition is satisfied, the more nearly does the result hold.] The radiation inside the enclosure must be that of a black body at the same temperature, for any one of the bodies might be a black one. Let us suppose that one of the bodies absorbs one definite radiation only, and allows all others to pass freely through it. (Solutions of didymium salts approximately possess this property.) This body must emit the same kind of radiation as it absorbs, and that to precisely the same extent ; otherwise its temperature would vary. This proves the law as stated for any definite radiation. Many experimental illustrations of the truth of the law were given by Stewart. Thus, a piece of red glass, held in front of a fire, appears red because it absorbs the green and blue rays. If placed in the fire it becomes colourless when its temperature becomes equal to that of the fire for it then still allows the red rays to pass through it, and, in addition, itself emits the rays which it absorbed. If taken out of the fire and held in a dark room it emits bluish-green light precisely that which it absorbed. Again, Stewart, and also Kirchoff (who arrived at the results under consideration independently of, though somewhat later than, Stewart), showed that a plate of tourmaline, cut parallel to the- axis of the crystal, emits, when heated, light which is polarised (Chap. XIX.) perpendicularly to that which it allows to pass, that is, it emits the rays which it absorbs when cold. RADIATION AND ABSORPTION. 231 204. Spectrum Analysis. In order to examine the luminous radiation emitted by a given body, we may place in front of the body a narrow vertical slit, which is situated at the principal focus of a convex lens. The light diverging from the slit is thus condensed into a parallel beam which is passed through a prism (usually of dense glass) and so gives rise to a spectrum. This spectrum is magnified by means of a telescope. Such an arrangement essen- tially constitutes the instrument called a spectroscope (or spectro- meter, if a graduated circle and vernier are attached for the purpose of determining the angular positions of the telescope when it is directed towards different parts of the spectrum). Let us suppose that we are examining the light emitted from a highly-heated lime-ball, and that this light, before falling on the slit, passes through a Bunsen. flame in which metallic sodium is being vaporised. The lime-ball alone would give a continuous spectrum. The sodium-tinged flame alone would give a discontinuous spectrum consisting of two bright yellow or orange lines situated close together. The spectrum actually observed is continuous, but has two bright lines in the same position as those in the spectrum of the sodium flame. If we vaporise the sodium in the flame of a spirit lamp instead of in a Bunsen flame, everything else remaining the same, a con- tinuous spectrum, crossed by two dark lines in the positions of the former bright ones, will be seen. This was pointed out and explained by Kirchoff. The cause lies in the difference of the temperatures of the Bunsen flame and the flame of the spirit lamp. A line will appear bright, or dark, according as the intensity of the radiation of that particular kind from a black body at the temperature of the flame exceeds, or falls short of, the intensity of the radiation of that kind which is emitted by the source. For, if R be the intensity of the given radiation as emitted from the source, while B'=^?R is the intensity of the light of that kind emitted from a black body at the tem- perature of the flame, and p is the radiating power (or absorptive power) of the flame for that radiation, the intensity of the given kind of light which reaches the eye is R joR+p^>R = E[l-f-,o(_p--l)] . This quantity exceeds, or falls short of, R, according as p is greater, or less than, unity. If the source were a black body, p could not exceed unity unless the temperature of the flame were greater than that of the source ; but, the source not being a black body, p may exceed unity, although the temperature of the flame is below that of the source, i.e., bright lines may be visible. If, however, the differ- ence of temperature of the source and flame be sufficiently great, the lines will appear dark. 232 A MANUAL OF PHYSICS. The spectrum of sunlight was found by Wollaston and (later) Fraunhofer to exhibit a number of persistent dark bands. In accordance with the above principles we conclude that these lines are due to absorption. Some of them can be shown to be due to absorption in the earth's atmosphere, but the great majority are produced by absorption in the (comparatively) cold vapours sur- rounding the hot body of the sun. Sufficient matter to produce such absorption does not exist in the space between the earth and the sun. Now we can experimentally determine the kinds of radiation emitted by the hot vapours of the various elementary substances. And if it is found that any of these radiations are absent from the spectrum of sunlight it is to be inferred that the vapours of these substances (provided the cause is not terrestrial) are present in the regions immediately surrounding the sun. In this way it is found that a. great many substances existing on the earth's surface are FIG. 122. present in the sun in the form of vapour. The lines A, B (Fig. 122) are due to oxygen, but have been shown to be caused by absorption in the earth's atmosphere. The lines C and F are due to hydrogen. The (double) line D is caused by sodium vapour. The (triple) line b is produced by the vapour of magnesium. Some hundreds of lines are caused by the presence of the vapour of iron. In the same way the light emanating from any star, comet, or nebula, etc., may be examined, and the chemical constitution of the luminous body inferred. The various stars may be classified into several groups according to the nature of their spectra, and this classification indicates approximately their relative age. Generally speaking, the more recent stars have bright-line spectra, while the older stars exhibit continuous spectra crossed by numerous dark lines. The spectra of nearly extinct stars, however, resemble those of recent stars to a considerable extent. If the slit of the spectrometer is wide the various coloured images overlap and produce an impure spectrum ; and, under the same con- dition, a bright light broadens out and becomes indistinct. But, however narrow the slit may be, a line even if due to light of one definite refrangibility alone has always some finite breadth. The RADIATION AND ABSORPTION. 233 reason is that the radiating molecules are in violent motion some moving towards, others moving from, the observer. If n be the number of vibrations produced in the ether per second, and, if the molecule emitting the light be at rest relatively to the observer, the wave-length, X, of the disturbance is given by the equation V=nX, where V is the speed of light. But, if the molecule be moving, relatively to the observer, with speed v, the wave- length will be given by the equation V .v = n\' ; and the apparent change of wave-length is Hence, the molecules of a luminous body having all possible speeds included between the limits + v and -v, a bright line in the spec- trum of its light will possess finite breadth, even when it corresponds to one definite kind of radiation alone. This principle has been applied to determine the rate of rotation of the sun on its axis, the speed of projection of gases in a solar eruption, and the rate of motion of stars to or from the earth. 205. Law of Absorption : Body Colour. Dicliroism. Let R be the amount of radiation of some definite kind which falls upon an absorbing medium. Let p (called the absorption co-efficient) be the percentage of this radiation, which is stopped by a plate of the medium of unit thickness. The quantity which passes through the given plate is therefore R(l -p). A second plate of the substance, also of unit thickness, will stop the fraction, p of this quantity ; so that the amount, E(l p) 2 , passes through a plate the thickness of which is two units. And, generally, the quantity which passes through a plate, the thickness of which is n units, is E(l - p)". This practically vanishes, however small p may be (provided only that it is finite), when n is sufficiently great. Conversely, the amount of radiation, of the given kind, from a sufficient thickness of such a substance, is equal to that of a black body at the same temperature. [The expression R(l-p)"is only true on the assumption that p is constant for all radiations considered. If it is not so, we must write 2 . E(l-p)" instead.] A substance which absorbs (say) red light, will appear bluish- green when viewed by transmitted light. And, the greater the thickness of the substance through which the light passes, the denser will be the apparent colour, until, finally, practically no light can pass. Hence a substance into which light penetrates for a short distance and is then reflected out, will appear to be coloured, provided that selective absorption takes place, and its colour will be the same as 234 A MANUAL OF PHYSIC^. that of the light which it transmits. This colour is termed the body colour of the substance. For example, a mixture of blue and yellow pigments appears to be green because, if white light falls upon it, the particles of the blue pigment absorb the rays of small refrangibility, while the par- ticles of the yellow pigment absorb the rays of large refrangibility. The green rays alone are partially reflected by both substances, and so the mixture appears to be green. (A mixture of blue and yellow lights is of a purplish colour. This may be seen by rotating rapidly a disc, divided into sectors, some of which are coloured yellow, and some blue.) Now suppose that some substance absorbs (say) red light and green light, and let the coefficient of absorption for red light be much greater than the coefficient of absorption for green light. Suppose also that, in the incident light, the red rays are more intense than the green rays. It is obvious that, while the intensity of the red rays in the transmitted light will exceed the intensity of the green rays so long as the thickness of the substance is small, after a certain thickness is reached, the green light will be trans- mitted in greater intensity than the red light. The colour of such a substance will therefore change from a reddish hue to a greenish hue, as seen by transmitted light, as its thickness increases. This phenomenon is known as dickroism. The accompanying diagram illustrates these facts graphically. The abscissae of the curves represent the thicknesses of the absorbing medium ; the ordinates of one set of curves represent the intensities of the transmitted light of one kind, and the ordinates of the other set indicate the intensities of the transmitted light of another kind, corresponding to the various thicknesses. The numbers accompanying the curves indicate different values of the coefficients of absorption. At the point p the high absorptive power (0'7) of the substance for the originally more intense light has diminished the intensity of that light to the same value as that which is exhibited by the originally feeble light, for which the co-efficient of absorption is only O'l. Many such bodies occur in nature. For example, glass, coloured with a cobalt salt, while it transmits blue light when its thickness is small, appears red by transmitted light when its thickness is suffi- ciently great. The law of absorption, above stated, must be true (neglecting such extraneous effects as internal reflection or scattering of light) so long as the^coefficient of absorption does not depend upon the intensity of the light. Such experiments as have been made to test this point furnish confirmatory evidence. RADIATION AND ABSORPTION. 235 206. Surface Colour: Metallic Reflection. Some substances reflect from their surface certain rays only; thus gold reflects yellowish rays, and copper reflects reddish rays. The colour pro- duced by this ' metallic reflection ' is called surface colour. The light which is transmitted by a thin film of such substances is complementary to that which is reflected, that is, the transmitted light and the reflected light together make up a light of the same FIG. 123. composition as that which was incident upon the surface. The reflected light cannot be plane -polarised at any angle of incidence (Chap. XIX.). Many substances, besides metals, exhibit surface colour for example, thin films of rose aniline, or of blue aniline, etc., appear of different colours according as they are viewed by transmitted, or by reflected, light. Such films may be prepared by placing a layer of 236 A MANUAL OF PHYSICS. an alcoholic solution of the aniline on a plate of glass and allowing the alcohol to evaporate. The colour, as seen by reflected light, varies somewhat with the angle of incidence. The light reflected from such films cannot be entirely polarised at any angle of incidence. It consists of two parts a part which can be plane -polarised at a certain angle of incidence and is identical with the transmitted light (which, in fact, constitutes the body colour of the substance) and a part which cannot be plane-polarised, and so resembles the surface colour of metals. The polarisable part may be got rid of by suitable means, so that the remaining part may be examined alone. In the case of permanganate of potash, Stokes found that the surface colour seemed to be due to precisely those rays which were absent from the transmitted light, or, which is the same thing, the body colour. Hence, the colour of the light trans- mitted through this substance is due only to a very slight extent, if at all, to absorption. The spectrum of the transmitted light has five dark bands in the green part ; the reflected light is green, and the spectrum of the surface-colour portion of it consists of five bright bands, which correspond to the dark bands in the spectrum of the transmitted light. 207. Anomalous Dispersion. In close association with the existence of dark absorption bands appears the phenomenon of anomalous, or abnormal dispersion. In general, the rays of greater wave-length suffer refraction, on passage through a prism, to a smaller extent than the rays of shorter wave-length. But, in many substances, this rule does not hold. Such media are said to possess the property of anomalous dispersion. Fox Talbot was the first to observe the phenomenon, but he did not publish his observations for about thirty years. In the mean- time Le Koux has observed that iodine vapour refracted red light more than it refracted blue light. Christiansen, Kundt, and others have widely extended our know- ledge of such substances. Kundt has shown that the property of anomalous dispersion is possessed by all substances which exhibit surface-colour. If a continuous spectrum, such as may be given by a glass prism, be examined through another prism of a substance which exhibits abnormal dispersion, the spectrum will no longer be continuous but will present one or more dark bands. If the second prism be now turned so as to have its edge at right angles to the edge of the glass prism, the parts of the continuous spectrum will be displaced from their original positions to an extent depending upon the refractive RADIATION AND ABSORPTION. 237 index of the substance for each kind of light. The displacement of the rays, in a part of the spectrum close to a dark band, but of smaller wave-length than the absorbed rays, is abnormally small; and the displacement of the rays of slightly larger wave-length than those which are absorbed is abnormally great (Fig. 124). FIG. 124. The general law, as given by Kundt, is that the rays of slightly less refrangibility than the absorbed rays have their refrangibility abnormally increased, while the rays of slightly greater refrangi- bility than the absorbed rays have their refrangibility abnormally diminished on passage through the absorbing medium. 208. Fluorescence. The phenomenon of fluorescence is also necessarily associated with the absorption of light. Brewster observed that the path of a beam of white light through a solution of chlorophyll glows with red light, and he termed the phenomenon ' internal dispersion.' Then Herschel noticed that the surface of a solution of sulphate of quinine upon which sunlight falls is of a bright blue colour. He named this appearance * epi- polic dispersion '; but Brewster showed that the blue colour could be manifested in the interior of the liquid if the beam of sunlight were sufficiently concentrated, and so he concluded that the phenomenon was of the same kind as that which he had already observed in the case of chlorophyll and fluorspar, etc. Stokes has shown that a great many ordinary substances, such as bone, white paper, etc., possess this property, to which (avoiding any reference to dispersion) he gave the name of fluorescence, from its being noticeable in fluorspar. His method of observation con- sisted in allowing a beam of light to enter a darkened chamber through a plate of blue cobalt-glass. This beam fell partly upon a white non-fluorescent body (white porcelain), and partly upon the body under examination. The light reflected from the two sub- stances was then examined through a slit and prism. In this way the fluorescent light was compared with the light which produced it. Stokes found that the light which was emitted by the fluorescent body was always of lower refrangibility than the light which pro- 238 A MANUAL OF PHYSICS. duced it. The fluorescence of sulphate of quinine is due to the extreme violet rays of the spectrum, and to invisible rays of still higher refrangibility. Hence, by means of such a solution, the absence of rays beyond the visible part of a spectrum may be deter- mined. For, if the spectrum be thrown on a screen damped with this solution, fluorescence is produced, beyond the usual visible part, except when rays of certain refrangibilities may be absent. In the case of chlorophyll the light which produces the effect is chiefly in the visible spectrum. The explanation of the phenomenon given by Stokes is that the ethereal vibrations are absorbed by the fluorescent matter, which is set in vibration, the period of its vibration being usually longer than, never shorter than, the period of vibration of the ether. The vibrating matter now reacts upon the ether, and sets up in it vibra- tions which are generally longer still, but are never shorter than those induced in the molecules of the matter. This explains the lowering of refrangibility. Dynamical illustrations of such interaction can be given. The following is due to Stokes. Ships at rest on a calm sea may be set in vibration by waves of definite period propagated from a distance. The natural period of oscillation of each ship will not generally agree with that of the waves. Any ship which is thus set in vibra- tion will, by its vibrations, produce waves which spread outwards from it ; but the period of these waves will generally be greater than that of the original waves, and can never be less than it. If Stoke's's explanation be true it is to be expected that the light which gives rise to fluorescence will be absent from the absorption spectrum of the substance. This is invariably the case. Phosphorescence is precisely the same phenomenon as fluores- cence. The only difference which subsists between the two is a difference of duration. Phosphorescence (so-called) frequently lasts for hours after the stimulating radiation is removed ; fluorescence is maintained usually for only a small fraction of a second after the light ceases to fall on the substance. Becquerel demonstrated and measured the finite time of duration of fluorescence in many substances. His apparatus consisted of a box with perforated revolving discs at either end. The perforations were so arranged that one end of the box was closed, while the other was open. The substance which was to be tested was placed inside the box, and, on the discs (which had a common axis) being rotated, an intermittent beam of light passed through the substance. No light could pass out at the end of the box opposite to that at which the light entered unless the substance were fluorescent. RADIATION AND ABSORPTION. 239 But, that condition being satisfied, light could pass through when the speed of rotation of the discs was sufficiently great. The duration of fluorescence is exemplified in the above dyna- mical illustration by the continued oscillation of the ships for some time after the cessation of the disturbance which originates it. 209. Theories of Dispersion. Cauchy was the first to advance a dynamical theory of dispersion. He ascribed it to the coarse-grained- ness of the matter of which the dispersing substance is composed. The great difficulty of this theory is that, in order to account for the observed values of the refractive indices of substances such as glass, etc., the number of molecules of matter existing side by side in the length of a wave of light must be assumed to be much smaller than, from other considerations ( 146), can possibly be admitted. Sir W. Thomson has recently shown that Cauchy's hypothesis can be so modified as to enable it to surmount this difficulty. This hypothesis leads to an expression for the refractive index, H, of any substance of the form where a, &, and c, etc., are constants, and X is the wave-length. This formula shows that the refractive index increases as the wave- length diminishes. Its results accord very well with experimental observations within the range of the visible spectrum, but it does not apply well to the invisible rays at the less refrangible end of the spectrum. The various terms rapidly diminish in numerical magnitude. Briot generalised Cauchy's investigation somewhat, and deduced the expression . which agrees better with experimental observations than the former does, and applies to a much greater range of wave-lengths. The term x\ z depends upon the direct action assumed to exist between the ether and matter. Modern theories (for example, that of v. Helmholtz) have regard, not so much to space relations between wave-length and molecular distance >as to time relations -between the period of vibrations in the ether and the period of free oscillation of the material molecules. V. Helmholtz assumes the existence of a viscous resistance to the motion of the molecules. When the periods of the ethereal and the molecular vibrations are identical, or approximately identical, 240 A MANUAL OF PHYSICS. absorption takes place, and, because of the viscosity, the vibrational energy takes the form of heat. Thomson's results differ from those of v. Helmholtz chiefly because he purposely avoids the assumption of the existence of viscosity. He obtains the equation when ft is the refractive index, r is the period of vibration of the ether, and xi, x 2 , etc., are the natural periods of oscillation of the molecules arranged in ascending order of magnitude. So long as T is considerably greater than xi and considerably less than x 2 , this equation will correspond to the case of ordinary refraction. As "7" approaches x in value the refractive index is abnormally increased. When r is less than x it ^ is at first negative, but afterwards becomes positive, though abnormally small, as T still further decreases. This explains the existence of anomalous dispersion. Negative values of /* 2 , which accompany anomalous dispersion, indicate the existence of absorption or metallic reflection. Thus the high reflecting power of silver is, on this theory, due to the fact that each one of all the kinds of radiation which are observed to be reflected from it has a vibrational period which is smaller than the smallest of the natural periods of oscillation of the mole- cules of silver. Again, when r has such a value that n is positive, but is less than unity, the particular radiation, of which r is the period, will pass through the substance more quickly than it passes through air. The energy of the rapid vibrations of the molecules is gradually transmuted into energy of the slow vibrations. This explains fluor- escence and the radiation of heat from a body which has absorbed light. The molecule may be so constituted that the fluorescence (or phosphorescence) may last for a very long time. CHAPTEE XVIII. INTERFERENCE. DIFFRACTION. 210. Principle of Interference. If light consists of undulations, propagated through the ether, the effects of which, at any point of the ether, are superposed in precisely the same way as are the effects of separate simple harmonic motions ( 52), we should expect that conditions might occur under which the resultant motion at that point would be null while, under other conditions, the resultant motion might be exceptionally great. We already know that, for a similar reason, when waves are propagated along the surface of water from two different sources, no resultant disturb- ance of the surface may exist at certain points. So also sounds from two different sources may be totally unheard by an ear placed at certain positions within hearing distance.. In order to produce continuous interference at given points it is absolutely necessary that the waves diverging from two sources should be of precisely the same period, as otherwise the resultant disturbanoe would vary from a minimum to a maximum alternately. Thus, in the case of sound, difference of period gives rise to beats which may be observed by the ear. Now the phase of the vibration emitted from one point of a name has absolutely no relation with the phase of that emitted from any other point ; and hence we cannot expect observable inter- ference between rays coming from different luminous sources. Inter- ference of course does occur between such rays constantly, but, in general, the alternations between maximum and minimum effects will succeed each other so rapidly that the eye can perceive no varia- tion of intensity. Therefore we conclude that, in order that persistent interference] effects may be observable, the two interfering rays must originally 1 have proceeded from a common source. More than two centuries ago Grimaldi observed that, when rays of light from two sources overlapped each other and fell 16 242 A MANUAL OF PHYSICS. upon a screen, the portion of the screen which was illuminated by the two rays appeared to be darker than when it was illuminated by one ray alone. He allowed sunlight to enter a darkened chamber through two small apertures in the shutter. But these apertures were illuminated by light coming from all portions of the sun's disc, and so the effect which Grimaldi observed, to whatever cause it may have been due, could not have been produced by inter- ference. Grimaldi, indeed, was not looking for interference pheno- mena this was not thought of until 150 years later he wished to prove that light was not material, since two portions of light appar- ently destroyed each other. And this reasoning is practically con- clusive, for the conditions which would have to be assumed, in order to make an explanation of these phenomena by the emission theory possible, would be so arbitrary and artificial that no one could seriously advance them. 211. Young's Experiment. Young was the first to observe true interference effects. He admitted light through a single small aperture in a shutter behind which he placed another shutter pierced by two small openings. In this way he obtained two rays of light which proceeded originally from a common source the single opening in the first shutter. That portion of a screen which was illuminated by both rays was crossed by alternately- arranged dark and bright bands. Young observed that the bands became narrower when the distance between the holes in the second screen was increased. He also noticed that the effect disappeared if either opening were closed. The wave theory affords a ready explanation of the phenomena which Young observed. Let A, A' (Fig. 125) represent the two openings in the screen, and let AP, AT be two rays which each illuminate the point P of the I .XI FIG. 125. screen PN. M is the central point of AA', and MN is drawn per- pendicular to AA' and PN. Denote the length of AM (or A'M) by a and the length of MN by b, and let x represent the distance PN. LIGHT : INTERFERENCE, DIFFRACTION. 243 The waves, which travel along AP and A'P, start from A and A' respectively in the same phase. Consequently the point P will be bright or dark according as AP - A'P is an even, or an odd, multiple of half a wave-length. Now AP 2 =(a+o0 2 +& 2 and A'P 2 = (a-o0 2 +& 2 . Therefore AP 2 - A'P 2 = (AP+A'P) (AP- A'P) = 4ax. But (a and x being very small in comparison with b) AP-f-A'P is approximately equal to 26, and so the condition gives 2ax X and the point P is bright or dark according as n is an even or an odd integer. This formula indicates that, to the degree of approximation with which we are dealing, the locus of P, when b varies and n is con* stant, is the straight line MP. The eaact locus is a hyperbola, of which A and A' are the foci. This follows at once from the con- dition AP A'P=a constant. A and A' may, of course, represent narrow luminous strips with their length perpendicular to the plane of the paper. The point P then corresponds to a dark or bright band also perpendicular to the plane of the paper. By measuring the quantities a, b, and x, and by counting the number, n, of the particular band under observation, we can calculate the value of X. The distance between the n' h and the (n+l)'* band is independent of n, and is therefore constant when a, b, and X are fixed. 212. Fresnel's Experiment. In Young's experiment the beams of light passed through apertures cut in a solid. Hence the observed effects might have been due to diffraction ( 224) . The result was that Young's explanation was not generally accepted ; but a modification of his experiment, made by Fresnel, completely settled the matter. Light, diverging from the point fi (Fig. 126), is reflected from two mirrors, OB, OS, which are hinged: together at 0, and are inclined to each other at a very small angle. After reflection the rays appear to diverge from A and A', the images of B in OS and OB respectively. Hence A and A' act as two sources of : light, the radiation emitted from each of which is similar in all respects to that emitted from the other. The light has nowhere passed through an aperture, so that the objection made to Young's form of the experiment does not apply, and yet the same effects are observed to occur. The points A', A, and ^f obviously lie on a circle, the Centre of which is at ; and the lines OB and OS are respectively perpendi- 162 244 A MANUAL OF PHYSICS, cular to A'B and AB. Hence the angle A'BA is equal to the angle of inclination of the mirrors = 9 (say) ._ But A'OA = 2A'B A = 20 ; and FIG. 126. OM is practically equal to OB = r (say). Therefore, if we denote ON by r\ the formula of last section becomes X Very accurate adjustments are necessary in order to obtain good results from this form of the experiment. 213. Lloyd's Experiment. Lloyd repeated the above experiment with only one mirror. A ray of light diverging from a slit, A' (Fig. 127), is reflected in part, at grazing incidence, from a mirror N FIG. 127. KS. We thus obtain two rays of light, one actually diverging from A', and the other apparently diverging from A, the image of A' in BS ; and these rays produce interference effects as formerly. Yet one distinct difference is observable. In both forms of the experiment previously described the point N is brightly illuminated, for AN - A'N = 0. In Lloyd's experiment N is dark, and the whole system of bright and dark bands is shifted by the breadth of one band. In explanation of this Lloyd suggested that the phase is altered by 180 in the act of reflection. LIGHT : INTERFERENCE, DIFFRACTION. 245 In all cases the slit through which the light passes should be narrow ; but this is not of so much importance in the present case as in the previous cases. For the slit A is the inverted image of A', and so M is the centre of all corresponding parts of A and A', the part of A which is nearest to M being the image of the part of A' which is nearest to M, and so on. Hence the effects of all the parts are strictly superposed at P. But, in the two previous cases, since there is no inversion of A' with respect to A, the part of A' which is nearest to M corresponds to the part of A which is farthest from M (M being taken as the middle point of the line joining the central parts of the slits), and so on. Hence the systems of bands due to the light from the various corresponding parts of the two slits are not exactly superposed, and the definition is in consequence less accurate. 214. Fresnel's Biprism. A second form of the experiment, to which also the objection taken to Young's experiment does not apply, is due to Fresnel. ES (Fig. 128) is a glass prism of very obtuse angle. It is placed with its flat face towards M, the source of light. Each half of the prism forms an image of M, so that the rays emerge from the other faces of the prism as if they proceeded from points A and A', which are practically situated on a straight line, through M, drawn perpendicular to the flat face of the prism. If i^ r 1? are the angles of incidence and refraction at the flat face of the prism, while i,, r 2 , are the similar angles at the opposite face, the total deviation of the ray ME, i.e., the angle A'EM is ( 192) i^n+i^-r^ These angles being small, ii and i. 2 are respectively equal to pr^ and \ir^ \i being the refractive index of the substance of which the prism is composed. Hence the deviation is (p -1) (r 1 +r 2 ) = ( / t 1), where a is the acute angle of the prism. This gives A'M( = AM) = &(/*- l)a approxi- mately, b being the distance of M from the biprism, and so the formula giving the value of x becomes l)ax \, ~ where b' is the distance between the prism and the screen. 246 A MANUAL OF PHYSICS. 215. Coloured Interference Bands. In the immediately pre- ceding sections we have assumed the wave-length to be constant. But the breadth between two adjacent bright or dark bands is pro- portional to X, and so the band situated at N is the only one which is colourless. All other bands are coloured, the first red band being about twice as far from t N as the first violet band. About a dozen of these bands can be fairly well distinguished when ordinary white light is used ; but the succeeding bands of different colours are so superposed that all traces of interference effects practically disappear, and the screen seems uniformly illuminated. If the quantity a in the formula of 214 were variable and pro- portional to X, x would be constant for all wave-lengths, that is, the bands would be colourless. This effect may be attained by the use of a diffraction grating ( 233). In the biprism method the distance between the points A and A' depends upon the wave-length, and is greater the shorter the wave- length is. The result is that the coloured bands are more widely separated than they usually are. The introduction of a coloured glass, which diminishes the number of different kinds of light in the interfering beams, produces a very marked increase in the number of bands which are visible. As many as 200,000 bands have been counted when a flame, tinged deeply orange by burning sodium, was employed as the source of light. When the difference between the lengths of the paths travelled by the two interfering rays is a very large multiple of the wave-length, the nature of the vibrations may have completely altered in the interval of time between the setting out, from the source, of the two waves which simultaneously reach P, so- that no interference could occur. But the fact that no more than 200,000 bands have ever been counted does not prove that no more than 200,000 vibrations of the ether at a given point are sufficiently nearly similar to pro- duce continued interference, for we can neither obtain absolutely monochromatic light nor use an infinitely narrow slit. Yet the converse statement, that 200,000 successive vibrations are practically similar, is true. 216. Displacement of Bands by Befracting Media. If a dense medium be placed in the path of one of the two interfering rays, the whole system of bands will be displaced towards that side of MN on which the medium is placed. For if t be the thickness of a medium of refractive index //, which is traversed by the ray, the effect is the same as if the ray had traversed a thickness, pi, of air. Thus the effective length of the path of that ray is increased by the amount (/i 1) t. LIGHT : INTERFERENCE, DIFFRACTION. 247 Let L (Fig. 129) represent the medium interposed in the path of the ray AT. The effective length of A'P is increased, and so the length of AP must be increased. In other words, PN must increase. Suppose now that L is removed, and that we shift A' back from the screen through the distance (^ - 1)^/2 into the position A/. Let also A be moved towards the screen, through the same distance, FIG. 129. into the position A x . In this way the effective length of A'N is increased, relatively to that of AN, by the amount (n l)t ; and the central band, originally at N, will now be found at Q, which is such that MQ is perpendicular to AiA'j. But QN/MN = AAj/AM = (/* l)/2a. Hence the displacement of the central band (if we are dealing with monochromatic light) is When the light is not monochromatic the displacement of the central (which is then the brightest) band could only be given by this for- mula if the refracting substance did not produce dispersion, i.e., if ^ were independent of X. The brightest effect will really be- pro- duced at a place where the rate of variation of QN with X is a minimum, for, at such a place, the various adjacent coloured bands are most nearly superposed. By means of the formula the refractive index of the interposed substance may be found with extreme accuracy. The method is specially applicable to the determination of the refractive indices of gases. 217. Interference Bands in Spectra. If a beam of white light, diverging from a narrow slit, be made parallel by a suitable lens, and then be refracted by a prism, the usual continuous spectrum will be obtained. But if a plate of a refracting substance be inter- posed in the path of one half of the beam, the spectrum will be 248 A MANUAL OF PHYSICS. crossed by dark bands. The reason is that one half of the rays are retarded relatively to the other half, and so interference effects are produced. Those rays, the relative retardation of which amounts to a semi-wave-length, are obliterated. Various forms of this experiment are described by Powell, Fox, Talbot, Brewster, and Stokes. 218. Colours of Thin Plates. Reflected Ltght.Thm films of transparent substances are frequently observed to be brilliantly coloured. The colours vary with the angle of incidence and with the thickness of the film. Familiar examples occur in the cases of a soap bubble, of the wing of the common house fly, and of highly tempered steel, etc. In the latter case the thin film consists of an oxide formed on the FIG. 130. surface of the steel at a high temperature. Very old glass vessels frequently exhibit these colours from the partial splitting away of thin films at the surface of the glass. The wave theory gives a complete explanation of these pheno- mena. Let AB represent a film, of (small) thickness , of a substance the refractive index of which is //. A ray, ab, falling upon the upper surface of the plate is partially reflected along be and in part is refracted along bd. The refracted ray suffers partial reflection in the direction db' and finally emerges from the substance in the direction b'c' parallel to be. If perpendiculars b'm and b'n be dropped from b' upon be and bd, the parts bm, bn of these paths intercepted between b and the feet of the perpendiculars are described in equal times. Hence the effective difference of path described by the two rays is nd-\-db', which is equal to 2 cos ?, where r is the angle of refraction. And this portion is described in a substance of refractive index ^ so that the equivalent path in air is 2/* cos r. It might, therefore, be expected that the effects of the two rays would be mutually intensified when this quantity is an integral LIGHT : INTERFERENCE, DIFFRACTION. 249 multiple of a wave-length, that they would mutually annul each other when it is an odd multiple of a semi-wave-length, and that, when the thickness of the plate is much smaller than a semi-wave- length of violet light, all rays would be intensified, so that white light would be reflected. But the exact; reverse of these effects are observed. When the thickness of the plate is very small no light is reflected, and, when the quantity ^\ii cos r is an odd multiple of half a wave-length, the light is strongly reflected. These results are precisely those which would occur if, in the acts of reflection at the upper and under faces, a difference of phase of half a period were introduced. The conditions under which the two reflections take place are exactly opposed to each other. In the one case the light is passing from a rarer into a denser medium : in the other it is passing from a denser into a rarer medium. Hence, reasoning by analogy from the effects of impact of two elastic balls of different masses, Young pointed out that the relative acceleration of phase which seems to be required ought to be produced. [The propagation of waves along a ^ rope composed of two parts of different linear densities, is precisely , analogous. A wave propagated along the less dense portion is in part reflected from the junction with a complete reversal of phase. (As an extreme case imagine the rope to be fixed at the junction. Jj This corresponds to infinite density of the second part.) A wave travelling along the more dense portion is partly reflected at the junction without change of phase.] Young pointed out that if his explanation were correct an entire reversal of the effects should occur when the reflecting plate was intermediate in density between the media on either side of it. Further, he carried out such an experiment, and found that his pre- diction was verified. Lloyd's experiment ( 213) furnishes another verification of the correctness of Young's explanation. The effective difference of path, 2ju cos r, decreases as the angle of incidence increases, and therefore the wave-length of the reflected light decreases as the angle of incidence increases. If the refractive index and the thickness of the plate be sufficiently large the series of colours may be repeated a number of times, but, if ordinary white light be used, partial overlapping will occur between all the series above the second, for the wave-length of the extreme red light of the spectrum is approximately double of that of the extreme violet light. 219. The above explanation of the reflection of light from thin plates is not quite complete. The intensity of the reflected ray, be, is always greater than that of the ray b'c', and so complete 250 A MANUAL OF PHYSICS. annulment of light is not accounted for. But complete annulment does take place. A complete treatment of the problem was given by Poisson, who pointed out that all the various rays which emerge at V must be taken into account. The ray which enters at b and suffers one internal reflection at d before it passes out of the plate at b' \AA/V fed FIG. 131. has the greatest effect in producing the final result ; but the ray which suffers two such internal reflections (at e and d) before emergence also has a considerable effect. Similarly, those which have under- gone three, four, etc., internal reflections, have each an appreciable, though rapidly diminishing, share in the ultimate result. The effective difference of path between the ray which has suffered n such internal reflections, and the ray which is once reflected externally at b is ^n\it cos r+X/2, the semi- wave -length being added in order to take account of the acceleration of phase produced in the act of reflection at b. This being taken into consideration it is found that the intensity of the light reflected from the plate does vanish when 2/i cos r is an even multiple of X/2, and that it is a maximum when 2/j.t cos r is an odd multiple of X/2. 220. Colours of Thin Plates. Transmitted Light The light which is transmitted through the plate is complementary to that which is reflected from it ; that is, the kinds of light which are absent from the reflected beam are precisely those which are present in the transmitted beam. The intensity of the reflected light is never equal to that of the incident light, and so the intensity of the transmitted beam never entirely vanishes. Also, since the minimum intensity of the reflected light is zero, the maximum intensity of the transmitted light is equal to the intensity of the incident light. 221. Newton's Rings. Newton observed the colours produced by the interference of rays reflected from both sides of a thin film of air enclosed between two pieces of glass. One of the pieces of glass had a plane surface ; the surface of the other was convex and spherical. LIGHT : INTERFERENCE, DIFFRACTION. 251 The thickness of the film of air at a distance, d, from the point of contact is approximately d 2 /2B where E is the radius of the spherical FIG. 132. surface. Hence the condition that light of wave-length X shall be intensified is u'd? r R - cos r n being any integer, and so the point of contact is surrounded by a series of bright rings. In this formula, r is the angle of refraction from glass into air and n' is the reciprocal of the refractive index of glass. The radii of successive bright rings are therefore given by sec r (2w+l)/2 . X, where /i is the refractive index of the glass referred to air. It follows from this formula that the replacement of the film of I \ air by a denser substance would cause all the rings to close in some- 1 1 what towards their common centre. This result is proved by experi-l | ment, and hence we get another proof of the fact that light travels slower through a medium such as water than it does through air. The successive radii are proportional to the square roots of the natural numbers of the even numbers in the case of the dark rings, and of the odd numbers in the case of the bright rings, and so successive rings enclose equal areas. The radii also increase as the wave-length increases, and so the first red ring is farther from the centre than the first blue one is. Lastly, d increases when the angle of incidence increases. When the two pieces of glass are pressed sufficiently close together a black spot appears at the centre. The central thickness is then very small in comparison with the wave-length of any visible light, and so the reflected light vanishes ; for the effective length of the paths traversed by rays which emerge at a given point after internal reflection is practically the same as that of the light which is directly reflected at the same point without entering the thin film, and so the two sets of rays practically differ in phase by half a period. The transmitted light is complementary to that which is reflected. The central portion is therefore white. 252 A MANUAL OF PHYSICS. Theory indicates that if the refractive index of the film be inter- mediate between the indices of the two transparent media which bound it, the rings seen by reflection should commence from a white centre. Young verified this prediction by means of a film of oil of sassafras enclosed between a lens of crown glass and a lens of flint glass. 222. Colours of Mixed Plates. If a bright object be viewed through an intimate mixture of two media of different refractive indices (e.g., a mixture of oil and air enclosed between glass plates), colours are observed to which Young gave the name of ' colours of mixed plates.' The colours are arranged in rings precisely as in the case of those seen by transmission of light through a thin homo- geneous plate, but the whole system is on a larger scale. The phenomenon is due to the interference of the rays which pass through the different media and so suffer relative change of phase. When the incident light is oblique and a dark object is placed behind the plates, the system resembles that which is ordinarily seen by reflection, for one of the interfering portions is reflected and undergoes the usual acceleration of phase. 223. Colours of Thick Plates. Brewster observed that, in certain circumstances, interference may be produced by means of plates the thickness of which is not small in comparison with the wave-length of light. AB and BC (Fig. 133) represent two such plates of parallel glass, which are precisely equal in thickness, and are inclined to each other at a small angle, . A pencil of light, Pm, falls perpendicularly upon the plate BC, and, passing through it, is partly reflected from the first surface of FIG. 133. AB and in part is refracted into the plate AB. A portion of the refracted part is reflected at ra. If r be the angle of refraction in AB the effective difference of path so produced between the two LIGHT : INTERFERENCE, DIFFRACTION. 258 portions of light is Ipt cos r = 2/i cos sin- 1 (l/^ . sin a), where /* is the refractive index and t is the thickness of the glass. A similar action occurs at the plate BC, and the rays which were reflected from the first surface of AB sustain, relatively to the other rays, an effective increase of path to the amount Ipt cos r' = 2^ cos sin- 1 (l//z . sin 2), r' being the angle of refraction in BC. The effective difference of path of the two rays, pq, which finally emerge from the side of AB remote from P, is therefore 2ju(cos sin" 1 ^ sin ) - cos sin ( sin 2a) ). Interference occurs when this quantity is sufficiently small. Jamin has applied this principle to the construction of an extremely sensitive instrument for the measurement of refractive indices. Newton observed interference effects when he allowed light to fall upon the surface of a concave glass mirror which was silvered behind. The mirror was everywhere of uniform thickness and the light was admitted through a small opening in a sheet of white paper the opening being situated at the centre of curvature of the mirror. A few broad coloured rings, resembling those due to light transmitted through a thin plate, were seen on the paper. All these rings were concentric with the opening through which the light passed. The origin of these colours is totally different from that of the colours which Brewster observed. The rings are due to the inter- ference of light, ordinarily reflected at the silvered surface of the mirror and then scattered by particles of dust on the first surface, with light, also reflected from the silvered surface, but which had been previously scattered (or, rather, diffracted} by particles of dust upon the first surface of the mirror. When the mirror is slightly inclined the centre of the coloured rings is situated midway between the opening in the paper and the image of it which is formed upon the paper. This central spot is alternately bright and dark (when homogeneous light is used), as the distance between the opening and its image increases ; it undergoes a rapid variation of colour when the incident light is white. 224. Diffraction. The principle by means of which Huyghens explained the rectilinear propagation of light has already been given ( 186). The following remarks of Stokes on this subject are specially worthy of note. 254 A MANUAL OF PHYSICS. ' When light is incident on a small aperture in a screen, the illu- mination at any point in front of the screen is determined, on the undulatory theory, in the following manner. The incident waves are conceived to be broken up on arriving at the aperture ; each element of the aperture is considered as the centre of an elementary disturbance, which diverges spherically in all directions, with an intensity which does not vary rapidly from one direction to another in the neighbourhood of the normal to the primary wave, and the disturbance at any point is found by taking the aggregate of the disturbances due to all the secondary waves, the phase of vibration of each being retarded by a quantity corresponding to the distance from its centre to the point where the disturbance is sought. The square of the co-efficient of vibration is then taken as a measure of the intensity of illumination. Let us consider for a moment the hypothesis on which this process rests. In the first place it is no hypothesis that we may conceive the waves broken up on arriving at the aperture : it is a necessary consequence of the dynamical principle of the superposition of small motions, and if this principle be inapplicable to light, the undulatory theory is upset from its very foundations. The mathematical resolution of a wave, or any portion of a wave, into elementary disturbances must not be confounded with a physical breaking up of the wave, with which it has no more to do than the divisions of a rod of variable density into differential elements, for the purpose of finding its centre of gravity, has to do with breaking the rod in pieces. It is a hypothesis that we may find the disturbance in front of the aperture by merely taking the aggregate of the distubances due to all the secondary waves, each secondary wave proceeding as if the screen were away ; in other words, that the effect of the screen is merely to stop a certain portion of the incident light. This hypothesis, exceedingly probable, a priori, when we are only concerned with points at no great distance from the normal to the primary wave, is confirmed by experiment, which shows that the same appearances are presented, with a given aperture, whatever be the nature of the screen in which the aperture is pierced ; whether, for example, it consist of paper or foil, whether a small aperture be divided by a hair or by a wire of equal thickness. It is a hypothesis, again, that the intensity in a secondary wave is nearly constant, at a given distance from the centre, in different directions very near the normal to the primary wave ; but it seems to me almost impossible to conceive a mechanical theory which would not lead to this result. It is evident that the difference of phase of the various secondary waves which agitate a given point must be determined by the difference of their radii, and if it should LIGHT : INTERFERENCE, DIFFRACTION. 255 afterwards be found necessary to add a constant to all the phases the results will not be at all affected. Lastly, good reasons may be assigned why the intensity should be measured by the square of the co- efficient of vibration.' 225. Huyghens' construction, if rigorously carried out, would indicate the existence of a wave running back towards the source as well as a wave which travels forwards. Analogy points to the conclusion that the part of the construction which leads to a reverse wave must be ignored. For example, the investigation of 73 shows that no wave can travel backwards from a disturbance which runs along a stretched cord. But Stokes, in his paper on the Dynamical Theory of Diffraction, of the introduction to which the above quotation forms part, has shown from purely dynamical prin- ciples, that the disturbance in a secondary wavelet is a maximum in the direction of the wave-normal, and that it diminishes constantly as the direction considered is inclined more and more to the normal, ultimately becoming zero in the direction opposite to" that in which the primary wave travels. He then shows that the result of the superposition of all the secondary effects is the same as if the wave (assumed to be practically plane, i.e., of radius which is large in comparison with the wave-length, a condition always satisfied in experiment) had not been supposed to be broken up into a series of separate centres of disturbance, and that no back-wave is pro- pagated. 226. Effect of a Rectilinear Wave. We have already stated that Fresnel showed that Huyghens' principle, according to which the new wave front is found to be the envelope of the secondary wave- fronts, should be explicitly associated with the principle of inter- ference if it is to give a complete explanation of the rectilinear pro- pagation of light. The envelope is the locus of points each of which is simultaneously reached by more than one secondary disturbance the phases of which are identical. It is, therefore, the locus of points at which the light has great intensity. The necessity for the introduction of the principle of interference will appear very evidently from the following investigation of the effect of a rectilinear wave at any external point. Let AB (Fig. 134) represent a portion of a linear wave which extends to infinity in both directions, and let P be the point at which we have to determine the effect of the wave. Draw PM perpendicular to AB and take points m, m r , m", etc., such that Pra PM = Pm' Pm = Pm"-Pm' = etc., = X/2, where X is the wave- length of the light emitted from the various points of AB. The length of the half-period element Mm is J 256 A MANUAL OF PHYSICS. where a, is the length of PM. When X is so small in comparison with the other length involved that it may be neglected, this becomes ^/a\. Similarly Mm', Mm", etc., are respectively equal to M FIG. 134. v/3X, etc. Hence the lengths of the successive half-period elements, from M outwards, are *Ja\, ^/a\ ( */% - 1), */a\ ( *J3 ^/2), etc., and the limit to which they ultimately approach is X/2. If we divide each element into the same number of infinitesi- mal portions, the light sent out by the first portion of the first element differs in phase from that emitted by the first portion of the second element by one half of a period. Similarly, the light emitted by the second portion of the first element differs in phase by one half of a period from that emitted by the second portion of the second element, and so on. Now the effects at P of the various parts of the first element are not quite compensated by the effects of the corresponding parts of the second element. For the breadth of the parts of the first element is rather greater than that of the parts of the second ; and the inclination of each part to the line joining it to P, and also its distance from P, increase as the part is more remote from M. But the difference between the effects of the corresponding portions of the ri k and the n+V k elements is vanishingly small when n is large. Now, as X is a very small length it follows that a very large number of half -period elements are included in a small portion of AB in the near neighbourhood of M, and, consequently, only a small part of the wave near M produces any effect at P. Hence, a small opaque object placed on the line PM would entirely prevent the wave AB from producing any effect at P. Hence the propagation of light is practically rectilinear when X is LIGHT : INTERFERENCE, DIFFRACTION. 257 so small that its square may be neglected in comparison with the other quantities involved. Let e 2 , 20 cos i sin r /ox *- ^nlFHT (8) ' y=-*ssl- -^ %p cos i sin r (10). sin ft+r)cosft-rf ' If 0=0, so that the incident light is polarised in the plane of incidence, and if the incidence is perpendicular, (7) shows us that the ratio of the intensity of the reflected, to that of the incident, light is , ^' 2 _sin 2 (i~r}__ /i r\ z _ //i--l\ 2 The same equation shows that, when i = 90, the whole of the incident light is reflected, LIGHT : DOUBLE REFRACTION, POLARISATION. 275 Again, by (8), when = 0, ^ = 0, we see that the intensity of the refracted light bears to that of the incident the ratio n* _( 2?- V 8 / 2 \ 2 ~*\i+r) ~ + l/ ' Similar expressions may be obtained from (9) and (10). All these results have been verified experimentally. Let the plane of polarisation of the reflected light make an angle tf> with the plane of incidence. From (7) and (9) we get - = tan 8 . q' qcos(ir) cos (i r) Hence the plane of polarisation is rotated by reflection so as to more nearly coincide with the plane of incidence. The angle is equal to at perpendicular incidence ; and it diminishes as i increases, until when i+r = 90 i.e., at the polarising angle it becomes zero. When i increases beyond this value, becomes negative ; and, at grazing incidence, 0= -9. So long as and Q have the same sign, the difference of phase of the two components of the reflected vibra- tion is zero : at the polarising angle the difference changes suddenly to TT. If $ be the angle which the plane of polarisation of the refracted light makes with the plane of incidence, (8) and (10) give ^ = - sec (i - r) = tan 9 sec (i r). tan The rotation of both the planes of polarisation may be increased by successive repetitions of the same process. Since sec (ir) = sec (r-i), refraction through a parallel plate gives tan 1^2 = tan 9 sec 2 (i-r). Equations (5) and (6) above express the condition that there shall be continuity of displacement of the ether parallel to the bounding surface. No account has been taken of the displacement perpen- dicular to the surface. If we replace Fresnel's assumption of uniform rigidity by the condition of no normal discontinuity, we find p = p'. Hence Fresnel's third assumption is inconsistent with normal continuity of displacement. Making the assumption that the ether is of uniform density in all media, Maccullagh and Neumann deduced expressions for the ampli- tudes of the components of the vibration of the reflected ray, in which the quantities on the right-hand sides of (7) and (9) are 182 276 A MANUAL OF PHYSICS. simply interchanged. Hence, on this theory, we must assume that the vibrations are in the plane of polarisation. A similar interchange occurs in the expressions for the components of the vibration in the refracted ray ; and, in addition, the magnitudes are altered. But this alteration occurs in such a way that the amplitude of vibration in the first medium, after refraction through a parallel plate, is identical on both theories. And, further, the rotations of the planes of polarisation are of the same magnitude and sense on the two theories. Therefore none of the phenomena with which we are now dealing are capable of furnishing a test between the assumptions of uniform density and uniform rigidity. 241. Plane, Circular, and Elliptic Polarisation. In the special examples considered in last section, the difference of phase of the two rectangular components of the resultant vibration was or ?r. But the resultant of two rectangular simple harmonic motions is, in general ( 52), elliptic motion. Not only is this true of two rec- tangular components ; it is true of any number of simple harmonic components in lines inclined at any angles to each other. Hence, if we can assume that the vibrations of a particle of the ether are simply harmonic when polarised light (such as we have hitherto considered) is passing, we must conclude that the most general vibration of such a particle, when subject to various simul- taneous disturbances, is elliptical. This assumption is justified by the fact that no phenomena of light, which are not due to simple superposition of displacements, are observed. The resultant elliptic path is described continuously so long as the amplitudes, phases, and periods of the components remain con- stant. In this case the light is said to be elliptically polarised. As a particular case, when all the components can be compounded into two rectangular components, equal in amplitude and period, but differing in phase by 7r/2, the ellipse becomes a circle, and the light is circularly polarised. Ordinary polarisation for example, that produced by reflection occurs when the components can be reduced to two which differ in phase by any multiple of TT. This is usually termed plane polarisa- tion, in order to distinguish it from the above forms. 242. Nature of Common Light. Common light exhibits no trace of polarisation of any description. But this is known to be true also of plane polarised light if its plane of polarisation be made to rotate very rapidly so rapidly that, in little more than one-tenth of a second, the directions of vibration have been distributed uniformly in all possible orientations perpendicular to the ray. Hence we may conclude that ordinary light consists of elliptically polarised LIGHT : DOUBLE REFRACTION, POLARISATION. 277 light, the magnitude, form, and position of the ellipse being in a constant state of rapid change. But the phenomena of interference of light show that practically no change occurs in the course of some thousands of vibrations, for many thousands of interference bands can be counted when homo- geneous light is used. On the other hand, since light travels at .the rate of 186,000 miles per second, while the length of a wave is, on the average, about one forty-thousandth part of an inch, many millions of millions of vibrations must take place per second. But, again, as Stokes has pointed out, from the facts that every source of common light consists of a practically infinite number of points, and that the light emanating from each of these points is, in general, totally independent of that issuing from any other in respect of direction of vibration and also in respect of phase, we cannot expect anything else than an average effect in which there is no manifesta- tion of ' sides.' It follows that a beam of common light must necessarily be divided into two beams of equal intensity when it is transmitted through a doubly refracting substance. From the formulae (7) and (9) of 240, we see that the total intensity of the reflected portion of a beam of unit intensity which falls on a transparent substance is I/sin 2 (i-r) tan 2 (i r)\ 2V sin 2 (i+r) "*~ tan 2 (i+r))' since the two oppositely polarised parts into which the incident beam is supposed to be divided are of equal intensity. Now the second term in this expression is, in general, smaller than the first, and so, in the reflected beam, there is an excess of light polarised in the plane of incidence. The second term vanishes when i + r = 90, from which it follows that the reflected light is entirely polarised in the plane of incidence at the polarising angle. Similarly, we can show that the refracted beam contains an excess of light polarised perpendicularly to the plane of incidence, that it is entirely polarised in the perpendicular plane at the polaris- ' ing angle, and that its intensity is then equal to that of the re- flected beam. At all angles of incidence there are equal amounts of polarised light in the two beams. The known laws of the polarisation of common light by reflection and refraction are therefore consequences of the undulatory theory. 243. Metallic Reflection. Mains observed that light is never completely polarised by reflection from the surface of metals, but 278 A MANUAL OF PHYSICS. that the polarisation attained a maximum at a certain angle of incidence. He also observed that polarised light appeared to be completely depolarised by reflection from a metallic surface when its plane of polarisation was inclined at an angle of 45 to the plane of incidence. Brewster verified, and extended, these results. He showed that the reflected portion of a beam of common light might be com- pletely polarised by a sufficient number of reflections under like conditions a result previously inferred by Biot. He found also that, when the incident ray is polarised in, or perpendicular to, the plane, of incidence, the reflected ray is still polarised in the same plane ; that, when the original polarisation is in any plane other than these, partial depolarisation seems to take place ; and that the depolarisation is greatest at the angle of maximum polarisation. Further, a second reflection, in the same plane, and at the same angle, repolarises the light ; and the new plane of polarisation lies on the opposite side of the plane of incidence and makes a different angle with it. The ' depolarisation ' above spoken of does not mean restoration to the condition of common light. The originally polarised light may be decomposed into two parts, one polarised in the plane of incidence, the other polarised in the perpendicular plane. The amplitudes of these parts may suffer change by reflection, which ( 240) produces a rotation of the plane of polarisation. The phases may also be altered, and this will give rise to elliptic polarisation. Jamin's experiments on this subject show that Brewster's ' depo- larisation ' is really elliptical polarisation, and also show the nature of the variations of amplitude and phase. The laws of change of amplitude are the same as those already found in 240. The difference of phase increases from perpen- dicular incidence to grazing incidence by the total amount TT, the phase of the ray polarised in the plane of incidence being accelerated with reference to that of the other. The change of phase is ex- tremely slow, except in the immediate neighbourhood of the angle of maximum polarisation, between near limits on either side of which the total change occurs. The difference of phase ought, according to Fresnel's theory ( 240), to increase suddenly by TT at the polarising angle. Extending his observations to transparent bodies, Jamin found that the difference between them and metals is only a difference of degree. In all cases elliptic polarisation is produced, and the maximum ellipticity occurs at the angle of maximum polarisation, which co- incides very closely with the angle deduced from Brewster's law. LIGHT : DOUBLE REFRACTION, POLARISATION. 279 Jamin found that some transparent substances differ from metals with respect to the sign of the difference of phase which is produced by reflection. Substances whose refractive index is less than 1'46 retard the phase of the component which is polarised in the plane of incidence ; substances which have a refractive index exceeding T46 resemble metals in accelerating the phase of this component ; substances in which the refractive index is equal to 1'46 obey Fresnel's laws. He also found that, in metals, the angle of maximum polarisation decreases as the wave-length of the light increases ; from which we see that metals, if they obey Brewster's law, must refract light of long wave-length more than light of short wave-length. Eecent experiments on refraction through thin metallic prisms seem to confirm this conclusion. When light polarised in the plane of incidence is reflected from a metallic surface, the intensity of the reflected beam is a minimum at the angle of maximum polarisation. Maccullagh pointed out that transparent substances, the refractive index of which exceeds 2 + v3, possess (according to his theory) a minimum reflecting power at a definite angle of incidence. 244. Double Eefraction by Biaxal Crystals. Brewster dis- covered that most doubly refracting crystals possess two optic axes. In uniaxal crystals the axis is equally inclined to the three edges which meet at an obtuse-angled corner of the crystal. In biaxal crystals the lines which bisect the two angles contained by the axes, and the line at right angles to these two, have a definite relation to the crystalline form. Fresnel has proved, theoretically and experimentally, that neither of the two rays in a biaxal crystal obeys the ordinary law of refrac- tion. By means of certain assumptions, he investigated the problem of the propagation of waves of transverse vibration in a non-isotropic elastic medium. Crystalline substances are known to be non- isotropic, and presumably the property is impressed upon the ether which pervades them, and which is known to be hampered, as regards its free oscillation, by the presence of material particles. The complete laws of double refraction may most readily be studied from the point of view of his theory, which we now proceed to give. 245. Fresnel's Theory of Double Eefraction. Fresnel under- took his investigation when the discovery of double refraction in biaxal crystals made it apparent that Huyghens' construction for the wave-surface was not applicable in all cases. In a non-isotropic substance, the resultant force which opposes 280 A MANUAL OF PHYSICS. the displacement of a particle does not in general act in the direction of the displacement. But Fresnel showed that there are three directions, at right angles to each other, in which the force, called into existence by the displacement, acts so as to move the particle directly back to its position of equilibrium. He showed that if, from any point in the interior of the substance, lines be drawn with lengths proportional to the square roots of the elastic forces which resist displacement in the directions in which the lines are taken, the extremities of these lines will lie on an ellipsoid (called the ellipsoid of elasticity). The three principal axes of this surface are in the directions in which the force tends to move the displaced particle directly back to its undisturbed position. The speed of wave-propagation in an elastic medium is propor- tional (cf. 168) to the square root of the elastic force (or distor- tional rigidity), and hence the radii of the ellipsoid are proportional to the speeds of propagation of waves when the vibrations are along the given radii. Consider a plane wave passing through the medium. Fresnel proved, from the fact that the intensity of a beam of light which is compounded of two beams polarised at right angles to each other is independent of the phase of either component, that the vibrations must lie in the wave-front. If, therefore, we regard a central section of the ellipsoid of elasticity by the wave-front, we see that there are only two directions those of the two axes of the section in which a displacement will give rise to a reverse force acting in a plane which is normal to the wave and which passes through the line of displacement ; for Fresnel showed that the force acts in the normal to that central section of the ellipsoid which is conjugate to the direc- tion of the displacement. In general, the force will have a compo- nent perpendicular to the wave-front ; but, according to assumption, this produces no effect in the way of wave-propagation. Corresponding to any given plane wave-front, there are therefore only two directions of vibration such that the elastic force developed by the displacement has an effective component entirely in the direction of the displacement. But this condition is essential to the propagation of a permanent wave. Hence a plane wave, incident upon such a medium, is, in general, broken up into two waves, which are propagated in different directions with speeds which are proportional to the radii of the ellipsoid drawn in these directions. [The following extract from Stokes' ' Lectures on Light ' will aid in the formation of clear ideas on this point : ' Now we have not far to go to find a mechanical illustration of LIGHT : DOUBLE REFRACTION, POLARISATION. 281 such a mode of action. Imagine an elastic rod terminated at one end, and extending indefinitely in the other direction. Let the rod be rectangular in section, the sides of the rectangle being unequal, so that the rod is stiffer to resist flexure in one of its principal planes than the other. Let this rod be joined on to a cylindrical rod form- ing a continuation of it which extends indefinitely. Conceive the compound rod as capable of propagating small transverse disturb- ances, in which the axis of the rod suffers flexure. Imagine a small disturbance, suppose periodic, to be travelling in the cylindrical rod towards the junction. It will travel on without change of type, even though the flexure of the axis be not in one plane. But to find what disturbance it excites in the rectangular . rod, we must resolve the disturbance in the cylindrical rod into its components in the principal planes of the rectangular rod, and consider them separately. Each will give rise in the rectangular rod to a disturb- ance in its own plane, but the two will travel along the rod with different velocities. This illustrates the sub-division of a beam of common light falling on a block of Iceland spar into two beams polarised in rectangular planes, which are propagated in the spar with different velocities. Again, suppose the original disturbance in the cylindrical rod confined to one plane. If this be either of the principal planes of the rectangular rod, the more slowly or the more quickly travelling kind of disturbance, as the case may be, will alone be excited in the latter ; and if the plane of the original dis- turbance be any other, the components into which we must resolve it in order to find the disturbance excited in the rectangular rod will in general be of unequal intensity, their squares varying with the azimuth of the plane of the original disturbance in accordance with Malus's law. This illustrates the sub-division of a beam of polarised light incident on Iceland spar into two of unequal intensity polarised in rectangular planes, and their alternate disappearance at every quarter of a turn. We see with what perfect simplicity the theory of transverse vibrations falls in with the elementary facts of polarisation discovered by Huyghens, standing in marked contrast in this respect with the conjecture by which Huyghens himself attempted to account for double refraction.'] In any one direction in the interior of the substance two plane waves may be propagated, generally with different velocities ; and the vibrations in these waves are necessarily at right angles to each other. But there are two directions in which the speed of propaga- tion of a wave is independent of the direction of vibration. These directions are parallel to the normals to the two sets of circular sections of the ellipsoid of elasticity ; for, all the radii of a circular 282 A MANUAL OF PHYSICS. section being equal in length, the same force of restitution is called into play by a given displacement along any radius. The resolution of an incident beam into two rectangularly polarised beams which obey Malus's law, and the existence of two directions in which a polarised beam is transmitted without modifi- cation, are therefore consequences of Fresnel's theory. The planes of polarisation pass through the normal to the wave and through the major and minor axes of the section of the ellipsoid of elasticity by the wave. But the lines in which the circular sec- tions cut the given plane section are equally inclined to the principal axes. Hence the planes of polarisation bisect the dihedral angles which are contained by the planes which pass through the normal to the wave and the optic axes. The form of a wave which spreads out through the medium from any centre is formed by the following construction, which is due to Fresnel: Along the normals to any central plane section of an ellipsoid, similar to the ellipsoid of elasticity, measure from the centre lengths which are proportional to the principal axes of the section. The wave-surface is the locus of the points so found, and consists of two sheets. The directions of the refracted rays are found from this surface by Huyghens' construction. The tangent plane drawn to one sheet gives the direction of the one ray ; that drawn to the other sheet gives the direction of the second ray. The wave-surface possesses symmetry with respect to the three principal planes of the ellipsoid from which it is derived. Its traces upon these planes consist respectively of a circle and an ellipse. Let OA, OE, OF (Fig. 146) be the three principal axes. AB is a circle of radius OA equal to the mean principal axis of the ellipsoid ; CD and EF are circles, the radii of which are respectively equal to the least and the greatest principal axes. BC is an ellipse, the principal axes of which are equal to the two least axes of the ellipsoid, and so on. When the ellipsoid of elasticity is one of revolution, the wave- surface consists of two separate sheets a sphere and an ellipsoid which have one axis in common. Hence Fresnel's theory explains the double refraction of uniaxal crystals. It is very desirable to note that Fresnel's results, although they are all seemingly in complete accordance with observed facts, are not rigorous deductions from his assumptions. Green and Neumann proved that a strict investigation, based upon these assumptions, will lead only approximately to Fresnel's Laws. Green then deduced these laws as rigorous consequences of an originally more general theory, the generality of which was subsequently limited by LIGHT : DOUBLE REFRACTION, POLARISATION. 283 imposing a very probable condition ; but this theory made it neces- sary to suppose that the direction of vibration is in the plane of polarisation. He then showed that a still more general theory would, by means of suitable assumptions, lead to the same laws, and to the conclusion that the vibrations are perpendicular to the plane of polarisation. Other theories also give like results; for example, Maccullagh's theory, which is identical in its results with Green's first theory. On this point Stokes remarks that the principle of transverse vibrations is common to all these theories, while Fresnel's Laws are the simplest which can suit the phenomena ; so that their mutual agreement can merely be regarded as a confirma- tion of that principle, while none of the special assumptions made as to the nature of the luminiferous medium in the interior of FIG. 146. crystals can be regarded as being proved solely by the correctness of the results to which they lead. 246. Conical Eefraction. Sir W. E. Hamilton showed that the tangent plane QR (Fig. 146) touches the wave-surface at all points of a circle of contact, so that the point P is a ' conical point.' Four such points, one in each of the quadrants in the plane EOA, exist. They are, of course, the four points of intersection of the circle APB with the eUipse DPE. The line OQ is perpendicular to the plane QR. But the perpen- dicular on the plane which touches the wave-surface represents the speed of propagation of the plane wave to which it is a normal ; and the direction of the rays in the crystal are those of the lines joining O to the point of contact. Hence a plane wave, incident upon the crystal in such a direction that RQ is its front after refraction, 284 A MANUAL OF PHYSICS. gives rise to a cone of rays which proceed in the directions of the lines joining to the various points of the circle of contact. Hamilton's theoretical prediction of this phenomenon, which is known as internal conical refraction, was verified experimentally by Lloyd. Both sheets of the surface are touched by the plane KQ, and hence there exists only one wave velocity in the direction OQ, which is therefore one of the optic axes. The other optic axis is the image of OQ in the plane EOF. An infinite number of tangent planes may be drawn to the sur- face at the point P, and therefore a ray which proceeds through the crystal in the direction OP, gives rise on emergence to a thin conical sheet of rays. This external conical refraction was also predicted theoretically by Hamilton, and found, as the result of experiment, by Lloyd. Since the radii of the wave-surface represents the velocities of the rays, we see that OP is a direction of single ray-velocity in the sub- stance, for the two sheets of the wave-surface intersect at the point P. The other axis of single ray-velocity is the image of OP in EOF. The axes of single ray-velocity never deviate much from the axes of single wave-velocity, i.e., from the optic axes. 247. Interference of Polarised Light. The laws of interference of polarised light were investigated experimentally by Fresnel and Arago. Two rays of light, which are polarised in perpendicular planes, do not give rise to phenomena of interference under circumstances in which rays of ordinary light would interfere. The planes of polarisation of two such rays may, by suitable means, be made to coincide ; but no interference will take place unless the two rays were originally parts of one polarised beam. Eays polarised in the same plane will always interfere under the conditions in which rays of ordinary light would interfere. When rays which have been polarised by double refraction pro- duce interference, the phenomena exhibited are such as to necessitate the assumption that the phase of one component has been accele- rated by the amount TT relatively to that of the other. The reason for this is not far to seek. Let OP (Fig. 147) represent the vibration in a ray which falls upon a crystal of Iceland spar the principal section of which is AA'. OP will be broken up into two components On and Om, respectively along and perpendicular to the axis. Let each of these be again resolved in the directions aa' and bb'. The components along aa' LIGHT : DOUBLE REFRACTION, POLARISATION. 285 are in the same phase, but the components along W are necessarily in opposite phases. 248. Colours of Crystalline Plates. Let us suppose that a beam of plane polarised white light is obtained by means of reflection, and let a second reflector be so placed as to extinguish the beam. If a thin crystalline plate be now interposed between the two reflectors in the path of the beam, intensely coloured light will in general be reflected from the second surface. The light disappears whenever the principal section of the crystalline plate coincides with, or is perpendicular to, the plane of reflection from the first surface. If the plate be turned round, in its own plane, from this position, some reflection from the second surface will be evident, and the reflected light will vary in intensity, FIG. 147. but not in colour, as the plate is rotated. (The same appearances, colour excepted, would, of course, be manifested even if the plate were thick.) On the other hand, if the plane of reflection at the second surface be varied, the crystal being fixed, the colour changes gradually into the tint which is complementary to the former. When any two successive positions of the second plane of incidence differ by 90 from each other, the reflected tints are complemen- tary. The colours depend upon the thickness of the crystalline plate, and vary with the thickness in the same way as the colours seen by reflection from a thin plate of air. The explanation of the phenomenon is simple. The originally plane polarised light is divided in the crystal into two beams, which are oppositely polarised, and which traverse the plate with unequal speeds. The phase of the one beam is therefore retarded relatively 286 A MANUAL OF PHYSICS. to that of the other, and so interference may take place if the vibra- tions in the two portions are again resolved in a common direction. The tint is due to the cutting out of some rays, and the intensify- ing of others, by interference. The light which emerges from the plate is elliptically polarised, since it is compounded of two rectangularly polarised beams which differ from each other in phase. In particular, when the difference of phase is any odd multiple of a quarter of a period, the light is circularly polarised ; and, when the difference is a multiple of half of a period, the light is plane polarised. If the difference is an even multiple of a half-period, the plane of polarisation coincides with the original plane ; it makes an equal angle with the principal section, on the opposite side, when the difference is an odd multiple of a half-period. FIG. 148. Let POP' (Fig. 148) represent the direction of vibration in the incident beam of light which falls upon the plane surface P M of a doubly refracting plate, and let p, // represent the principal section of the crystal so that the vibrations in the ordinary ray are in the direction M. If the amplitude of vibration along P be unity, the amplitude of the vibrations in the ordinary ray is cos 9, where 9 = P M ; and the amplitude of those in the extraordinary ray is sin 9. Let v v' represent the principal plane of a second doubly refracting crystal through which pass both the beams into which the original beam is divided. The vibrations along M and ^ give rise to two sets of vibra- tions along N, the amplitudes of which are cos 9 cos (0 0) and sin 9 sin (00) respectively, where = P O N. These two vibra- tions are the components of the vibrations of the ordinary ray which emerges from the second crystal. LIGHT 1 DOUBLE REFRACTION, POLARISATION. 287 The vibrations along M and p also give rise to two sets of vibrations along v, the amplitudes of which are cos 9 sin (0 0) and sin 9 cos (9 - 0) respectively. And these two vibrations are the components of the vibrations of the extraordinary ray which emerges from the second crystal. The intensity of the ordinary beam is therefore cos 2 9 cos 2 (9 - 0) + sin 2 9 sin 2 (9-0) +2 cos 9 sin cos (0-0) sin (0-0) cos 2 TT_' where I is the effective difference of path of the two rays in the thin crystalline plate, and X is the wave-length. Similarly, the intensity of the extraordinary beam is cos 2 9 sin 2 (9 - 0) + sin 2 9 cos 2 (0 - 0) + 2 cos 9 sin 9 cos (0-0) sin (0-0) cos 2 vL \ These two expressions easily reduce to cos 2 - sin 2 sin 2 (0 0) sin 2 TT _ , /Y and sin 2 + sin 2 sin 2 (0 - 0) sin 2 TT- X When light of various wave lengths is used, the sum of all quan. tities of the form sin 2 TT lj\ must be taken. From these expressions we can deduce all the observed effects. The sum of the two intensities is unity, and therefore the colours of the ordinary and extraordinary beams are complementary. The second term in each expression is the quantity upon which the coloration depends. Hence all colour vanishes when = or |, that is, when the principal section of the crystalline plate is parallel or perpendicular to the original plane of polarisation. It also vanishes when 0-0 = or-, that is, when the principal sections of the thin plate and the second doubly refracting crystal are parallel or mutually perpen- dicular. The reason for this is that, in each of these four cases, one of the components of both the extraordinary and the ordinary beams necessarily vanishes : and the reason for the colours of the two beams being complementary is that the difference of the phases of the components of the extraordinary beam necessarily A MANUAL OF PHYSICS. differs from that of the components of the ordinary beam by the amount ir. The coloration is greatest when sin 2 9 sin 2 (9 0) is a maximum. As the maximum value is unity, this necessitates 9 = = or that is, the principal section of the crystalline plate must make an angle of 45 with the original plane of polarisation, and the prin- cipal section of the second doubly refracting crystal must be parallel, or perpendicular, to that plane. The colours change through regularly recurring cycles as the quantity Z, and therefore as the thickness of the plate, increases by successive equal stages. 249. Special Cases. Hitherto we have assumed that the incident beam of light is parallel. We shall now suppose that a diverging beam of polarised light traverses a uniaxal crystalline plate the parallel plane faces of which are perpendicular to the axis. A ray O P (Fig. 149) which passes perpendicularly through the plate suffers no change. It will pass through, or be stopped by, a second plate which is cut parallel to its optic axis according as the principal FIG. 149. section of the second plate is parallel to, or perpendicular to, the original plane of polarisation. Any other ray will undergo change according to its inclination to the optic axis, and according to the angle which the plane passing through it and the axis makes with the plane of polarisation. Let A B' A' B (Fig. 150) be perpendicular to the axis of the cone of rays, and let A A' and B B' represent respectively the original plane of polarisation and the perpendicular plane. All rays which LIGHT : DOUBLE REFRACTION, POLARISATION. 289 emerge from the crystal in these planes will be allowed to pass through, or will be stopped by, the second plate, according as its principal plane is parallel, or is perpendicular, to the original plane of polarisation. The field will therefore exhibit a light or a dark cross. (Haidinger's Brushes are observed when polarised light is ex- amined by the naked eye. The phenomenon consists of two yellowish- brown patches of light forming a brush the axis of which is parallel to the plane of polarisation ; and two other bluish or purplish patches occur in the angles between the yellow patches. The appearance is due to a polarising structure which is most highly developed in dark eyes. It appears that the yellow spots of the eye are doubly re- fracting and absorb the extraordinary ray to a greater extent than the ordinary ray. Helmholtz finds that the effect only appears with blue light. The brushes soon disappear unless the plane of polarisa- tion be changed at intervals.) At any other point, such as Q, the vibration of the ray is resolved into its two components, polarised parallel and perpendicular to the plane through PQ. Thus the incident ray will be divided into two, which traverse the crystal with different speeds and so give rise to interference. The retardation of the phase of the* one component relatively to that of the other is constant so long as the distance PQ is constant, and becomes greater and greater as PQ increases in length. Hence the field exhibits a series of alternately light and dark circles surrounding the point P. The circles are brilliantly coloured if white light be used ; and the FIG. 151. colours seen when the principal section of the second crystal occu- pies any definite position are exactly complementary to those which are seen when this plane is rotated through a right angle. 19 290 A MANUAL OF PHYSICS. These effects are produced in comparatively thick crystals, since the difference between the speeds of the two rays, in directions not greatly different from the optic axis, is comparatively small. The squares of the radii of successive circles are nearly propor- tional to the natural numbers. For it has been proved that the difference of the squares of the speeds of propagation of the two waves is proportional to the square of the sine of the angle which the ray within the crystal makes with the optic axis, and also that it is proportional to the thickness of the plate, and to the interval of retardation conjointly. Hence the retardation varies as the square of the sine of- the angle between the ray and the axis. But this angle is very nearly equal to the angle QOP (Fig. 149), or to the distance QP. Consider now a parallel plate cut from a biaxal crystal in a direc- FIG. 152. tion perpendicular to the line bisecting the optic axes. The interva of retardation is, in this case, proportional to the product of the sines of the angles which the wave normal makes with the axes. And these sines are approximately proportional to the distances of the point of emergence of the ray from the points in which lines drawn parallel to the axes, from the point of incidence on the first face of the crystal, meet the second face. Hence the locus of P is an oval of Cassini. LIGHT : DOUBLE REFRACTIONj POLARISATION. 291 250. Artificial Production of the Doubly Refracting Structure. Fresnel showed that glass and other singly refracting substances become doubly refracting when subjected to stress ; and Brewster showed that unequal heating of the substance is sufficient to pro- duce the requisite strain. Compression gives rise to a structure which resembles that of Iceland spar in producing an extraordinary ray the refractive index of which is less than that of the ordinary ray. Expansion produces an opposite effect. The optic axis, in these cases, is fixed in position as well as in direction, unless the strain be homogeneous. If an elliptic cylinder of glass be suddenly heated uniformly over its surface, a biaxal refracting structure is developed. This structure is also usually seen in unannealed glass. Analogous changes may be produced in substances which naturally exhibit double refraction. Clerk-Maxwell showed that a viscous liquid becomes doubly refractive while subjected to shearing stress. 251. Eotatory Polarisation. Quartz is a doubly refracting sub- stance in which the refractive index of the extraordinary ray is greater than that of the ordinary ray. The wave-surface consists of a sphere and a spheroid, but the spheroid lies entirely within the sphere. It follows from this that the speeds of propagation of the two rays along the optic axis are not the same. But, further, the vibrations in the two rays are not rectilinear. They are elliptical in general, and the ellipses are described in opposite directions in the two rays. When the rays are propagated along the axis, the vibrations become circular. Now the resultant of two uniform motions, of equal period, in opposite directions in the same circle, is rectilinear motion. For, if A, A' (Fig. 153) represent simultaneous positions of the oppositely moving points, the resolved parts of their motion perpendicular to the line PQ, which is drawn from the centre of the circle through the middle point of the arc joining A, A', destroy each other. Hence the resultant is simple harmonic motion in the line PQ. But if A' be retarded relatively to A, the line PQ will take a new position, bisecting the arc between A and the new position of A'. And, if A' is continuously retarded, PQ will revolve con- tinuously round in the direction A A'. On the other hand, if A be retarded relative to A', PQ will revolve in the direction from A' to A. The two circularly polarised rays in quartz therefore produce plane polarised light on emergence ; but, because of the retardation of one ray relatively to the other, the plane of polarisation has been 192 292 A MANUAL OF PHYSICS. rotated through an amount which is proportional to the thickness of the quartz. The rotation is right-handed in some specimens of quartz ; left- handed in others. Amethyst consists of alternate layers of right- handed and left-handed quartz. The amount of rotation is dependent upon the wave-length. Biot, and subsequently Broch, proved that it is approximately inversely proportional to the square of the wave-length. The first three terms of the formula furnish a much better approximation, p being the rotation, and a, 6, c being constants. The spectrum produced by plane polarised sunlight which has passed through a plate of quartz is indistinguishable from that which is produced by ordinary sunlight. But a profound modification takes place if, previous to its passage through the refracting prism, the light be passed through an apparatus arranged to polarise light in a plane at right angles to the original plane of polarisation. Were the quartz plate taken away, no light could pass under these circumstances. The effect of the quartz is to restore the light, with the exception of such rays as have had their plane of polarisation rotated through a multiple of two right angles. Consequently, the spectrum is crossed by dark bands. By rotating the second polarising apparatus any one of the dark bands may be caused to occupy any desired position in the spectrum, and so the total rota- tion for any particular kind of light (and, consequently, the wave- length of that light) may be measured with extreme accuracy. LIGHT I DOUBLE REFRACTION, POLARISATION. 293 The dark bands are not sharply marked, for rays near those which are totally extinguished are necessarily partially extinguished. And, since portions of the light are cut out, the finally emergent light is coloured. The coloration disappears when the length of the quartz is so great that the portions which are cut out are dis- tributed with practical uniformity throughout the spectrum. Many liquids, solutions, and even vapours, possess this rotatory power the rotation being in some cases left-handed, in others right- handed. As a rule, the rotation produced by a given thickness of a liquid is much smaller than that produced by the same thickness of quartz. Neither dilution by an inactive substance nor vaporisation alters the rotatory power of -a liquid, except in degree ; on the other hand, Herschel showed that quartz is inactive when in solution ; and Brewster showed that it is inactive when fused. Hence it is inferred that, while the rotation in quartz depends upon the crystalline structure, the rotation in liquids and vapours is a molecular phenomenon. All rotatory polariscopes, or saccharimeters, depend essentially upon the above principles. In some of them, equal intensity of two beams of light is used as a test ; in others, the test is furnished by the equality of tint of two beams. A spectroscopic test, such as that which has been described above, is by far the most delicate. If the light which has passed through quartz be reflected back, so as to retraverse it in the opposite direction, the rotation will be un- done. Kotation of the plane of polarisation may be produced in a magnetic field ; but, in this case, reversal of the path of the light will not (Chap. XXXII.) be accompanied by an undoing of the rotation. 252. Polarising Prisms. Many methods are used for the pro- duction of polarised light. Plane polarisation by reflection has been already described. Any doubly refracting crystal, of course, furnishes us with the means of producing plane polarised light, provided that we can separate the two beams. Some substances, such as tourmaline, strongly absorb one of the two rays into which the incident beam is divided, and so furnish, when of sufficient thickness, a ready means of obtaining plane polarised light. A block of Iceland spar separates the rays in proportion to its length, but it is extremely difficult to obtain large blocks which are fr/e from internal flaws. In Nicol's prism, therefore, one of the two rays is got rid of by total reflection. A long block of the spar is divided into two parts by a plane which is perpendicular to its 294 A MANUAL OF PHYSICS. principal section. The two portions are then cemented together, in their original position, by means of Canada balsam, the refractive index of which is intermediate in magnitude between the indices of the spar for the extraordinary and the ordinary rays. Consequently, when the inclination of the dividing plane to the path of the ray is sufficiently great, the ordinary ray suffers total reflection at the sur- face of the film of balsam, and the extraordinary ray alone is transmitted. FoucaulVs prism is essentially similar, but the film of balsam is replaced by a film of air. A shorter block of the spar suffices in this method, but considerable loss of light takes place by reflection at the surface of the film. Considerable separation of the rays may be obtained by -the use of a prism of Iceland spar, or of quartz, which is achromatised by means of a prism of glass. In EocJion's prism the edge, and one of the faces, of the prism are perpendicular to the optic axis. The rays therefore pass through the prism without separation until they emerge at the opposite face. The prism is achromatised by means of a second prism, of the same substance, the refracting edge of which is parallel to the axis. The ordinary ray proceeds through the second prism without alteration of direction, and is therefore uncoloured ; but the extraordinary ray is considerably refracted, and is coloured at its edges, since the refraction depends upon the wave- length. Greater angular separation of the rays may be obtained by Wollastori } s prism. In this arrangement the refracting edge of the first prism is perpendicular to the optic axis, but the face upon which the light falls perpendicularly is parallel to the axis. In all other respects the arrangement is the same as in Eochon's prism. Both rays are coloured at their edges, for both are deviated from their original direction. Methods of producing elliptically, or circularly, polarised light have been already indicated. They all depend upon the introduction of a difference of phase between the two rectangularly polarised com- ponents of a beam of light. This difference may be produced by transmission through a doubly refracting plate. When the plate is of such thickness as to produce a difference of phase of a quarter- period, the light is circularly polarised provided that the two beams are of equal intensity. Such a plate is termed a quarter-wave plate. The difference of phase may also be produced by reflection or refraction as in FresneVs rhomb. This consists of a parallelepiped of St. Gobain glass the faces of which are so inclined that a ray enters and emerges perpendicularly at opposite faces after two LIGHT : DOUBLE REFRACTION, POLARISATION. 295 internal reflections at an angle of 54 37'. Each reflection produces a difference of phase of 45, and therefore the light emerges circu- larly polarised if it were originally polarised in a plane inclined at 45 to the plane of internal reflection. Conversely, Fresnel's rhomb shows the existence of circularly polarised light by changing it into plane polarised light. This will also occur in the case of elliptically polarised light when either axis of the ellipse is inclined at 45 to the plane of internal reflection. But the two cases may be distinguished by means of a Nicol's prism ; for rotation of the Nicol produces no alteration of intensity if the beam which is passing through it is circularly polarised, but it does produce variation of intensity if the beam is elliptically polarised. Though similar variation of intensity takes place if the incident beam is partially plane polarised, a quarter-wave plate, or a Fresnel's rhomb, enables us clearly to distinguish between these two cases. CHAPTEE XX. THE NATURE OF HEAT. 253. Radiant Heat. Its Identity with Light. We are accustomed to speak of the heat which we receive from the sun just as we speak of the light which we receive from it. And so the term ' radiant heat,' as applied to the heat which comes to us from distant bodies, apparently without the intervention of ordinary matter, has come into scientific use. The non-intervention of ordinary matter in the process of radia- tion by which heat, like light, passes from one body to another at a distance from it, can readily be proved. There is no sufficient amount of ordinary matter in interstellar or interplanetary space to account for the transference : and a hot body cools in vacuo almost as readily as when surrounded by air indeed, in certain cases, a hot body wih 1 cool less rapidly when surrounded by a material medium than it will otherwise. Luminous bodies (those which exhibit fluorescence or phosphor- escence being alone excepted) radiate heat as well as light, and the hotter they are the more luminous do they appear. We might, therefore, naturally conclude that the difference between light and radiant heat is not a difference of kind, but only a difference of degree. This inference is fully borne out by many facts. One point in which they completely resemble each other is recti- linear propagation in free space or in a homogeneous medium. The heat-shadow which any obstacle casts is identical with the shadow which it produces with regard to light proceeding from the same source. And, since the path of light is rectilinear, this proves that radiant heat also moves in straight lines. Heat, like light, is not propagated instantaneously ; and an, even more fundamental point of resemblance than the above appears in the fact that their speeds in vacuo are identical. This is made evident by the simultaneous disappearance and re-appearance of the light and heat when a total eclipse of the sun takes place. THE NATURE OF HEAT. 297 The laws of reflection of the two are identical, for the focus of a mirror for heat-rays is the same as its focus for light-rays. A thermo-electric pile, placed at the focus of a reflecting telescope, can make evident the heat radiated from a star. Their laws of refraction are also the same ; although, at first sight, a difference appears because the focus of a lens for heat is farther off from the lens than its focus for light. But this only strengthens the analogy, for, unless the lens be achromatised, the focus for red rays is farther off than the focus for blue rays. Both are governed by the same laws of interference and of polarisation. The phenomena of interference prove the existence of periodicity of motion, and show that the vibrations are transverse, as in the case of light. And the usual methods based upon interfer- ence, diffraction, and refraction, enable us to measure the wave- length, which is found to be greater than that of luminous rays. We therefore conclude that light and radiant heat are one and the same thing ; that the latter differs from the former only as red light differs from blue light ; and that it is not evident to the sense of sight because the eye is so constituted that it cannot respond to the slower vibrations. We already know that some eyes can perceive rays at the red end of the spectrum (and also at the blue end) which are totally invisible to other eyes. And colours the word being used by analogy appear in heat just as they do in light. Rock-salt is very transparent to heat rays just as glass is transparent to light ; but, on the other hand, ordinary glass is very opaque to heat rays, i.e., it absorbs them to a great extent. It acts to heat just as coloured glass acts to light ; and many other substances act similarly. The law of absorption, with varying thickness of the medium, is the same as that which holds in the case of light ( 205). 254. Heat in Material Bodies. Hypothesis of Molecular Vor- tices. Since hot bodies give out radiation, and since the propagation of radiation involves motion of the particles of an inert medium, we might infer that the particles of a hot body must be in rapid motion, and that the communication of heat from one body to another depends upon the intercommunication of motion. It is scarcely a century since Rumford and Davy arrived at this result upon experimental grounds. Previous to their investigations heat was supposed to be a form of matter which was occluded in the interior of substances, but which was looked on as imponderable since it did not add to the weight of a body. This imponderable substance was termed Caloric. According to the caloric hypothesis, a body was hotter or colder 298 A MANUAL OF PHYSICS. in virtue of its having absorbed a greater or a smaller quantity of caloric ; and when, by any means, the capacity of a body for caloric was diminished, it gave out heat (or rather caloric). In the year 1798 Rumford was engaged in the boring of cannon, and observed (what had often previously been noticed] that there was a rise of temperature in the process of the reduction of the solid metal to the state of filings. But, according to the caloric hypothesis, the rise of temperature implies a diminution of the capacity of the substance for caloric ; and, conversely, an increase of the capacity of a body for caloric would be accompanied by a fall of tempera- ture unless additional caloric were supplied. Hence, in Eumford's experiments, the rise of temperature of the filings signifies, on this hypothesis, a diminution of the capacity for caloric. Bumford, therefore, sought to determine by experiment whether or not the filings had less capacity for heat than the solid metal had. He heated equal weights of the solid metal and of the filings to the same high temperature, and dropped them into equal quan- tities of water at the same low temperature. He found that the same changes of temperature were produced in both cases, and con- cluded that the capacity of the substance for caloric had not been reduced when the body was broken up into smaller portions. He was not aware that there was a distinct difference between the physical states of the two specimens of the substance which might have caused his experiment to indicate a wrong result. For the filings might have been so strained as to contain a considerable amount of latent heat which would only appear on their complete recovery from the state of strain. His conclusion was nevertheless correct, for it is entirely supported by experiments based upon accurate principles; and his observations therefore prove that the caloric hypothesis is incorrect. But Kumford did not stop at this stage. He observed that the quantity of heat which was developed in the process of boring was independent of the amount of metal which was abraded that a blunt borer and a sharp borer, though producing very different amounts of filings, caused the same development of heat if the same amount of work were spent in driving them. And, further, there seemed to be practically no limit to the amount of heat which might be produced. He therefore reasoned that heat could not be a material substance, and stated that he could scarcely conceive of anything but motion 'which could be excited and communicated in the manner observed. Almost at the same time (in 1799) Davy was experimenting in precisely the same direction. He showed that two pieces of ice THE NATURE OF HEAT. 299 might be melted simply by rubbing them together. Now heat is required to produce this change of state, and so, on the caloric hypo- ( thesis, the capacity of water for heat must be less than the capacity of ice f&r heat. But it is well known that the exact reverse is true. The necessary heat might have been furnished by surrounding bodies, and therefore Davy enclosed the two pieces of ice by other portions of ice and placed them in the exhausted receiver of an air- pump. Under these conditions heat could only reach them by first melting the surrounding ice. Davy was entitled to conclude, from the result of his experiments, that heat was not a form of matter, but, at the time, he merely said, 'Friction, consequently, does not diminish the capacities of bodies for heat.' In 1812, when again discussing the point, he spoke of heat as ' a peculiar motion, probably a vibration, of the corpuscles of bodies tending to separate them,' and said that ' The immediate cause of the phenomenon of heat, then, is motion, and the laws of its com- munication are precisely the same as the laws of communication of motion.' The second statement is rigidly correct ; the first that heat is motion is only true if properly interpreted. Heat, since it is not matter, must be energy ; and so the true meaning of Davy's state- ment is that heat consists in the energy of motion of the particles of a material body. But the word * energy ' was not introduced into science at that time. Davy illustrated this heat-motion, which he termed ' repulsive motion,' by the analogy of the orbital motion of the planets. If the speed of motion of any planet were increased, the orbit would become larger just as if a repulsive force had acted. In Davy's statement we have therefore the complete foundation of the whole kinetic theory of gases (Chap. XIII.), and, more generally, of the modern dynamical theory of heat. Since heat is a form of energy we might infer the possibility of its existence in a potential form, and the use of the common term latent heat bears out the inference. In working out a dynamical theory of heat Rankine advanced a hypo- thesis of ' molecular vortices.' He supposed that the motions which constitute heat in bodies are vortical motions in atmospheres around nuclei, and that radiation consists in the propagation of vibratory motions of the nuclei under their mutual forces. The energy of the vortices is the amount of heat which bodies possess ; and the absolute temperature of any body is the quotient of this energy by a definite 330 A MANUAL OF PHYSICS. constant for each substance. The elastic pressure, according to dynamical laws, must be directly proportional to the vortical energy, and must be inversely proportional to the volume which the vortices occupy, except in so far as the mutual nuclear forces which exist in all non-perfect gases modify this result. Latent heat is the equiva- lent of work done in varying the dimensions of the vortical orbits when the volumes and shapes of the spaces which they occupy are altered. Specific heat is the equivalent of work spent in varying the vortical energy. CHAPTEK XXI. RADIATION AND ABSORPTION OF HEAT. 255. Prevost's Theory of Exchanges. The fact that the laws of communication of heat are precisely those of the communication of motion leads to the conclusion that motion of the particles of a hot body is not confined to the surface alone ; and this conclusion is confirmed by the greater radiation which takes place from a thick plate, than from a thin plate, of a transparent substance, when both plates are at the same temperature. It indicates also that the radiation from a hot body is dependent upon the state of that body alone, and is not influenced by the presence of any other body, except in so far as it may cause an alteration in the thermal state of the former. This is the essence of the Theory of Exchanges which was advanced by Prevost of Geneva, under the title of a theory of ' movable equilibrium of temperature.' According to Prevost, two bodies, which are of different tempera- tures, and are placed in an enclosure which is impervious to heat, will both radiate heat. The hotter body will radiate at a greater rate than the other, until, by absorption, the temperatures of the two are equalised. After this, each will still radiate, but at precisely the same rate ; so that the heat which one loses by radiation is balanced by that which it gains by absorption. This is the condition which Prevost termed a condition of movable equilibrium (or, as we would now call it, of kinetic equilibrium) of temperature. 256. Stewart's and Kirchhoff" s Extension of Prevost' s Theory. As in Chap. XVII., we define the absorptive power of a body, undei given conditions, for any definite radiation, as the fraction of the whole incident radiation of that kind which it absorbs ; and we also define the emissivity of a body, at a given temperature, for any given radiation, as the ratio of the quantity of that radiation which 302 A MANUAL OF PHYSICS. it emits to the quantity of it which is emitted by a black body under the same conditions. And Stewart's proof, as given in Chap. XVII., leads to the result that the emissivity and the absorptive power of a body, at a given temperature, for any radiation, are equal. It is needless to enter into any discussion of special cases further than those which have already been given ( 203). Suffice it to say that, whatever be the quality and the quantity of the radiation emitted by any body under given conditions, absorption must exactly balance emission, in respect both of quality and quantity, if the given conditions are to be maintained. Previously to Stewart's investigations, it was known from the experiments of Leslie, De la Provostaye, and Desains, that the radiating and the absorbing powers of any one body were proportional to each other ; that is to say, it was known that a good radiator was a good absorber, and that a bad radiator was deficient in absorbing power. 257. Laws of Radiation of Heat at Constant Temperature. Early in the history of the subject, it was known that the radiation from a body at a given temperature depended upon the nature of the surface of that body. (This furnishes an additional analogy between light and radiant heat.) Leslie constructed a hollow metal cube, one side of which was polished, while another was rough. A third side was covered with lampblack, and the fourth was coated with white enamel. Although the surfaces of the two latter sides were so very different, Leslie found that both radiated about equally well when the cube was filled with hot water. The radiation from the bright metallic sur- face was much smaller than that from any of the others, and the radiation from the rough metallic surface was considerably less than that from the enamelled and the blackened surfaces. It has already been proved ( 205) that the amount of any given radiation, which is emitted from a sufficient thickness of a substance of given radiating power, is equal to that which is emitted from a black body at the same temperature. It was shown that the amount which is transmitted through a plate of the substance, n units in thickness, is K(l p)", where E is the total incident radia- tion and p is the absorption co-efficient. Hence the amount which is stopped by the plate is and therefore, by definition, the absorptive power and, conse- quently, the emissivity is l-(l-p)'. RADIATION AND ABSORPTION OF HEAT. 303 As the temperature of a body rises, radiations of shorter and shorter wave-length are emitted, and the energy in each pre- viously existing kind is increased. 258. Heat Spectra. If the radiation from a luminous body be passed through a slit and a prism in the usual manner, a spectrum is obtained which enables us at once to discover the nature of the radiation. And we may measure the amount of energy in any given portion of the spectrum by allowing that radiation to fall upon a medium which entirely absorbs it, and is consequently raised in temperature to a measurable extent. But, in any such experiment, it is necessary first to make certain that absorption does not take place, to any appreciable extent, in the substance of the prism. We might also indirectly analyse the radiation emitted by a given body by means of determinations of the absorption which the body exercises upon radiations of different wave-lengths; but such measurements are of little value unless the substance used is of definite chemical composition and physical structure. Some gases (e.g., olcfiant gas) exert powerful absorption on the heat-rays ; others -exhibit very little. The absorption produced by water vapour seems to be largely due to the dust nuclei ( 277). These remarks apply to the invisible portions of the spectrum also, whether these consist of rays of higher refrangibility, or of rays of lower refrangibility, than those which form the visible parts. It is found that the invisible portions of the spectrum possess characteristics which are precisely analogous to those which appear in the visible parts. We cannot, by means of the thermopile ( 328), or of the bolometer ( 343), which are the two most suitable instruments for the present purpose, determine the energy of the radiation of one definite wave-length. All that can be done is to measure the energy of the total radiation which is contained between rays of known wave-length, for the face of the thermopile and the metal strip of the bolometer necessarily possess considerable breadth. In the case of a refraction spectrum, besides the difficulty regard- ing absorption by the substance of the prism, there is the difficulty of the crowding together of the rays towards the less refrangible end of the spectrum according to an unknown law. If a diffraction spectrum be used, absorption may take place in the substance of the grating if it be made of glass ; and, if it be a metallic grating, great uncertainty exists with regard to the nature of its action upon the invisible rays. And, although the dispersion be ( 233) practically proportional to the wave-length, so that equal 304 A MANUAL OF PHYSICS. breadths of the spectrum correspond to equal differences of wave- length, we must remember that the measuring instruments ought to determine the energy contained in portions of the spectrum which are bounded by rays the wave-lengths of which are in a constant ratio. Further, since we are dealing with radiations of all wave- lengths, we see that, at any one part of a diffraction spectrum, an infinite number of different radiations are superposed, because of the existence of an infinite number of spectra of different orders. The difficulties of the problem have been largely overcome by Langley, to whom is chiefly due our recent great increase of know- ledge regarding radiations of large wave-length. He has detected, by means of the bolometer, traces of heat in portions of the solar spectrum corresponding to wave-lengths about twenty-four times greater than those of the least refrangible part of the visible spectrum. He has shown that the wave-length at which the maximum of energy in the spectrum exists diminishes as the temperature is raised. This result has been deduced from theoretical considera- tions by Michelson ; and the numerical deductions from the theory accord very well with Langley's observations. The energy at a given part of the spectrum dies away in amount very rapidly when we pass from the maximum in the direction of decreasing wave-length; it dies away much more slowly as we pass in the opposite direction along the spectrum. Cauchy's formula ( 209) connecting the refractive index of a sub- stance for a given radiation with the wave-length of that radiation does not agree with Langley's observations on the heat-rays. Briot's formula agrees better, but it ultimately differs from them in the opposite direction to that in which Cauchy's differs from them. 259. Radiation at Different Temperatures. Hitherto we have assumed that the radiating bodies with which we have dealt have been kept at constant temperature so that their radiating powers remained unaltered. We must now consider the relation between emissivity and temperature. Let us suppose that the hot body is placed inside an enclosure which is kept at a constant temperature t. Let t-\-Q be the temperature of the hot body, and let no heat pass from it except by radiation. If f(t) represent the rate of loss of heat from the hot body at tem- perature t, the rate of loss of heat under the assumed conditions will be /(* + 0)-/(0; and, since we have no adequate knowledge of the mechanism of RADIATION AND ABSORPTION OF HEAT. 305 emission, the form of this expression must be determined from experiment. According to Newton, the rate of loss of heat is proportional to the excess of the temperature of the hot body over that of its sur- roundings ; or, in symbols, But, since we must regard the rate of emission as independent of the surrounding bodies, this is equivalent to f(t) = at + b, where a and & are constants, and t is any temperature. The above law is not even roughly accurate unless the differences of temperature are small, and it becomes less and less applicable the greater the differences are. Dulong and Petit made an elaborate series of experiments with the view of discovering a more correct law. They found that, when the temperature difference was kept constant, the rate of loss in- creased in geometrical progression as the temperature of the sur- roundings of the body increased in arithmetical progression ; and the ratio of the geometrical series is independent of the tempera- ture excess so long, at least, as the excess is not greater than 200 C. When the temperature excess vanishes, the loss of heat is zero. Dulong and Petit therefore expressed the law in the form which agrees very closely with the result of their observations when t varies from up to 80 C., and 9 does not exceed 200 C. Since this formula may be written in the form /(* + e ) -/(*) = +*-<& < we see that the absolute rate of radiation* independently of the surroundings, is f(t) = aa'+ b. The constant a depends on the nature of the radiating surf ace t but the constant a is practically an absolute constant, and is equal to 1-0077. The law of Dulong and Petit, therefore, asserts that, when the temperature excess is constant, the rate of loss of heat is proportional to (1*0077)*, where t is the absolute temperature of the 20 306 A MANUAL OF PHYSICS. bodies to which the heat is radiated. It asserts also that the rate of loss is proportional to (1*0077) 1 when t is constant. The equation where r represents rate of loss, might not have been borne out by experiment. The fact that it is so borne out furnishes, as Balfour Stewart pointed out, an independent proof of the truth of Prevost's Theory of Exchanges, of which it is a necessary consequence. We may write the expression a 1 in the form (1 -{- _>)# 1, which, by expansion, becomes But, by Newton's law of cooling, the rate of loss should be propor- tional to 9 ; and hence, substituting the values 9 11, p = 0-0077, we see that Newton's value is fully 4 per cent, too small when the temperature excess is 11. More recent experiments by De la Provostaye and Desains verified the accuracy of the law of Dulong and Petit within the limits already assigned. According to Hopkins the radiation per square foot per minute, from glass at 100 C. to an enclosure at C., is 0-176 heat units, the unit being the amount of heat which is required to raise the tem- perature of one pound of water from C. to 1 C. Under the same conditions the radiation from unpolished limestone is 0'236 units ; and that from polished limestone cut from the same block is 0'168 units. Dulong and Petit's law seems to be applicable only to the total radiation from a body, and not to each definite radiation of which the whole is composed. The rate of emission of particular radia- tions from a black body seems to increase rapidly at first, and then more slowly, as the temperature of the black body is raised. 260. Solar Radiation. Pouillet was the first to make fairly accurate measurements of the amount of radiation which is received by the earth from the sun in a given time. For this purpose he invented the Pyrliclioinctcr. This instrument consists of a flat cylindrical metallic vessel, the surface of which) with the exception of one end which is covered with lamp-black, is highly polished. The bulb of a thermometer is RADIATION AND ABSORPTION OF HEAT. 307 inserted in the cylinder, and its stem lies along the axis. The cylinder is filled with water or mercury, and its blackened face is directed towards the sun. In order that this may be done with accuracy, a metal disc, the diameter of which is exactly equal to that of the cylinder, is fixed on the end of the axis of the instru- ment remote from the cylinder. The shadows of the disc and the cylinder will not coincide unless the face of the latter be accurately turned towards the sun. The area of the blackened face, and the amount of heat necessary to raise the temperature of the cylinder and its contents to a given extent, are exactly determined. If the temperature of the instrument is the same as that of the air, and if its blackened face, carefully shaded from the sun, be turned towards the sky, radiation will take place, and the temperature will fall 9 (say) in t units of time. Now let the apparatus be turned towards the sun for t units of time ; after which let it be turned towards the sky as before, during an equal time, and let 9 f be the fall of temperature. It is concluded that the deficiency introduced into the total rise of temperature which took place when the instrument was exposed to the sun, is for, during this exposure, the cylinder was steadily rising in tem- perature because of the heat which it absorbed from the sun, and was at the same time steadily falling in temperature because of its own radiation. Hence the full rise of temperature, had no radiation taken place from the cylinder, would have been -. where 6 is the rise of temperature which was actually observed. And so, from the known constants of the instrument, the quantity of heat which was received, in a given time, by unit area at the earth's surface could be calculated. Pouillet gave the expression for the rate of rise of temperature under the above, conditions at different times of the day. The quantity I is the thickness of the earth's atmosphere which was traversed by the sun's rays. The constant 6 varies from day to day, but the constant a does not so vary. Pouillet concluded that the expression would be applicable 202 dOH A MANUAL OF PHYSICS. even if there were no atmosphere, in which case the constant a would represent the rate of rise of temperature ; and he calculated that, in the event of no atmospheric absorption, the quantity of heat falling on a square centimetre of the earth's surface, in one minute, would raise the temperature of 1*76 grammes of water by 1 C. From the average values of the constant e, Pouillet inferred that about one -half of the total incident solar radiation is absorbed in the earth's atmosphere. Sir W. Thomson concludes, from the above data, and those independently given by Herschel, that the rate of radiation from the sun's surface is about 7,000 horse-power per square foot, or thirty times the amount which is radiated, per square foot, from the furnace of a locomotive. The Actinometer is another instrument used to determine the magnitude of solar radiation. It consists essentially of a metal enclosure (blackened internally and kept at constant temperature) at the centre of which the bulb of a thermometer is placed. By means of an opening in the enclosure, the sun's rays are allowed to fall upon the bulb for a given time, after which the bulb radiates to the enclosure. Various experimenters have used this apparatus in one or other of its forms. By its means Violle has found that the radiation from the sun, which would fall per minute on a square centimetre of the earth's surface did no absorption occur, would raise the temperature of 2-54 grammes of water by 1 C. Langley's more recent measurements also indicate that Pouillet's estimate is too low. He finds that the quantity of unabsorbed heat falling on a square centimetre would raise the temperature of 1-81 grammes of water by 1 C. Did no absorption take pl&ce, this would become 2'8 grammes, and the amount of heat radiated from the sun per square foot of its surface would be fifty times greater than that radiated from a square foot of the surface of a loco- motive. From Langley's data we can calculate that the yearly radiation from the sun to the earth would, if spread uniformly over the surface, melt a uniform crust of ice fully 150 feet in thickness. 261. Radiation from Moving Bodies. The following considera- tions regarding the motion of radiating bodies, first advanced by Balfour Stewart, are specially worthy of notice apart from their intrinsic value as an example of the useful employment of the principle of Conservation of Energy. Let us suppose that ultimate equality of temperature has been arrived at inside the enclosure, impervious to heat, which was pos- tulated in 203 ; and let us further assume that one of the bodies in that enclosure suddenly commences to move about with a speed RADIATION AND ABSORPTION OF HEAT. 309 which is comparable with that of light. A direct application of Doppler's principle shows us that any other body in the enclosure, towards which the motion may be directed, will receive energy from the moving body at a greater rate than will one which is so situated that the motion is directed from it. It follows that relative motion of radiating bodies is inconsistent with ultimate equality of temperature. But persistent inequality of temperature would imply a perpetual source of energy; and we therefore conclude that the relative motion of radiating bodies must gradually cease. CHAPTEK XXII. EFFECTS OF THE ABSORPTION OF HEAT: DILATATION AND ITS PRACTICAL APPLICATIONS. 262. Temperature. Increase of temperature usually accompanies the application of heat to any substance. The fundamental distinc- tion between heat and temperature is very clearly brought out by a consideration of the meaning of the words hot and cold as applied to different material substances. The bodies which are said to be hot or cold may really be at the same temperature. Thus a mass of iron and a mass of wood, though their temperatures be equal, feel very differently to the touch. If the temperature of the hand be higher than that of the two masses, the former feels cold, and the latter feels warm ; while, if the temperature of the hand be lower than that of the bodies, these conditions are reversed. The reason s that the physical properties of the substances are such that, of the two, iron is the one which most rapidly abstracts heat from, or supplies heat to, the hand. It is sufficient for our present purpose that we regard temperature as that condition which determines the flow of heat from one body to another. (See 254, 150.) If two bodies, which differ in temperature, be placed in contact with each other (or in thermal communication of any sort), heat passes from the body which is at the higher temperature to the body at the lower temperature. And two bodies, between which, on the whole, there is no transference of heat when thermal communica- tion is established, are said to have equal temperatures. It is an experimental fact that any two bodies, which have each the same temperature as a third body, are themselves at equal temperatures. Various methods, which will be described later, are used for the determination of the difference of temperature which subsists between two bodies, or between a body in one given physical state and the same body when in another state. DILATATION AND ITS PRACTICAL APPLICATIONS. 311 We may premise, for present purposes, that the temperatures of ice-cold water, and of boiling water, respectively, are called degrees and 100 degrees ; that a given number of degrees, at any part of our scale, corresponds to a constant interval of temperature ; and that we call the degrees Centigrade degrees the letter C being used to discriminate them from the degrees of other scales. 263. Dilatation of Solids. One of the most obvious effects of the application of heat to a substance is expansion. In a homogeneous isotropic solid, the expansion is equal in all directions. On the other hand, in a non-isotropic solid, the expansion is different in different directions ; but, in this case, three rectangular directions (called the principal axes, cf. 245), can always be found such that in one the expansion is a maximum ; in another, it is a minimum ; and, in the third, its value is intermediate between those along the other two being, in fact, a maximum-minimum. When the expansions, along these directions, which accompany a given rise of temperature, are known, the expansion in any other direction can be found. We have, first of all then, to consider the laws of linear dilatation of a solid. [The requisite measurements are readily made by direct micrometric methods, which give accurate determinations of the lengths of a bar at different known temperatures.] 1. It is found that the increase of length of a given bar is pro- portional to the length of the lor at the initial temperature. 2. The alteration of length is proportional to the increase of temperature. These laws are symbolically represented by the equation where l t and l u are respectively the lengths of the bar at the higher and lower of the two temperatures of which t is the difference, and k is a constant called the co-efficient of linear dilatation. This constant is obviously the increase of length, per unit rise of tempera- ture, of a bar the original length of which is unity. From the three equations where the letters Z, 6, and d denote the length, breadth, and thick- ness of a rectangular parallelepiped of the substance, and & & 2 , Tc are respectively the co-efficients of dilatation measured in the direc- tion of the length, breadth, and thickness, we can obtain an expression 312 A MANUAL OF PHYSICS. for the cubical dilatation of the solid. Let us suppose that these equations refer to the principal axes ; then or where the Vs represent volumes. In all cases in nature, the quantities ls v & 2 , & 3 are so small that their squares and products may be neglected ; hence, to a sufficient degree of approximation, if K represents the co-efficient of cubical dilatation. This gives K = M-*a+fcs; and so the co-efficient of cubical dilatation (which is the fractional increase in bulk of unit volume per unit rise of temperature) is equal to the sum of the three principal co- efficients of dilatation, When the substance is isotropic, this latter equation becomes K = 3&; which asserts that the co-efficient of cubical dilatation of a homo- geneous isotropic solid is three times the co-efficient of linear dilata- tion. If the edges of the rectangular parallelepiped be not in the direc- tions of the three principal axes, the effect of the application of heat will be to change the inclination of the faces, so that the parallelepiped will cease to be rectangular. When the three edges are all originally equal, and the diagonals of one of the square faces of the cube are parallel to two of the principal axes, the application of heat changes the square face into a rhombus such that the tangent of half its obtuse angle is equal to 1 + (7ci Jc 2 )t, where t is the increase of temperature and the original length of the diagonals is assumed to be unity. This affords a ready means of determining the co-efficient of expansion along one of the two principal axes, provided that we know its value along the other ; for the change of angle can be measured with extreme accuracy by optical methods. Fizeau introduced a specially delicate method of measuring the change of length of a rod. Newton's rings ( 221) are produced by the interference of light reflected at the surface of a film of air contained between two plates of glass, of which one at least is slightly curved ; and the colour of any particular ring depends on the thickness of the film, which can be calculated when the radius DILATATION AND ITS PRACTICAL APPLICATIONS, 313 of the ring is known. If one of the two plates be fixed, while the other is attached to the expanding rod, the slightest change of length of the rod causes a diminution of the thickness of the film of air which can be easily calculated by means of the optical changes which are simultaneously produced. He gives the following empirical formula connecting the co- efficient k with the temperature t expressed in Centigrade degrees : fc = a +a'(t- 40). The constants were determined by observations at three tempera- tures, viz., 10, 45, and 70. The cubical dilatation of a solid is usually determined directly by heating it in a liquid, of known expansibility, which is contained in a vessel made of a substance whose co-efficient of linear (and, there- fore, also of cubical) dilatation has been found. Let V be the volume of the vessel at C., and let KX be its co-efficient of dilata- tion. The volume at t C. is Similarly, the volume at t C. of the liquid which filled the vessel at C. is If we now place a solid, the volume of which at C. is v oj and at t C. is the volume of liquid which overflows from the vessel at tempera- ture t is V,(1+IM) - or VXKj from which expression the value of K 3 may be found. It is needless to enumerate the various practical applications of the dilatation of solid bodies when their temperature is raised, or of their contraction as the temperature falls. The well-known processes of shrinking the tires on wheels, and of drawing together the walls of a building when these have bulged outwards, will sufficiently serve as an indication of their nature. One particular application to the construction of a compensation- pendulum or balance-wheel of a watch merits special notice. The ordinary compensation-pendulum is constructed upon the principle that the difference between the lengths of two rods, of different expansibilities, will remain constant, however the tempera- ture may be altered within allowable limits, provided that the lengths 314 A MANUAL OF PHYSICS. of the rods be made in inverse proportion to the co-efficients of dilatation of the substances of which they are composed. This obviously affords a means of keeping constant the distance between the bob of a pendulum and its point of support. If two equal bars, of different expansibilities, be soldered together throughout their length, a rise of temperature will cause the com- pound bar to bend in such a way that the less expansible bar is on the concave side ; for this is the only way in which the tendency to- wards unequal expansions can be satisfied. This fact is made use of in the construction of compensated balance wheels. When the temperature of an uncompensated wheel increases, the expansion of spokes carries the rim of the wheel further out from the centre, and the consequent increase of moment of inertia produces an increase in the period of oscillation. Hence a watch, or chronometer, fitted with such a wheel will go too slow in warm weather and too fast in cold weather. But if the rim of the wheel be divided into a number of independent parts, each part being carried by a separate spoke, and if the rim be compound as above described, matters may be so arranged that the throwing-out of the weight from the centre because of the expansion of the spokes is counterbalanced by means of the inward bending of the segments of the rim. It is worthy of note that one or two of the principal expansibilities of a solid may be negative, i.e., the substance may contract in at least one direction when its temperature is raised. In such a case a series of directions may be found in the substance such that a change of temperature does not give rise to any alteration of length of the substance in any of these directions. [These directions may be found by imagining a sphere to be drawn in the unheated body and finding its intersection with the ellipsoidal surface into which it becomes distorted by the application of heat. All lines drawn in the body parallel to the lines joining the points of the curves of inter- section to the centre of the sphere are unchanged in length.] Hence a rod cut from the substance in any such direction may be used as the rod of a compensated pendulum. Brewster pointed out that a rod of marble might be so used. An interesting example of contraction of a solid when its tempera- ture is raised is seen in the case of india-rubber under tension. The experiment may readily be performed by blowing steam through a hollow tube of the substance, which is fixed at its upper end, and is extended by means of a weight attached to the lower end. The subjoined table contains Fizeau's determinations (see above) of the values of the constants in his formula for the co-efficients of linear dilatation of a few well-known substances. The constant a is, DILATATION AND ITS PRACTICAL APPLICATIONS. 315 of course, the value of the co-efficient at 40. When more than one value is given these refer to the various principal dilatations. a. a'. Carbon (retort) ... 0-00000540 ... 0*0000000144 Platinum ...... 0*00000905 ... 0*0000000106 Steel ...... 0-00001095 ... 0-0000000124 Iron (compressed) ... 0-00001188 ... 0'0000000205 Copper (native) ... 0-00001678 ... 0-0000000205 Silver ...... 0*00001921 ... 0-0000000147 Lead ...... 0-00002924 ... 0'0000000239 Ouartz f 0-00000781 ... 0-0000000205 10-00001419 ... 0-0000000348 T i -, (0-00002621 ... 0-0000000160 (0-00000540 ... 0-0000000087 (0*00003460 ... 0-0000000337 Arragonite ...... j 0*00001719 ... 0-0000000368 1 0-00001016 ... 0-0000000064 An average value of the co-efficient of expansion of glass is 0-0000085, 264. Dilatation of Liquids. In liquids it is merely the cubical dilatation with which we have to deal. This may readily be deter- mined if we know the cubical dilatation, K', of the substance of which the containing vessel is composed. Let K be the unknown co-efficient, and let an amount of the liquid, of weight W, fill the vessel (which must have a narrow neck) at a temperature, , while the weight of the quantity which fills the vessel at is W . Now the weight of the quantity which would have filled it, had the liquid been inexpansible, is W (1 + K') ; but, since the liquid is ex- pansible, this amount is diminished in the ratio of unity to Hence we get which determines K. A very simple method, by means of which the co-efficient of dilata- tion of a liquid may be found without any knowledge of that of the substance of which the containing vessel is composed, was devised by Dulong and Petit. In all essential particulars the apparatus con- sists of a double U-tube, abed (Fig. 154). The portion be is occupied by air, which separates the portions of the liquid contained in ab and cd while preserving continuity of pressure between them. Equal atmospheric pressure acts at the points a and d, and the air in be is necessarily at uniform pressure throughout. Hence, when 316 A MANUAL OF PHYSICS. equilibrium is maintained, the pressure per square inch due to the difference of level ab must be equal to the pressure per square inch due to the difference of level dc. Let us suppose that the limb cd d* FIG. 154. is raised to temperature t, while the limb ab is kept at 0. average expansibility throughout the range of temperature is dc- ab The If dv be the increase of volume produced by a small rise of tern perature dt, while v, is the total volume of the liquid at 0, the quantity dv v dt is usually taken as the co-efficient of dilatation at the temperature t. The true co-efficient at temperature t is obviously given by the ratio dv vdf where v is the volume of the liquid at that temperature. The values, given by Regnault, of the ordinary co-efficient of dila- tation of mercury at temperatures varying from C. to 850 C. are well represented by the formula K - 0-0001791 +0-0000000504*. Water presents a marked peculiarity as regards its change of volume with rise of temperature. Between its freezing-point, 0C., and a temperature of almost exactly 4 C., its volume diminishes as the temperature increases. At all higher temperatures the volume DILATATION AND ITS PRACTICAL APPLICATIONS. 317 increases as the temperature is farther raised. Thus water is in a condition of maximum density at a temperature of about 4 C. The existence of the temperature of maximum density is readily shown by means of Hope's experiment. The necessary apparatus consists of a cylindrical glass vessel the central portion of which is surrounded by a metal casing in which a freezing mixture may be placed. Two thermometers are inserted horizontally through the glass vessel, one near the top and the other near the bottom, so that their bulbs are at the axis of the cylinder. The vessel being filled with water, a freezing mixture is placed in the casing. Very soon the temperature marked by the lower thermometer begins to diminish, which shows that the cold water is of greater density than the warm water near the top of the vessel. This goes on until the lower thermometer registers a temperature of 4 C., at which it remains. Soon afterwards the temperature of the water at the top of the vessel begins to fall, which shows that the colder water is now ascending and must therefore be expanding ; and this process goes on until the water at the top freezes at C. The maximum-density point is lowered by pressure to the extent of about 3 C. by a pressure of one ton's weight per square inch. Kopp's determinations of the co-efficient of dilatation of water are fairly well represented between C. and 20 C. by the formula 72,000 The more recent experiments of Pierre, Hagen, and Mathiessen indicate that the denominator of this fraction is about 5 per cent, teo large ; but Eossetti's results agree better with those of Kopp. Different experimenters have determined the co-efficients of expansion of various liquids when under pressure sufficient to keep the liquid in equilibrium with its vapour at temperatures above the ordinary boiling-point. Drion gives the subjoined values of the co- efficient of dilatation of sulphurous acid : Temp. C. Co-efficient. Temp. C. Co-efficient. 0-00173 70 0-00318 10 0-00188 90 0-00415 30 0-00219 110 0-00592 50 0-00259 130 0-00957 From these results it appears that the co-efficient of dilatation of this liquid, at about 120 C., is double of that of air. 318 A MANUAL OF PHYSICS. Him gives the volume of water at different temperatures as follows : Temp.'C. Volume. Temp. C. Volume. 4 ... 1-00000 140 ... 1-07949 100 ... 1-04315 160 ... 1-10149 120 ... 1-05992 180 ... 1-12678 Consequently, at 180 C., the expansibility of water is nearly one half of that of air. 265. Dilatation of Gases. A gas must be kept in an enclosed space when experiments are to be made upon it with regard to its alterations of, volume under varying conditions of temperature. But we know that, so long as the temperature is" maintained at a constant value the volume and pressure of the gas are in inverse proportion to each other. Hence we may investigate the effect of variation of temperature in two ways : we may measure the expan- sion under constant pressure ; or, we may measure the change of pressure at constant volume. By such measurements Charles (and, subsequently, Gay Lussac) was led to the conclusion that the volume of a given quantity of any gas, under constant pressure, increases by a constant fraction of its amount for a given increment of temperature. This state- ment is known as Charles' Law, and is represented symbolically in conjunction with Boyle's Law by the equation pv=C (l + oO, where c and a are constants, and the meaning of the other quanti- ties is obvious. The pressure being kept constant, the volume increases by the fraction a of its amount at per unit rise of temperature ; a is therefore the co-efficient of dilatation under constant pressure. Again, if the volume be kept constant, the pressure increases by the fraction a of its amount at per unit rise of temperature. Thus the fractional increase of volume 'under constant pressure, and the fractional increase of pressure at constant volume, have the same numerical value if the above equation be rigidly true. The magnificent series of experiments carried out by Eegnault have shown that, while the above law is very nearly true in the cases of the most permanent gases, marked discrepancies are ex- hibited when the more readily liquefiable gases are employed. His experiments on the undernoted gases gave as the value of the dilata- tion between C. and 100 C. : DILATATION AND ITS PRACTICAL APPLICATIONS. 319 Hydrogen 0-3667 0-3661 Air 0-3665 ... ... 0'3670 Nitrogen 0'3668 0'3670 Carbonic oxide ... 0'3667 ' 0*3669 Carbonic acid 0'3688 0'3710 Cyanogen 0-3829 0-3877 Sulphurous acid ... 0'3845 0-3903 The figures in the second column represent the value of the co- efficient as determined at constant volume; those in the third column are the values determined under constant (atmospheric) pressure. With a single exception (in the case of hydrogen) the latter number exceeds the former. When the initial pressure of gases at is increased, the dilata- tion between and 100 increases. Some of Regnault's results for air are : 109-72 ... 149-31 ... 0-3648 374-67 ... 510-35 ... 0-3659 760-00 ... 1038-54 ... 0'3665 1692-53 ... 2306-23 ... 0-3680 3655-66 ... 4992-09 ... 0'3709 The numbers in the first two columns represent respectively the pressure per unit area at and at 100 expressed in terms of the weight at of a column of mercury of unit section and one milli- metre in height. For carbonic acid Begnault gives the similar results : 758-5 ... 1034-5 ... 0-36856 901-1 ... 1230-4 ... 0-36943 1742-9 ... 2387-7 ... 0-37523 3589-1 ... 4759-0 ... 0-38598 The variation in the case of this gas is therefore greater than in the case of air. The results, for the same two gases, under constant pressure, are : Air 760 ... 0-36706 2525 ... 0-36944 Carbonic acid ... 760 ... 0-37099 2520 ... 0-38455 Regnault's apparatus consisted essentially of a glass bulb D (Fig. 155), which contained the gas and communicated through the tube E with a reservoir AB which was filled with mercury. 320 A MANUAL OF PHYSICS. A tube F, open to ths atmosphere, also communicated with the reservoir. When the co-efficient was to be determined at constant volume, a plug C was screwed in until the mercury stood at the points E and F in the tubes E being a fixed point. D was, under these conditions, surrounded successively by melting ice, and by the steam from boiling water, and the pressure in each case was found A B FIG. 155. from the difference of level of the mercury in the two tubes and the known barometric pressure. Suitable corrections had, of course, to be applied for the expansion of the bulb by heating or by pressure. When the dilatation was observed under constant pressure, the plug C was, at each temperature, screwed out until the mercury stood at the same level in the two tubes. The pressure was then equal to that of the atmosphere. Other volumetric, or gravimetric, methods have been employed by Regnault and various experimenters. 266. Absolute Zero of Temperature. We may write the equa- tion which expresses Boyle's and Charles' Laws in the form pv- Now as the second term within the brackets represents temperature, the first term must also represent a temperature, for the dimen- sions ( 27) of all the terms of a physical equation must be iden- tical. And the numerical value of that temperature is about 273, since a = 0*003665. Hence we must suppose that the zero of the Centigrade scale corresponds to a temperature of 273 degrees on this DILATATION AND ITS PRACTICAL APPLICATIONS. 321 new scale, which we may call a scale of absolute temperature, since its magnitude is independent of the particular gas employed. To look at the matter from another point of view, we observe that, as t diminishes to zero and then becomes an increasing negative quantity, the product pv constantly diminishes, and finally becomes zero when t = - 273. If the volume is constant, this means that the pressure vanishes when t has this value. But the pressure of a gas is due to the motion of its particles, and hence, when t= - 273, the particles cease to move, and the gas is therefore totally deprived of heat. The above equation may therefore be written in the form pv = IU, where R = Ca and t represents absolute temperature. We shall see later ( 303) that this estimate of the position of the absolute zero on the Centigrade scale is confirmed by thermo- dynamical considerations. 267. Measurement of Temperature. The most usual method of measuring temperature is by means of the expansion of a liquid or a gas. Mercury is generally used in the former case, air in the latter. A glass tube of narrow, and as nearly as possible uniform, bore is first chosen. If the bore be not quite uniform, its variations are determined by the process of calibration. This consists in running a small quantity of mercury along the tube from one e*nd to the other, and measuring its length at the various parts. The quotient of the weight of the mercury by its specific gravity and th length of the column at any part gives the mean section of that portion of the tube. Next, a bulb is blown, on one end of the tube, of a size which is determined by the bore of the stem, the expansibility of the liquid to be used, and the required length of a scale division. The bulb is now slightly heated to expel some air, and the instru- ment is then inverted in a vessel of the liquid with which it is to be filled. As the bulb cools, some of the liquid enters it. This liquid is then boiled in the bulb, and its vapour expels the remaining air. A repetition of the process of inversion of the bulb and stem in the liquid will result in both being entirely filled. So much of the liquid is then run out that the remainder scarcely fills the stem when it is boiled once more. The vapour drives out the air which entered, and the tube is then hermetically sealed. Two fixed points must now be determined on the stem. As we shall see afterwards ( 273, 275), this may be done by means of melting ice, and of the steam coming from boiling water in a nearly" closed vessel. 21 322 A MANUAL OF PHYSICS, But, before these points are determined, a considerable time should be allowed to elapse, for the bulb will not shrink quickly to its final volume. The process of shrinkage usually goes on for years, though, by careful annealing, the effect may be considerably lessened. To determine the lower fixed point of the scale, the bulb and part of the stem are surrounded by melting ice. The final position of the extremity of the liquid column is marked on the stem. The upper fixed point is determined by surrounding the bulb, and as much of the stem as possible, by the steam which is escaping from water boiling under a pressure of one atmosphere. The final position of the liquid is then marked on the stem. The distance between the two marked points is then divided into a number of equal parts. On the Centigrade seals, the lower fixed point is marked 0, and the higher is marked 100 ; on the Fahrenheit scale the lower is marked 32, and the higher 212. Thus, on the Centigrade scale, there are'100 divisions between the boiling-point and the freezing- point of water hence its name ; on the Fahrenheit scale there are 180 divisions between these points. The Fahrenheit zero was determined by a freezing-mixture of snow and salt, which gave the lowest temperature known at the time when Fahrenheit first con- structed his thermometers. The relation between the two scales is obviously given by the equation F-32^ C 180 "100* where F and C respectively represent the Fahrenheit and Centigrade scale readings. The same interval on Reaumur's scale is divided into 80 equal parts, the zero being the same as that of the Centigrade scale. As already remarked, the liquid generally used is mercury. For the measurement of temperatures below the freezing - point of mercury, alcohol is employed. The air thermometer is of great use in the determination of temperatures above those at which mercury can be employed ; and its readings agree pretty closely with those of the true absolute scale of temperature as determined by thermodynamical considera- tions. The following numbers, taken from -Regnault's results, show the difference between the Centigrade scale of the air thermometer and that of the mercury thermometer : Air ... ... 20 40 6 60 80 100 200 300 Mercury ... 19 o; 98 39 6< 67 59-62 79'7S 100 202-78 308'34. DILATATION AND ITS PRACTICAL APPLICATIONS. 323 Self-registering thermometers are frequently employed for the purpose of indicating the maximum, or the minimum, temperature attained between two given periods. In the usual form of the maximum thermometer, the expanding column of mercury pushes an iron index along the tube. This index is left behind when the column contracts, for mercury does not wet iron. In the usual minimum thermometer, alcohol is used along with a glass index. The liquid, when expanding, flows past the index ; when it contracts, it pulls the index with it, for its surface tends to take the smallest possible area ( 120), and this is attained when it occupies the space between the index and the walls of the tube. Continuously - registering thermometers are also used. The principle of one of the best forms of these instruments is identical with that of the Bourdon gauge. Let abed (Fig. 156) be a longitudinal section of a hollow metal receiver in its unstrained state, and let us suppose that the receiver is filled with a liquid. As the temperature FIG. 156. rises, the pressure increases ; and, since cd is greater than ab, the sum of the moments of the forces tending to straighten the receiver exceeds the sum of those which act contrariwise. A system of levers attached to such a receiver (fixed at one end) traces a con- tinuous record on a properly-graduated paper placed on a drum which revolves slowly at a uniform rate. Pyrometers are used for the rough determination of very high temperatures. In Daniell's pyrometer, a bar of platinum is slipped into a hole bored in a rod of graphite. A plug of graphite, or baked clay, rests on the top of the bar and fits tightly into the hole, or is otherwise kept tightly in position. When the platinum (which rests on the bottom of the hole) expands, the plug is pushed out, and remains in its position of maximum displacement when the temperature falls. From the increase of length of the platinum thus registered, we can calculate the temperature of any furnace in which it may be placed, on the assumption that the law of dilatation, determined throughout moderate ranges of temperature, holds up to the high temperatures of the furnace. Other methods are also employed for the determination of high temperatures. See 269, 343. 212 CHAPTER XXIII. EFFECTS OF THE ABSORPTION OF HEAT : CHANGE OF TEMPERATURE AND CHANGE OF STATE. 268. Unit of Heat. Specific Heat. Thermal Capacity. One of the most marked effects of absorption of heat is a rise of the tem- perature of the heated body. In some cases, no change of temperature takes place, and the effect which appears instead is a change of the physical state of the substance. But, before discussing these effects, we must consider the methods of measuring the amount of heat required to produce a given change ; and this, in turn, necessitates the adoption of a definite unit of heat. We may conveniently adopt as our unit the quantity of heat which is required to raise the temperature of one pound of ice- cold water to 1 C. A given quantity of heat may therefore be measured by the number of pounds of ice-cold water which it can raise in tempera- ture to 1 C. We might measure it also by the number of pounds of ice at C. which it is just able to melt ; for, as we shall see shortly, the number of units of heat which are required to just melt one pound of ice at C. is quite definite and measurable. More generally, we might adopt as our unit the amount of heat necessary to produce any definite physical change, and then we might deter- mine what fraction of the unknown quantity of heat could produce this change. The number of units in the unknown quantity would be the reciprocal of this fraction. We define the specific heat ; of a substance, under given condi- tions, as the quantity of heat which is required to raise the tem- perature of one pound of the substance by 1 C. From this definition, and from our previous definition of the unit of heat, it follows that we must consider the specific heat of one pound of water 'at C. to be unity. More strictly, we should define it as the rate at which the quantity CHANGE OF TEMPERATURE AND CHANGE OF STATE. 325 of heat supplied to the substance, per pound of its mass, varies with the temperature. But, in all actual cases, there is no practical difference between the two definitions. The mean, or average, specific heat of a substance, throughout a given range of temperature, is obtained by dividing the amount of heat which is required to raise one pound of the substance through the given range by the difference between the two extreme temperatures. The Thermal Capacity of a substance is the quantity of heat which is required to raise the temperature of unit volume of the substance by one degree. It is therefore equal to the product of the specific heat and the density of the substance. 269. Specific Heat of Solids and Liquids. Various methods are used for the determination of specific heat. In one method, use is made of the fact that the rate of emission of heat from a body at a given temperature depends only upon the nature of its surface. Hence, if we fill a thin metal globe successively with two different liquids, and observe the rate of cooling of each liquid at the same temperature, we can compare the specific heats of the two liquids. For, if m, s, r, and m', s', r', represent respectively the mass, the specific, and the rate of cooling of the liquids, we have msr=m r s'r f . If one of the liquids be water, so that s = l, we get In actual experiment, the liquids would be raised to a common high temperature and readings of their temperatures would be taken at equal intervals of time as they cooled. If a curve were then drawn the ordinates of which represented temperature, while the abscissas represented time, the rate of cooling would be given by means of the tangent to the curve. Thus (Fig. 157), to find the rate of cooling at a temperature 9, draw the tangent ab to the curve at the point P, corresponding to 9, and let it intersect the time-axis in the point a and the temperature-axis at the point b ; the rate of cooling is ob/oa. Another method of determining specific heat is known as the Method of Mixture. Let m pounds of one substance at tempera- ture t be mixed with m' pounds of another at temperature t', and let the specific heats of the two substances be respectively s and s r . If the temperature of the mixture be 9, the heat lost by the hotter body (say that of mass m') is m's'(t' - 9). Similarly, the heat gained 326 A MANUAL OF PHYSICS. by the colder substance is ms(9 - t). A]so, if ^ be the mass of the vessel which contains the substances (and we may regard the stirring-rod used to mix the substances as forming part of it) while 9 separates the region in which liquefaction can occur from that in which it is impossible. [Compare, with this diagram, Fig. 11, which is drawn, so as approximately to suit the case of carbonic acid, from theoretical considerations based, by Professor Tait, upon the kinetic theory of gases.] We may with great advantage, as Tait suggests, describe the substance as a true gas, or a true vapour, according as the tempera- ture is higher, or lower, than the critical temperature. The compressibility of the substance is dv/vdp, where v is the volume and dv, dp, represent respectively simultaneous small increments of the volume and the pressure. Now, the diagram shows that, at the commencement of liquefaction, the inclination of the isothermal to the axis of volume becomes greater and greater as the temperature rises, i.e., the ratio dv/dp decreases as the temperature rises. Hence, since we suppose unit volume to be taken in all cases, the com- pressibility of the vapour when it is upon the point of condensing decreases as the critical temperature is approached. Similarly, the value of dvjdp in the liquid state, when the substance has just been entirely liquefied, increases as the temperature rises ; and thus we see that the compressibility in the two states tends towards equality, simultaneously with the volumes, as the temperature rises to its critical value. For a considerable distance above the critical point the isother- mals exhibit two points of inflexion ; but these finally cease to be visible, and the isothermals closely resemble those of a perfect gas. [The illustration affords a good example of the use of contours. The isothermals may, as was stated in Chap. III., be regarded as the projections of the plane sections of a surface which represents the various simultaneous values of the pressure, volume, and tem- perature of the gas.] 279. Solution. Freezing Mixtures. The process of solution is extremely analogous to the processes of liquefaction. A gas which is dissolved in a liquid may be regarded to a certain extent as if it were liquefied, and latent heat is given out in the process of solution. Similarly, when a solid is dissolved in a liquid, latent heat is required, just as if the solid were directly liquefied ; but in some cases this absorption of latent heat is masked by the heat which is developed because of molecular action between the liquid and the solid. The amount of heat which is disengaged in the solution of a gas CHANGE OF TEMPERATURE AND CHANGE OF STATE. 841 342 A MANUAL OF PHYSICS. is frequently very marked. This is so specially in the cases of the more soluble gases, such as ammonia when water is the solvent. The amount of gas, under definite pressure, which a given liquid will dissolve, becomes less as the temperature is raised ; though, by careful treatment, a state of supersaturation may be induced which is analogous to the prevention of boiling at temperatures consider- ably over the ordinary boiling-point of a liquid. Supersaturation of a liquid solution of a solid may also take place notably in the case of a substance, such as acetate of soda, which dissolves in little more than its own water of crystallisation. If a crystal of the acetate be dropped into the supersaturated solution to act as a nucleus, crystallisation will take place rapidly with the development of latent heat. A crystal of any other substance of the same crystalline form will produce the same effect. Heat is frequently developed or absorbed when two liquids are mixed (mutually dissolved). If chemical action takes place to any extent between the two, heat will be developed unless other causes prevent. If the total bulk of the two liquids increases on mixture as in the case of bisulphide of carbon and alcohol heat tends to be absorbed ; and again, the water-equivalent of the mixture may be greater than the sum of the water-equivalents of its constituents which also necessitates absorption of heat. If the opposite effects to these take place, heat will be evolved. Energy may also be changed into heat in the process of inter-diffusion of the liquids. Disengagement or absorption of heat take place respectively accord- ing as the effects of the one or the other sets of actions preponderate. And, as the various actions depend upon the temperature, we find that the total effect is sometimes reversed when the original tem- perature of the two liquids is sufficiently varied. Two solids even may dissolve in each other, so to speak, with the absorption of latent heat. (Salt and snow furnish a well-known example.) This can (Jbviously only occur when the freezing-point of the resultant liquid fe lower than the original (common) tempera- ture of the solids. Part of the latent heat is obtained by cooling the solids, part by cooling the liquid, and part, it may be, by cooling surrounding bodies. If the whole be intimately mixed the action necessarily ceases when the freezing-point of the resultant liquid is reached. These remarks contain the explanation of the action of solid freezing-mixtures, which has been elaborately investigated by Professor Frederick Guthrie. 280. Dissociation and Chemical Combination. When the tem- perature is raised sufficiently high a compound dissociates, or breaks up, into its constituents. The change is not sudden but gradual. CHANGE OF TEMPERATURE AND CHANGE OF STATE. 843 It commences at a certain lower limit of temperature, and ends completely at a certain higher limit ; and at all intermediate tem- peratures a state of kinetic equilibrium is arrived at in which recom- bination precisely balances dissociation. The magnitude of the limits will, in general, depend upon the pressure. It is usual to speak of the temperature at which one-half of the substance is dis- sociated as the temperature of dissociation. Conversely, when the two (or more) constituents are mixed, com- bination does not occur until a certain temperature is attained ; but, if the combination results in the development of heat, the process, once started, will continue until the percentage of the mixture which remains uncombined corresponds to the temperature which the whole mass attains because of the heat which is set free. On the other hand, if work is done during the process, or if heat is lost by conduction or otherwise, the process will continue until combination is complete when the temperature falls to the lower limit. All chemical combination takes place in accordance with the two laws of thermodynamics, and therefore further treatment of this subject is deferred until we have considered these laws. (See 298.) 281. Many other effects of heat might be noted here, but it is preferable to leave their discussion to those special sections in which we have to treat of the properties which are affected by the applica- tion of heat. CHAPTEE XXIV. CONDUCTION AND CONVECTION OF HEAT. 282. Conduction. We have already discussed the transference of heat by the process of radiation, that is, the transference of heat without the mediation of ordinary matter. In the process of radia- tion the transferred energy may pass through a material substance without being communicated to it ; indeed, radiation ceases in so far as such communication is made. We must now consider its transference when ordinary matter is the medium through which it is transferred. The most marked difference between the two cases lies in the rate of propagation, which is extremely rapid when radiation occurs, while it is extremely slow in comparison when ordinary matter is the medium of transference. Two methods exist according to which heat (which consists in kinetic energy of molecular motion) may pass from one place to another by means of matter. The energy may pass from one por- tion of matter to another, which occupies a different position, by actual (or virtual) impact between the molecules of the two portions ; that is to say, it may be handed on from one portion to another : or again, it may pass, not from one portion of matter to another, but from one locality to another by motion of the hot body. The former of these processes is known as Conduction ; the latter as Convection. Both take place in liquids and in gases ; the former alone can take place in solids. 283. Conductivity. Different substances, under like conditions, conduct heat at different rates. Thus a bar of iron, one end of which is red-hot, may be too hot to grasp at the cooler end ; while a bar of wood, of the same length, which is burning at one end, may be easily handled at the other. The property in virtue of which such differences arise is termed Conductivity. Most experiments which are intended to illustrate the differences between the conductivities, or conducting-powers, of various sub- CONDUCTION AND CONVECTION OF HEAT. 345 stances for heat exhibit only the differences between the rates at which the temperatures of the substances, at a given distance from the source of heat, attains a definite value under given conditions. The well-known experiment of Ingenhouz is of this description. In it, a series of similar and equal rods, of different substances, project from the side of a metallic trough into which hot water is suddenly poured. Each rod is coated with a thin film of beeswax which melts at a definite temperature. The rate at which this definite temperature travels along each rod is plainly shown by the motion of the line of demarcation between the melted and the unmelted portions of the wax. But obviously this rate will only coincide with the rate at which heat is conducted along when the thermal capacities of the various substances are practically identical, for, other things being equal, the rate at which the temperature rises is inversely proportional to the thermal capacity. Fourier was the first to give an accurate definition of conductivity. The whole subject of heat-conduction was so fully and accurately developed by him that his work ' Theorie analytique de la Chaleur ' published in 1822, still remains the text-book on the subject. Let us suppose that the substance, the thermal conductivity of which we are considering, is in the form of a uniformly thick plane slab of practically infinite extent. Let 9 be its thickness ; and let one side be kept at uniform temperature t, while the other is kept at uniform temperature t' until a steady flow of heat takes place from side to side. The quantity of heat, h, which passes in T units of time through an area a of the surface of the slab is found ex- perimentally to be directly proportional to r, a, and t' t, while it is inversely proportional to 9. Hence we may write The quantity (' )/# is called the temperature gradient, and Jc is the conductivity. If the area, the temperature gradient, and the time, be all unity, the equation becomes &*, and so we obtain the following definition of the conductivity : The thermal conductivity of a substance, at any temperature, is the number of units of heat which pass, per unit of time, through unit of surface of an infinite slab of the substance, of unit thick- ness, the sides of which are kept respectively at temperatures half a degree higher, and half a degree lower, than the given tempera- ture. 346 A MANUAL OF PHYSICS. In making this definition we assume that the unit of length is not excessively small that it is (say) a centimetre, an inch, or a foot and that an ordinary temperature degree say, the Centigrade is used, so that the temperature gradient is not large. The necessity for these restrictions is apparent, if we consider that the conductivity may (it actually does) vary somewhat with the temperature ; for, in consequence of such variation, the temperature gradient could not be sensibly uniform from side to side of the slab, if the difference of the temperatures at the two sides were large. As a special case, let us suppose that the conductivity of a layer of the slab, of half its total thickness, is one-half of that of the remaining portion. Since the same flow of heat takes place through both portions, the dif- ference of temperature between the sides of the former portion must be double of that between the sides of the latter. Of course, even if the conductivity varies from point to point, whether from variation of temperature or from any other cause, the quantity &, determined from the above formula, will always repre- sent the average conductivity of the slab considered as a whole. But, quite apart from the question of such variation, we cannot assert that the quantity of heat which will pass through a slab, one unit in thickness, under unit difference of temperature, will be pre- cisely equal to the quantity which will pass through a slab, n times thinner, under a difference of temperature n times less, when n is a very large number, and all the other conditions are unaltered. 284. Measurement of Conductivity. In one form of experiment for the absolute determination of conductivity, a steady state of temperature is maintained throughout the substance. This method was used by Lambert, and subsequently, under greatly improved conditions, by Forbes. In Forbes' method a long bar of the substance of uniform cross- sectional area is used. One extremity of the bar is inserted in a bath of melted lead, or solder ; and the other extremity is exposed to the air, or, if necessary, is cooled by a current of water. Small holes, into which a little mercury is poured, are drilled in the bar at regular intervals; and these holes are lined with iron (if the bar itself be not made of iron) in order to prevent amalgamation. Thermometers, inserted in the holes (which are found not to appre- ciably affect the flow of heat along the bar) register the temperature of the bar in their immediate vicinity. If distance measured along the bar from the source of heat be laid off along ox (Fig. 160), and if ordinates be drawn at points such as j>, corresponding to the positions of the thermometers, and of lengths which are proportional to the readings of the thermometers at these CONDUCTION AND CONVECTION OF HEAT. 347 points, a curve drawn free-hand through the extremities of the ordinates will enable us to obtain the temperature gradient at any P FIG. 160. part of the bar. For the tangent of the angle of inclination of the line which touches the curve at the extremity of any ordinate is equal to the space rate at which the temperature varies, per unit of length, at the corresponding section of the bar ; i.e., it is equal to the gradient of temperature at that section. But the sectional area of the bar is known, and hence, if we can determine the quantity of heat which passes in a given time through the given section, we can determine the conductivity by calculation from the equation above. Now the heat which passes any section is entirely lost from the remaining portion of the bar by radiation, or otherwise ; and any heat which is given to the water employed in cooling the far end of the bar, if this is required, can be readily estimated by means of the rise in temperature of the water, while the heat which is lost by radiation and convection is found by a special experiment. During the above experiment, a thermometer, inserted in a hole in a small bar which is cut originally from the long bar, indicates .the temperature of the air in the neighbourhood of the large bar. In the second experiment, which is made for the purpose of deter- mining the rate of loss of heat, the small bar is heated uniformly to a temperature higher than the highest recorded in the former one. The bar is now allowed to cool, and the thermometer which is inserted in it enables us to determine the rate of loss of heat per unit of time, per unit of length of the bar ( 269). The mass of unit length is known, the specific heat is also determined, and the product of these quantities into the rate of fall of temperature gives the rate of loss of heat. This being known for all the various temperatures observed at the different parts of the bar in the first experiment, the total rate of loss of heat from the portion of the large bar, beyond any given section, is easily calculated. And so the conductivity, at particular temperatures, can be found. In the second experiment the temperature of the air is obtained 348 A MANUAL OF PHYSICS. by means of the long bar, so that the results in both cases can be compared, as is necessary, at the same excess of /temperature over that of. the surrounding air. But in addition to this, since the rate of cooling depends upon temperature and pressure of the air, it is necessary to perform both experiments under as nearly as possible the same conditions of temperature and pressure. The unit of heat which is employed in this method is obviously the amount of heat which is required to raise the temperature of unit volume of the substance by one degree, for the amount of the heat is measured in terms of changes of temperature in the bar. Con- sequently the quantity which is so determined is not the therma conductivity as above denned. Maxwell calls it the Thermometric Conductivity; Thomson calls it the Thermal Diffusivity. The thermal conductivity of any substance is obviously the product of the thermometric conductivity into the thermal capacity of that substance. Tait, who repeated and extended Forbes' experiments, gives the following values of Thermometric Conductivity. Temperature C. 0. 100. 200. 300. Iron ... 0-0149 0*0128 0*0114 0*0105 Copper, electrically good . . . 0'076 0*079 0*082 0-085 Copper, electrically bad . . . 0'054 0-057 0'060 0*063 German silver 0*0088 0-009 0*0092 0-0094 This table indicates that, with the exception of that of iron, the thermometric conductivity of all these substances increases as the temperature rises. Tait also gives, for the iron and the two specimens of copper (the units being the foot, the minute, and the degree C.), the following values of *** Thermal Conductivity. Iron 0-788(1-0-00002^) Copper, electrically good ... 4'03 (1+0'0013) Copper, electrically bad ... 2-84 (1+0-00140 It appears, therefore, that in general the thermal conductivity increases as the temperature rises. Also, the order of the metals with regard to conduction of heat is the same as their order with regard to conduction of electricity. Forbes had observed this fact, and had expected that, as in the case of electric conduction, the thermal conductivity would decrease as the temperature becomes higher. This, as we see, does not, in general at least, hold true. CONDUCTION AND CONVECTION OF HEAT. 349 Dr. A. C. Mitchell has recently, under Professor Tait's direction, repeated these experiments with the same bars, but under improved conditions. For one thing, all the bars were nickel-plated, so as to avoid alterations of the surface from oxidation at high temperature. His results are, on the whole," confirmatory of the previous results with this chief exception, that he found the temperature co-efficient for iron to be positive, as it is in all the other substances. A different method was employed by Angstrom. In his method one end of the bar is alternately heated and cooled during equal periods of time, the temperature of the source of heat being kept constant. The alternations of heating and cooling are. maintained until all the thermometers indicate practically periodical changes of temperature. Fourier's mathematical investigations show that, if the variations of temperature do not sensibly affect the conductivity and the specific heat, the conductivity can be calculated from the rate at which the range of temperature diminishes per unit of length of the bar, together with the observed speed at which the ' waves of temperature ' run along the bar, provided that the rate of surf ace - loss of heat is proportional to the excess of the temperature of the bar over that of its surroundings. 285. Conduction through the Earth's Crust. Angstrom's method has a direct application to the problem of the conduction of the diurnal and annual waves of solar heat downwards through the crust of the earth. In this investigation we may assume that the heated surface is practically an infinite plane, and that the flow of heat takes place in lines perpendicular to this plane. Let ab (Fig. 161) represent the surface, and let cd, ef, represent planes parallel to the surface, at distances x and x-\-dx, respectively, from it. If c be the thermal capacity of the substance through which ct f FIG. 161. the flow takes place, while v is its temperature, and t represents time, the quantity of heat which enters a portion of the substance, of small thickness dx and area a, in a small interval of time dt is JU. .... (1) 350 A MANUAL OF PHYSICS. for cadx represents the quantity of heat which must be abstracted in order to lower the temperature of the volume aSx of the substance by one degree, and dv/dt . t is the change of temperature in the time Si. If Tc be the conductivity of the substance, the quantity of heat which, in the time St, crosses in the positive direction the area a of the side of the slab nearest the surface is dx Similarly, the quantity which passes downwards through the area a of the surface which is distant from the former by the amount dx is (since dx is small) 7 dv t d (-. dv\ &j--Hjn&j- < dx dx\ dx) Consequently, the amount of heat which, on the whole, enters the volume a&x in the time dt is and hence we get c~ v = (%} (3) db dx\ dx)' ' since each of the quantities (1) and (2) represents the same amount of heat. Now ( 27, 67) let us consider this equation simply as an equa- tion of dimensions. The difference of temperature dv appears linearly on both sides of (3), and therefore the range of temperature does not appear in the dimensional equation. We get C _JL T~W ( ' where t represents time, and I represents length ; for dx, the dimen- sions of which are those of a length, appears twice as a factor in the denominator on the right-hand side of (3) ; while & and dt appear once only on the right-hand side and the left-hand side, respectively, of that equation ; and we must remember that the sign of equality indicates now equality of dimensions alone. From (4) we get M = V c' CONDUCTION AND CONVECTION OF HEAT. 351 which means that, if the times are altered in any fixed proportion p, the lengths must be altered in proportion to the square root of p in order that the flow of heat may take place under similar condi- tions in the altered circumstances. In other words, the distances at which similar effects are felt (for example, the distances below the surface at which the periodic variations of surface-temperature cease to be felt] are proportional to the square root of the period. Now the period of the annual heating and cooling is 365 times as great as the period of diurnal variations. Hence the effect of the summer's heat is felt about nineteen times as far below the surface as the effect of the diurnal heat is felt. Again, r r v ct' Here we may suppose that I represents the length of a wave, while t is the periodic time, so that the fraction on the right hand is pro- portional to the rate at which the wave of heat travels downwards. We see, therefore, that this rate is directly proportional to the square root of the conductivity, and is inversely proportional to the square root of the thermal capacity and the periodic time conjointly. It follows that, when the period is constant, the date at which the maximum temperature reaches any given depth is later than the date at which it left the surface in direct proportion to the depth. The law which regulates the diminution of the range of tempera- ture with increase of depth cannot be obtained from equation (3) in the way in which we have obtained the two laws just enunciated ; for the temperature does not appear in equation (4). But we may write (3) in the form ^dvdx_ d (->dv\ "dxdt ~dx( dx) ' and we may suppose that dxjdt is the speed with which the heat- wave travels downwards, in which case the equation becomes V ck dv_ d/,dv T ' dx- where T is the periodic time, and dv/dx now represents the rate at which (say) the maximum temperature changes as the wave passes down. This equation asserts that the rate of diminution of the rate of change of temperature with depth is proportional to the rate of change itself. In other words, the rate of change diminishes in 352 A MANUAL OF PHYSICS. geometrical progression as the depth increases in arithmetical pro- gression. And its rate of diminution is */cj v/^T. But, since the rate of diminution of the rate of alteration of the range is propor- tional to the rate of alteration itself, it follows that the rate of altera- tion bears the same ratio to the range. Hence the range diminishes in geometrical progression as the depth increases in arithmetical progression, the rate of diminution being directly as the square root of the thermal capacity, and inversely as the square roots of the conductivity and the periodic time conjointly. As the result of direct observations (begun by Forbes in Edinburgh in 1837) of the temperature at different distances below the surface of the earth, it is found that the annual heat-wave travels inwards at the rate of little more than sixty feet per annum, and that the range of temperature has diminished to a very small fraction of its original amount when half of that distance has been traversed. The diurnal heat is therefore inappreciable at a depth of at most about two feet. Of course, all these results depend upon the nature of the soil. The thermometers nearest the surface are affected by changes in the weather, but these disturbances rapidly die out. When a steady state of temperature is reached, dv/dt vanishes, and (3) becomes dfdv 'dx\ da This gives dv K-= = a (constant), dx which shows that the temperature-gradient varies inversely as the conductivity. This applies directly to the case of the earth regarded as a cooling body, and shows us that in strata throughout which the conductivity does not vary, the temperature increases uniformly per unit of depth. Of course, if the earth is regarded as a cooling body, the steady state of temperature is impossible ; but its rate of cooling is so slow that the time -variations of temperature may be neglected. Fourier's equations, when applied to past time as regards the earth or the sun, indicate a state of uniform high temperature throughout the mass a state which could not have arisen by any process of conduction. This suggests the production of the heat by the gravitation of separate masses ( 93). 286. Conduction in Crystalline Bodies. Crystalline bodies possess, in general, unequal conducting power in different directions. CONDUCTION AND CONVECTION OF HEAT. 353 The conducting power is symmetrical with regard to three rectangular axes called the principal axes of thermal conduc- tivity. If a single point- source of heat were placed in the interior of a crystal, the loci of constant temperature would be con- centric ellipsoids surrounding that point ; and Stokes has shown that the conductivities parallel to the axes of these ellipsoids ar proportional to the squares of the axes. Sections of such an ellipsoid can be obtained by means of thin plane plates of the crystal cut in different directions from the substance. If a copper wire be passed through a small hole drilled through such a plate in the direction of the axis of the ellipsoid conjugate to the plane, and if this wire be heated by an electric current, the heat conducted away by the plate may be made to melt a thin coating of beeswax on the surface of the plate. The boundary between the melted and the unmelted wax is a section of the above ellipsoid. 287. Conduction in Liquids and Gases. The thermal conduc- tivities of liquids (neglecting liquid metals) are small in comparison with those of solids ; and those of gases are smaller still. In experimental investigations on the subject, great care must be taken to avoid convection currents (see below), which would com- pletely invalidate the results. In the case of gases, the conductivity can be calculated from the kinetic theory. 288. Convection. Under gravity, all liquids and gases tend to arrange themselves in horizontal layers the densities of which de- crease as their distances from the earth's surface increase. This condition may be entirely disturbed because of variations of tem- perature ; for the consequent changes of density destroy the equili- brium, and currents are set up in the fluid so as to restore it. These are called ' convection-currents.' We have already discussed a typical case in dealing with the maximum density point of water. Very marked examples occur on a large scale in nature. The trade -winds are due to the ascent of hot air currents in equatorial regions, while colder air blows in from the polar regions to take its place. [The north-easterly or south-westerly direction of these winds is due to the rotation of the earth.] A considerable part, at least, of ocean circulation, is also of the nature of convection. Again, when- ever water evaporates, heat is absorbed to be evolved wherever condensation occurs. This is indeed one of the most fruitful sources of violent storms ; for, if sufficient heat is developed, the consequent increase of temperature causes a rapid up-rush of air, and so creates 23 354 A MANUAL OF PHYSICS. a partial vacuum which gives rise to a violent inflow of the sur- rounding air. [The air which comes from the south has (in the northern hemisphere) a greater eastward motion than that which comes from the north, and so a counter-clockwise vortex motion is produced. Thus the rotation of a cyclone is explained.] Practical applications of the principles of convection are seen in the usual methods of boiling water, of ventilation, etc. A hot body, which is cooling in a gas or a liquid, loses heat by convection as well as by radiation. The law of convective cooling in a gas has been elaborately studied by Dulong and Petit. Their results are expressed by the formula where r is the rate of cooling, a is a constant for a given gas and a given body, p is the pressure of the gas, 6 is a constant for any one gas, and 9 is the excess of the temperature of the cooling body over that of the gas. The rate is independent of the nature of the surface of the body but varies with its form and dimensions. CHAPTER XXV. THERMODYNAMICS : HEAT AND WORK. 289. Mechanical Equivalent of Heat. First Law of Thermo- dynamics. Heat, since it is & form of energy, may be transformed into mechanical work and into all other forms of energy ; but the former transformation is the only one with which we are at present concerned. Colding and Joule were the first, after Rumford, to make deter- minations of the amount of work which can be produced from a given amount of heat, i.e., of the mechanical (or, more properly, the dyna- mical) equivalent of heat. The most direct method of conducting such an investigation con- sists in spending a known amount of work in the production of heat by friction. This method was used by Joule, who caused a falling weight to drive a vane rotating in the interior of a calorimeter which contained a known amount of water. The amount of heat developed was determined by means of the observed increase of the temperature of the water, due precaution being taken to correct for the loss of heat by radiation, and for the heat developed by friction between the parts of the apparatus. Various other methods were used by Joule, Him, Regnault, and others. For example, Joule proved experimentally that the heat developed by the sudden com- pression of air is practically equivalent to the work spent in com- pression ( 294) ; and from this result, together with an accurate determination of the specific heats of air, he obtained the value of the mechanical equivalent. He also determined its value by means of the heat developed in a conductor by the passage of a current of electricity through it under given conditions ( 342). One of Hirn's series of experiments was made upon a heat-engine hi actual use. In another series, he experimented upon the heat developed by per- cussion. The latter of these gave a good result ; the former did not. Joule finally gave the number 772 (in foot-pounds at the latitude 232 356 A MANUAL OF PHYSICS. of Manchester) as the amount of work necessary to raise the tem- perature of one pound of water by one degree Fahrenheit. The equivalent of the heat-unit (Centigrade) which we have hitherto used, is therefore 1,390 foot-pounds. These experiments prove the law of conservation of energy in so far as heat and work are concerned. The statement of the law of conservation for these two forms of energy goes by the name of the FIRST LAW OF THERMODYNAMICS, which asserts that wJien equal quantities of mechanical effect appear from purely thermal sources, or disappear in the production of thermal effects alone, equal quantities of heat disappear or are produced. 290. Carnot's Complete Cycle of Operations Although, as is indicated in the previous section, we can determine, by experiment, the direct relation between given amounts of heat and work, we are not entitled to draw any conclusion regarding the relation between the heat which disappears and the work which appears in any given physical process, unless a certain condition be observed. The necessity for this condition (which has already been referred to in 254) was pointed out by Sadi Carnot. The condition is that the working substance must pass through a complete cycle of operations, i.e., a cycle at the end of which the substance has returned to its original physical state. Heat may have been expended in the given cycle, and work may have been produced ; but, unless the final state is the same as the initial state, we cannot say that the work and the heat are mutually equivalent. For example, carbonic acid gas is heated by compression, but the heat developed is not the equivalent of the work spent in compression, for work is done by the molecular forces during the process. In order to be able to reason correctly upon the connection between heat and work, Carnot assumed the existence of a heat-engine which can never be realised in practice. But this does not render his results any the less valuable, for, in order that we may be able so to modify Carnot's results that they may apply to the special case, we only require to know in what way, and to what extent, any given engine differs in its action from Carnot's. He assumed that his engine was furnished with a cylinder the sides and piston of which were absolutely impermeable to heat, while the bottom was a perfect conductor of heat. He assumed also the existence of two bodies, one hot and the other cold, the temperatures of which were kept constant. The former of these was to act as the source of heat ; the latter was to act as the con- denser. The working substance is supposed to be placed in the cylinder below the piston, and may be any substance whatsoever, THERMODYNAMICS : HEAT AND WORK. 357 with any properties whatsoever. But, for the sake of definiteness, we may assume that it acts as the steam in an ordinary engine would act. Let us suppose that the cylinder, with its contents at the tem- perature of the cold body, is placed on a non-conductor of heat. The contents will retain their temperature (to, say) so long as the cylinder remains on the non-conductor, for the working substance is now surrounded on all sides by non-conductors. Let the volume and pressure of the substance be denoted by v and p respectively. [Eemark here that the physical condition of a known mass of the substance is completely determinate when any two of its tempera- ture, volume, and pressure are given.] As the first operation of Carnot's cycle, the cylinder still re- maining on the non-conductor, press down the piston until the temperature of the substance rises to that of the hot body ( 1} say) and the volume and pressure become v l and p l respectively. As the second operation of the cycle, place the cylinder, with the condition of its contents unaltered, on the hot body, and let the substance slowly expand until its volume becomes (say) v z , and the pressure becomes p. 2 ( w the quantity of work which the reversible engine would produce under the same conditions. Make M work backwards between the same sources at the same high and low tem- peratures. A quantity of work w is all that has to be expended in order to make it take in the amount h of heat at the tempera- ture t andrestore the amount \ to the source at the high temperature. Hence an excess of work W w is gained in the double process, while, on the whole, no heat has been transferred either way. But this means the perpetual production of work from nothing, which is impossible. Hence the reversible engine is perfect. 360 A MANUAL OF PHYSICS. To adapt this reasoning of Carnot to the modern ideas of energy, we have only to argue thus : Work is performed in the double cycle, while no heat is on the whole taken from the source, therefore N must give less heat to the cold body than M takes from it, and so the double engine can only work by giving to the hot body heat which it has taken from a colder body, the temperature of which it constantly lowers ; but this is in opposition to all known facts, and the denial of its possibility may be safely taken as axiomatic. This adaptation is due to Sir W. Thomson, who re-introduced Carnot's work to the scientific world when it had been long disre- garded, and who applied his principles to the deduction of thermo- dynamical results of the highest importance. The statement that an engine which is reversible, in the sense that all its physical and mechanical actions are capable of exact reversal, converts, under given conditions, the greatest possible fraction of the heat which is supplied to it into useful work is known as the SECOND LAW OF THERMODYNAMICS. We shall find in 298 that its analytical expression is 2(hjt) = o. 293. Absolute Temperature. The efficiency of a heat-engine is the ratio of the quantity of heat which it utilises in the form of mechanical work to the total quantity of heat which is supplied to it. In the notation used above it is The value of this fraction is a maximum, under any given con- ditions, when a reversible engine is used. And all reversible engines, whatever may be the nature and properties of the working- substance, are equally perfect (i.e., they possess the same efficiency) when they work through the same range of temperature. This enables us, as Thomson pointed out, to obtain an absolute measure of temperature absolute in the sense that it does not depend upon the properties of any particular substance. The definition which Thomson finally adopted was framed so as to make the absolute scale coincide as nearly as possible with the scale of the air-thermometer. It is this : ' The temperatures of two bodies are proportional to the quan- tities of heat respectively taken in and given out in localities at one temperature and at the other, respectively, by a material system subjected to a complete cycle of perfectly reversible thermo- dynamic operations, and not allowed to part with or take in Jieat at any other temperature; or, the absolute values of two tempera- tures are to one another in the proportion of the heat taken in to THERMODYNAMICS : HEAT AND WORK. 361 the heat rejected in a perfect ther mo dynamic engine, working with a source and refrigerator at the higher and lower of the temperatures respectively.' 1 Expressed in symbols, this gives = A) V where ^ and t are the absolute temperatures of the source and the condenser, respectively. 294. The Indicator Diagram. The indicator diagram was introduced by Watt for purely practical purposes. It is, neverthe- less, as we shall shortly see, susceptible of numerous important applications in pure science. The diagram exhibits the relation between the pressure and the volume of the working substance used in any heat engine ; and hence it shows the work done in a complete stroke of the engine. aMN FIG. 162. Let the curve APQBK (Fig. 162) represent the relation between the pressure and the volume at all stages of the stroke pressure and volume being respectively measured along the rectangular axes op and ov. Let the co-ordinates of the point P be PM=p, OM. = v ; and let those of Q be QN=p', ON^v'. When P and Q are indefinitely near each other, the product (p+p'} (v'-v) represents twice the area PMNQ. But, neglecting small quantities of the second order, half this product may be written p(v'v}. This latter product therefore represents the elementary area PMNQ. Now the work done by a force /, acting through a distance s, is ( 62) fs ; and the whole force which acts upon the piston in the cylinder of the engine is pa, where a is the area of the piston, and p is the pressure per unit area, so that pas represents the work done 362 A MANUAL OF PHYSICS. when the piston moves through the small distance s under the action of the pressure pa. But as is the small change of volume of the contents of the cyclinder which is produced by that pressure ; and therefore the work done is represented by the area PMNQ. Let A be the point at which the volume has its smallest value, v , and let B be the point of maximum volume, v lt As the volume expands from v to v v the state of the substance being represented by a point which moves along the path APQB, the work performed is represented by the area Aa&BQPA. Similarly, when the volume is diminished from v 1 to -y , the point moving from B to A along the path BRA, the work expended in producing compression is repre- sented by the area BRAa&B. The difference of these two areas, viz., the curvilinear area APQBRA, therefore represents the work which is expended, on the whole, during a complete stroke of the engine. Consequently, when the path is described in the positive direction, work is expended on the whole. This case corresponds to the reverse working of a reversible engine. When the closed path is described in the negative direction, work is performed on the whole by the engine to an extent which is repre- sented by the total area enclosed by the path. [The actual path might consist of a number of closed loops. In this case, each loop is to be considered separately, and the sum of the areas each with the proper sign attached, according as it is described positively or nega- tivelyis to be taken in order to estimate the total amount of work performed.] 295. Applications of the Indicator Diagram. We shall now consider the application of the diagram to the discussion of the working of Carnot's engine. For this purpose we must give the closed curve a special form. In the second and fourth direct operations of Carnot's cycle ( 290), the substance was maintained at constant temperature ; in the first and third, no heat, as such, was allowed to pass out of it or to enter it. If, therefore, the points D, A, B, C (Fig. 163) represent respect- ively the values (p , V Q ; p lt Vi ; p. 2 , v. 2 ; p s , u<) of the pressure arid volume of the substance at the commencement of the first, second, third, and fourth operations respectively, all points on the line DA (which represents the varying state of the substance as it passes from the state (p 01 t> , t ) to the state (p lt v lt tj are characterised by the condition that no heat, as such, enters or leaves the working sub- stance ; all points on the line AB represent states in which heat is absorbed during the passage from the condition A to the condition B in order that the temperature may retain the value ^ ; all points on the line BC represent states in which, again, no heat enters or THERMODYNAMICS : HEAT AND WORK. 363 leaves the substance ; and all points on the line CD represent states in which the temperature of the substance is, by disengagement of heat, maintained at the value t . Lines such as AB or CD are therefore called "isothermals, while lines such as DA or BC are called adiabatics. , The latter name, which is due to Rankine, simply implies that no heat is absorbed or emitted. But although heat, as such, neither enters nor leaves the sub- stance in the adiabatic condition, mechanical work is performed or FIG. 163. expended ; and so, by transformation, the amount of heat contained in the substance is actually varying. If the amounts of heat contained in the substance, in the states A and D, were identical, and if the amounts in the states B and C were identical, the amounts of heat which leave the substance in the processes represented by AB and DC would necessarily be equal. But we know that these quantities are (by our definition) in the ratio tj_ and t being absolute temperatures. Now ABCD represents the work which is performed in the direct cycle. It therefore, by the principle of conservation of energy, represents the quantity of heat 7 1 7t . And, if we/arrange matters so that 7ii-7z = l, equation (1) shows not only that t 1 m^islQbe equal to unity, b^a^gp^that hi = t lt and h =t . V Let us therefore intersect the diagram (Fig. 164) by a series of isothermals A! A 2 A 3 ..., BiB 2 B 3 ..., etc., and by a series of adiabatics AiBiCj..., A 2 B 2 C 2 ..., etc. ; and let the temperature corresponding to . be one degree below that corresponding to 364 A MANUAL OF PHYSICS. while the adiabatics are so arranged that the magnitude of each area, such as A 1 A 2 B B 1 , is unity. If we could continue this construction down to the absolute zero of temperature, each area included between any isothermal, the isothermal of absolute zero, and any two con- secutive adiabatics, would be numerically equal to the temperature indicated by the higher isothermal. FIG. 164. The experimental data necessary to the correct completion of the diagram are wanting, but its correct completion is not necessary to our present purpose. The following method of completing it is due to Maxwell. Let us suppose that CiC 2 C^ ... is the lowest isothermal whose form is correctly known. Draw any line K 1 K 2 K 3 ...to represent the isothermal of absolute zero, and complete the adiabatics in such a way that each of the areas C^B^Co, C 3 K 2 K 3 Cj,, etc., is numerically equal to the temperature to which the line C^C.^ ... corresponds. In order to determine the position of the absolute zero, it is only necessary to find the ratio of any two areas such as AjKjE^A,, and CiKxKgCa. The method by which Thomson and Joule solved this problem is described in 303. 296. Applications of the Indicator Diagram. Entropy. The one set of lines with which we have intersected the diagram are characterised by constancy of temperature. The other set of lines are characterised by the condition that no heat shall be transferred from the working substance to its surroundings, or from the sur- THERMODYNAMICS: HEAT AND WORK. 365 roundings to it. But we have seen that heat does pass from, or into, the substance by transformation. We have still to inquire, therefore, What quantity remains constant during adiabatic expan- sion or contraction ? Equation (1) of last section enables us to answer the inquiry. It gives whatever values ^ and t may have ; that is to say, when the sub- stance passes isothermally from any one definite adiabatic state to any other definite adiabatic state, the quantity of heat which is absorbed, or disengaged, bears a definite ratio to the temperature at which the absorption, or disengagement, takes place. This constant quantity h/t may therefore be regarded as the amount by which some quantity, 0, which is characteristic of the substance in the adiabatic state, changes in passing from the one adiabatic condition to the other at constant temperature t. This suggests the extension of the meaning of the quantity h to signify the total heat contained in the substance, so that we may define by the equation H where H represents the total heat. The quantity was first called, by Kankine, the Thermodynamic Function. Clausius called it the Entropy, and this name has been generally adopted. If one substance parts with an amount of heat, h, at temperature t, to another substance at temperature t , the entropy of the former substance decreases by the amount hjt lt and that of the latter increases by the amount Jijt . The total gain of entropy by the system is therefore The quantity of heat which is absorbed in the second operation of Carnot's direct cycle is t^fa ), and the quantity which is given out in the fourth operation is t Q (

), which therefore represents the work done in the complete cycle. 297. Applications of the Indicator Diagram. Total, Available, and Dissipated Energy. We have no means of determining experimentally the total amount of energy in a given system. But, 366 A MANUAL OF PHYSICS. in any practical case, we only require to determine the change of energy which takes place in the given operations. The indicator diagram' enables us to represent the total amount of energy in a way which, though obviously incorrect, leads to a correct representation of the change of the total energy of the system which is produced by a given change in its physical condition. Let the points A and B (Fig. 165) represent respectively the initial and the final conditions of the system, and let the path AB represent the series of changes by which the final condition was arrived at. [The diagram is constructed so as to exhibit the case in which there M N S FIG. 165. is both disengagement of heat and performance of mechanical work.] Let BE' represent the (arbitrary) isothermal of absolute zero, and let it be continued so as to cut the axis of volume in the point S. As the point which traces out the diagram moves from A to B, external work, which is represented by the area ABNMA, is per- formed, and at the same time, 295, an amount of heat is dis- engaged which is represented by the area ABK'BA. The whole area AMNBB'BA, therefore, represents the total loss of energy which the working substance has sustained. We may, therefore, as Maxwell suggested, regard the areas AMSKA and BNSB'B, re- spectively, as representing the total amounts of energy which are contained in the working substance in the conditions indicated by A and B respectively. It is specially to be noticed that the amount of energy which is lost in proceeding from A to B is totally independent of the path AB, and depends only on the initial and final conditions of the substance. THERMODYNAMICS : HEAT AND WORK. 367 Let us assume now, as a special case, that BC represents the lowest available temperature. We may then cause the substance to pass from the state A to the state B along the path ACB. The work performed will then be represented by the area ACBNMA, which area therefore represents the total amount of energy which is available, under the given conditions, for the performance of me- chanical work. Similarly, BCEE'B represents the heat which is necessarily given to the condenser, i.e.. the amount of energy which is necessarily dissipated so far as the performance of work by the given system is concerned. The energy which is unnecessarily dissipated is represented by the area ACB. 298. Thermodynamic Motivity. We have seen that, when a quantity of heat, 7^, is given out by a body at temperature t lf the entropy diminishes by the amount 7* 1 / 1 . We may therefore repre- sent the total loss of entropy of a system which consists of a number of sources which are emitting heat at various temperatures by the symbol 2 Similarly, if the heat emitted by these bodies is given to other bodies included in the same system, we may denote the gain of entropy from this source by The total loss of entropy is therefore \Y^i_ * Deleting the suffixes, we may denote this simply as and our definition of absolute temperature shows that this vanishes wlien a perfect engine is used. When any other engine is used, heat is always lost by conduction or otherwise, "so that the heat which is given to the condenser (or is otherwise wasted) is greater than h . Hence, in all actual cases of the transformation of heat into work we must write A instead of sf^_: H 368 A MANUAL OF PHYSICS. where ~k is greater than unity, as the proper expression for the loss of entropy. This is equal to which is necessarily negative, since k is greater than unity ; and so we prove Clausius' theorem that the entropy of the universe tends to a maximum. The amount of heat utilised by a perfect heat-engine is ( 293) Or 7*, -^. fi If a number of such engines work between the various parts of a complex system, the heat which is not given to the condenser is where 2(7*,) now takes account both of the heat which is taken from each body by some of the engines, and of the heat which is given to it by others. In a perfect engine, the heat which is not given to the condenser is represented by 2(7*,) alone, and is entirely converted into work ; which again shows us that, for such an engine, t(* must vanish. In all other cases, 2(7*,) represents the part of the heat which is utilised in the performance of mechanical work, and the second term (which we must remember is necessarily positive) represents the portion which is unnecessarily wasted. The quantity therefore represents the heat which is dissipated in the process. Thomson has tailed the total energy which could be made avail- able for mechanical work by a perfect engine under given condi- tions, the Thermodynamic Motivity of the system. If we have a medium external to the given system, which may be used as a con- denser, the motivity is the whole amount of work which can be obtained by the perfect engine in reducing the temperature of the system to that of the external medium. If the engine works so as to equalise the temperatures of the various parts of the system, the motivity is the whole amount of work which can be so obtained. THERMOrYNAMICS : HEAT AND WORK. 369 Let t be the final temperature, and let h be a quantity of heat taken from a body at temperature t. The motivity, so far as this quantity is concerned, is and, to obtain the total motivity, we must sum all such quan- tities. The total energy, e, in any system is equal to the sum of the motivity, ra, and the dissipated energy of that system. If is the entropy, 9 the temperature, and J the mechanical equivalent of heat, the dissipated energy is J00. Hence, if the system passes from a state indicated by the suffix 1 to a state indicated by the suffix 2, we get m l - m. 2 = e l - e z No change can take place of itself in the system unless thereby the motivity be decreased that is, unless m 1 - m. 2 be positive. But m^m^ may be positive, although e l e% is negative, provided that 0. 2 < 2 is sufficiently greater than 0^. Hence we see that a given chemical action may take place of itself with absorption of heat, provided that a sufficient amount of energy be dissipated in the process. (See 280.) 24 CHAPTEK XXVI. THERMODYNAMICAL RELATIONS. 299. IN the course of the discussion, in last chapter, of the con- nection between heat and work, we were led to consider five quan- tities in terms of which the physical condition of a substance may be represented. These quantities were the energy, e ; the entropy, ; the pressure, p ; the volume, v ; and the temperature, t. But we were also led to see that the physical condition of the sub- stance was completely determined when two of these quantities, p and v, were given : for, on the indicator diagram, we could lay down lines of constant temperature, of constant entropy, and of constant energy. The total values of the two latter quantities were not, it is true, indicated ; but that was due to a defect in our know- ledge, and not to any defect necessarily inherent in the diagram. It at once follows that the variation of any one of the five quan- tities can be represented in terms of the simultaneous variation of any two of the rest. Thus we may write ..... (1). de=fdt + gdv, ..... (2). de = mdt+ndp, ..... (3). and so on. 300. We shall first consider equation (1). If the volume be con- stant we obtain de = ad<{>. But we know that, under constant volume, the energy increases by the amount (in dynamical units) of heat which has been supplied ; and we also know, by the results of last chapter, that this amount is td. Hence a = t. Similarly, if no heat be supplied, so that remains constant, (1) becomes de = bdv. But under these conditions the energy diminishes by the amount of external work which is performed, that is, by the amount pdv. Hence b= -p, and (1) becomes de = td-pdv ..... (4). / de\ , (de\ This gives ( = t, ) = -p. \dJ, \&>f+ THERMODYNAMICAL RELATIONS. 371 The suffixes denote respectively that the volume and the entropy are constant ; so that the former equation asserts that, at constant volume, the increment of the energy is equal to the heat supplied ; while the latter asserts that, under adiabatic expansion, the decre- ment of the energy is equal to the amount of work which is per- formed. These equations therefore express the conditions upon which we deduced (4) from (1). From them we get e_ = (dt \ d(j> \dv) dvd(j> \dv $ ddv~ W Me hgi ve ()--(*), ....... .(5) where we may dispense with the suffixes. If the right-hand side of this equation be simultaneously multiplied and divided by t, the denominator represents the amount of heat which is -supplied, at constant volume, in order to produce the variation, dp, of pressure. If dp and td are positive, dtjdv is essentially negative. Hence the equation asserts that substances which, at constant volume, have their pressure raised (or diminished) by the application of heat, will fall (or rise] in temperature during adiabatic expansion ; and the change of temperature, per unit change of volume, is numeri- cally equal to the product of the absolute temperature into the change of pressure per unit of heat supplied. We may now combine with equation (1) the equation d(pv) =p as the result of which we get d(e+pv) = td+vdp ......... (6). From this we deduce, as above, the result If dt and dp are both positive, dv and d are necessarily of the same sign. Hence, multiplying and dividing the right-hand side by t, we see that substances which expand (or contract), under constant pressure, when heat is supplied to them, rise (or fall) in tempera- ture when they are subjected to adiabatic compression; and the change of temperature, per unit increase of pressure, is equal to 242 372 A MANUAL OF PHYSICS. the product of the increase of volume at constant pressure, per unit of heat supplied, into the absolute temperature. If we now combine with (1) the equation we obtain d(e-t)=- Multiplying each side of (9) by t we see that substances, which absorb (or emit) heat when their volume increases isothermally, have their pressure, at constant volume, raised (or diminished) by increase of temperature ; and the change of pressure, per unit rise of temperature, is equal to the quotient by the absolute tempera- ture of the heat which is absorbed (or emitted). Let L represent latent heat, and let v' v represent the change of volume of unit mass of the substance when it changes its state. In this case (9) becomes t(v'-^v) : from which we can calculate the change of the melting-point, or the boiling-point, which results from a given change of pressure. Combining (8) with d(pv) =pdv+vdp, we find d(e-t$+pv)= -tydt+vdp, (10), which leads to This tells us that substances which expand (or contract) under con- stant pressure when their temperature is raised emit (or absorb) heat in order that their temperature may remain constant ivlien the pressure is increased; and the heat which is evolved (or absorbed) per unit increase of pressure is equal to the continued product of the temperature, the volume, and the expansibility. For the expansibility is Ifv . dv/dt. 301. In equation (2) the quantity/ represents the rate at which the energy increases, per unit increase of temperature, at constant volume. It therefore represents the specific heat at constant THERMODYNAMICAL RELATIONS. 373 volume, which we have already denoted by the symbol c. Hence we have from (1) cdt = td<}>, with the condition dv = 0. These equations may be written in the form = But the quantity is evidently the specific heat at constant pressure, hitherto denoted by k. Hence (12) becomes dp ........... (14). Now the condition dv=Q necessitates a certain relation between dp and dt ; but we can eliminate these quantities from (13) and (14). Thus fc-^/T^f^)- (dv\ \dtJp By (11) this becomes To apply this result to the case of a perfect gas we must find the values of (I) - d (?) ^dth \dpJi from the equation pv=Ht. m u . (dv\ R _ fdv\ Rt This gives ( - ) = , and I ) = - - , \dth p \dph p 2 whence fe-c = R. (16). 374 A MANUAL OF PHYSICS. The difference between the two specific heats is therefore constant ; and, since the values of both k and R can readily be found by experiment, c (the experimental determination of which is very difficult) can be calculated by means of this relation. (See 271.) 302. The equation connecting the pressure, volume, and tem- perature of a perfect gas is pv = R. It will be useful to determine the relation between the pressure, the volume, and the entropy of such a gas. We have where td^jdt is obviously the specific heat at constant pressure, and, by (11), d$ldp is equal to dv/dt at constant pressure, which again is equal to R/j?, i.e., to (7t c)/p. Hence dt n .dp ^ = & T -(&-c) , t p the integral of which ( 38) is /'* ) t+(1c-c)logp, where a (and therefore log a) is a constant. We may write this in the form But t is equal to pvfR, whence A being equal to aR*. Thus, instead of pv = constant, we must write * pv c = constant when adiabatic compression or expansion takes place. 303. When air is compressed by the sudden application of pressure, the heat developed is almost precisely equivalent to the work which is spent in producing the compression. Joule proved this by enclosing air in a strong vessel which could be placed in communication with another vessel, of the same size, which had been exhausted of air. Both vessels were placed in a large mass of water the temperature of which was accurately determined. When a stopcock in a tube connecting the two vessels was opened, the air rushed from the one vessel into the other so as to equalise the pressure throughout. The temperature of the vessel containing the expanding air was lowered, for work had been done during expansion, so that the air was cooled. That of the other THERMODYNAMICAL RELATIONS. 375 vessel rose, for the violent impact of the air which rushed into it caused the development of heat. But the amount of heat which was absorbed in the one case was almost precisely equal to that which was evolved in the other, for the surrounding water, which was well stirred, showed no appreciable change of temperature. A more accurate form of this experiment was subsequently adopted by Joule and Thomson in their researches on the thermo- dynamical properties of gases. The gas under investigation was made to pass very slowly through a tube, in which a plug of cotton wool was placed, and its pressure and temperature on both sides of the plug were observed. The preceding methods lead to a simple equation connecting the changes of temperature and pressure with the volume, the abso- lute temperature, and the expansibility of the substance. The expansibility may then be expressed, by means of Charles' Law, in terms of the temperature on the Centigrade scale, if the range of temperature be so small that the Centigrade and the absolute degrees are practically equal throughout its extent; so that the equation gives a direct comparison of the absolute and the Centi- grade scales. Boyles' and Charles' Laws give ( 266) for a perfect gas T = + l/a, where T is absolute temperature. The investigation just alluded to gives where ;//, as Thomson and Joule's experiments indicate, is, in true gases, a small quantity generally positive. The experiments showed that all the true gases except hydrogen were made colder by their passage through the plug, and indicated that the absolute zero is about 273'7 C. 304. The truth of the Second Law of Thermodynamics ( 292) rests entirely on the immensity of the number of particles con- tained in any portion of matter which is of a size comparable with the dimensions of our instruments and machines. An ordinary thermometer, placed in any position in a mass of air, might indicate uniformity of temperature ; while another thermometer, sufficiently small in size, might (rather, would) indicate rapid variations of temperature, and might even show that heat was passing from cold parts to hot parts of the given mass. For, the quickly moving mole- cules might occupy on the whole one portion of a volume so small as to contain only a few molecules, while the slowly moving molecules occupied the remainder ; and some of the slowest of the 376 A MANUAL OF PHYSICS. quickly moving molecules might be exchanged for such of the slowly moving molecules as were actually moving more quickly than they were. An average uniformity is preserved on the large scale, though, on a sufficiently small scale, it does not obtain. It is because of this average uniformity that the statement that a heat-engine cannot continually draw the heat which it transforms into work from a body colder than its condenser which Thomson made the basis of the Second Law of Thermodynamics is true. CHAPTER XXVII. ELECTROSTATICS. 305. Electrification by Friction. When a rod of glass is rubbed with flannel or, better, with leather coated with a paste of zinc amalgam it acquires the property of attracting surrounding bodies. Light bodies, such as pieces of paper, can even be raised up by it against the attraction of the earth. When in this state, the glass is said to be electrified or it is said that electricity has been developed upon the glass. If the glass had been rubbed with cat's-skin or with any one of several other substances, similar effects would have ensued ; but the extent to which electrification is developed depends upon the nature of the substance which is used as a rubber. The glass may be replaced by sealing-wax, resin, ebonite, etc., and the phenomena will still be exhibited to a greater or less extent. In all cases it is necessary for success that the substances shall be well warmed and dried. 306. Conductors and Non- Conductors. If, instead of a glass rod or a rod of sealing-wax, we take a metallic rod, no electrical effects are in general observable ; and many other substances also are incapable (unless special means are adopted, 324) of being electrified by friction. We are thus led to divide all substances into two classes according as they are or are not electrifiable by friction in the usual way. Those of the former class are called Insulators, Dielectrics, or Non-conductors; those of the latter class are called Conductors. The latter terms are applied because it is found that all substances which cannot usually be electrified by friction have the power of allowing electricity to flow along them, while the other class of sub- stances prevent such flow. 307. Fundamental Phenomena presented by Electrified Bodies. The substances which are attracted by an electrified body are not 378 A MANUAL OF PHYSICS. necessarily non-conductors. In order to investigate the subject further we shall suppose that a pith-ball (which is a conductor, and is at the same time very light, so that the effects to be observed are easily seen) is the body to be attracted, and we shall suppose it to be insulated by being suspended from a dry glass rod by means of a dry silk thread. If an electrified glass rod be brought into the neighbourhood of the pith-ball, the ball will be drawn towards it ; and this will also take place when electrified sealing-wax is presented. Now let the glass rod be brought so near that the ball comes in contact with it. Immediately after contact the ball is violently repelled by the glass ; but, if the ball be touched with the hand, attraction will again occur ; and the same phenomena will happen when electrified sealing-wax, or any other electrified body, is used. Still, though all electrified bodies produce this effect, a slight modifi- cation of the experiment will bring to view a profound difference in the nature of the electrification of different substances. Instead of touching the pith-ball when it is repelled by the glass, let the electrified sealing-wax be presented to it. Strong attraction becomes apparent. Similarly the electrified glass will attract the ball when the sealing-wax repels it. And all substances which can be electrified by friction can be classified according as they act in this respect like glass or like sealing-wax. 308. Positive and Negative Electricity. In order to explain these phenomena we make the following assumptions : 1st. There ,are two ' kinds ' of electricity ; 2nd. Like kinds repel each other, unlike kinds attract each other ; 3rd. The attraction and repulsion diminish as the distance increases ; 4th. An unelectrified body may be' looked upon as a body which contains equal amounts of both kinds of electricity, which can be separated, to a greater or less extent, by means of the action of electrified bodies. Let us distinguish the electricities developed on glass and sealing- wax as positive and negative respectively. When the positively electrified glass rod is brought near to the unelectrified pith-ball, we assert, in terms of our hypothesis, that the neutral electricities in the ball are separated, negative electricity coming to the side near the glass, positive electricity being repelled to the far side. The attraction between the unlike kinds is stronger than the repulsion between the like kinds, for the former are at a less distance apart. Hence the pith-ball moves towards the rod, for the electricity is confined to it and so cannot further alter its distance from the elec- tricity of the rod unless the ball moves. If contact takes place, the negative electricity, which has been induced (as the phrase is) in ELECTROSTATICS. 379 the ball, unites with a portion of the electricity of the rod, so that the ball is now charged with positive electricity, and is therefore repelled from the rod until it loses its charge (say, by repulsion to the ground when the observer touches the ball). The same reasoning, with the interchange of the words positive and negative, applies when sealing-wax is used instead of glass. Finally, the ball which has touched the glass rod is positively electrified, and is, therefore, attracted by the sealing-wax ; and the ball which has touched the sealing-wax is negatively electrified, and so is attracted by the glass rod. All the phenomena are thus ex- plained by means of our assumptions. In the fourth assumption it was stated that an unelectrified body contains equal quantities of both kinds of electricity. In accordance with this assumption, it may be proved, by the methods to be shortly described, that the rubber which is used to produce electricity by friction becomes electrified to exactly the same extent as the rod which is rubbed, but with the opposite kind of electricity to that which is developed on the rod. At one time it was customary to speak of electricity similar to that usually developed on glass as ' vitreous,' and of electricity similar to that which is produced on sealing-wax and other resins as ' resinous,' electricity. The mere fact that the so-called resinous electricity may be obtained from glass is sufficient proof of the undesirability of this classification. The terms positive and negative, as we have employed them above, are much preferable, for the words imply nothing but a distinction in kind. The phrase ' kind of electricity ' is very apt to be misleading. We do not yet know what electricity is. One would never dream of saying that the resultant positive and negative forces which con- stitute a stress are essentially different from each other, or that left-handed (positive) rotation is intrinsically different from right- handed (negative) rotation. Yet equal and opposite forces, and equal and opposite rotations, annul each other's effects. The terms ' positive electricity ' and ' negative electricity ' are merely adopted in order to enable us to consistently and concisely describe and (so far) explain certain phenomena. The use of the old expression ' electric fluid ' is to be carefully avoided. 809. The Gold-leaf Electroscope. An electroscope is an instru- ment which is used to indicate the existence of electrification. If, in addition, the instrument measures the magnitude of the electrifi- cation, it is called an electrometer. The gold-leaf electroscope is one of the most delicate of all electro- 380 A MANUAL OF PHYSICS. scopes. It consists of two pieces of gold-leaf a, a (Fig. 166), which are connected, by means of a metal rod, to a metal head h. The rod passes through the top of a glass vessel in the manner indicated in the diagram. The glass vessel is open at the bottom, and contains a FIG. 166. wire cage which surrounds the gold leaves. The cage can be placed in connection with the ground by means of the metallic con- nection 6. The use of this cage will appear afterwards ( 316) ; in the meantime we are merely concerned with the manner of using the instrument and the nature of its indications. If a positively electrified body be brought into the neighbourhood of the head h, negative electricity is drawn towards the head, and positive electricity is repelled into the leaves, which diverge, since they are similarly electrified. The closer the body is brought to the head, the more widely do the leaves diverge ; and, when the body is withdrawn, they collapse. If, while the leaves are still diverging because of the presence of the electrified body, the head h be momentarily touched by the hand, instant collapse of the leaves will ensue (for the positive electricity escapes from the leaves through the hand to the ground), and the state of collapse will continue so long as the electrified body is not withdrawn. But when the body is withdrawn from the neigh- bourhood of the head, the leaves once more diverge ; for the nega- tive electricity which was drawn to the head of the instrument now spreads in part through the metal rod into the leaves. The latter therefore are diverging with negative electricity. In this condition the instrument can be used to indicate the nature of the electrification of any body which is brought into the neighbourhood of the head h. If the body be negatively electrified, more negative electricity will be repelled into the leaves which will therefore diverge more. If it be positively electrified (or unelec- trified), the negative electricity is drawn from the leaves which then ELECTROSTATICS. 381 collapse; and if the positive electrification be sufficiently strong, some of the neutral electricity in the rod will be separated, and the leaves will diverge because of being positively electrified. The interchange of the words positive and negative in the above reasoning will enable it to apply to the case in which a negatively electrified body is originally brought near to the head of the instru- ment. It is obvious that these experiments are, in large part, merely a modified repetition of those which were discussed in 307. 310. Electrification by Contact and by Induction. Electric Quantity. In 308 we have spoken of the electricity which is in- duced upon a conducting body because of the presence of another electrified body. So long as contact does not take place between the two bodies, the total amount of induced electrification is zero, a certain amount being drawn to one side of the body, while an equal amount of the opposite kind is repelled to the other side. But whenever contact occurs, the attracted electricity unites with some of the electricity in the inducing body ; and so the conductor is electrified with the same kind of electricity as that which the induc- ing body possesses. It is then said to be electrified by contact. The total effect is the same as if the inducing body had given some of its electricity to the conductor, and it is usual to say that it has done so ; for we cannot distinguish one amount of electricity from any other equal amount. In the process of electrification by contact, the one body loses a certain amount of electricity, while the other gains an equal amount. This can be proved by means of measurements of the forces of attraction or repulsion which they exert upon an electrified body the electrification of which does not alter. In fact, we can electric- ally iveigli out equal amounts of electricity, just as we can gravita- tionally weigh out equal amounts of matter. We are therefore justified in speaking of electricity as a thing which can be doled out in measurable quantities ; and it is usual to say that an electrified body is charged with electricity, and to call the total quantity of electricity which it possesses its charge. Suppose, now, that we have a charged body charged positively, let us say. It is possible by its means to charge other bodies, either positively or negatively, to any desired extent and that without any reduction of its own charge. Let A (Fig. 167) be the positively charged body, and let B and C represent two of the other bodies, of wfcich B is to be negatively charged, while C is to be positively charged. Each of the three bodies being well insulated from other conductors,. place B and C in 382 A MANUAL OF PHYSICS. contact in some such position relatively to A as is indicated in the figure. Then let B and C be separated : B will be charged negatively, while C will have a positive charge. Greater effects would be produced, if necessary, by placing B and C at a con- siderable distance apart, and joining them by a thin conducting wire ; for the effect of the charge in A is largely counteracted by FIG. 167. the mutual attraction between the positive and negative elec- tricities in B and C ; and this mutual attraction is diminished as the distance between B and C increases. In practice, it is con- venient to let C be the earth, and to place B in connection with it by means of a metallic wire or other conductor. In this case the repelled positive electricity in C is practically at an infinite distance. We may then use the body B, instead of A, if we wish to charge any other body negatively. This process, in which the inducing body does not lose any of its charge, is called charging by induction. The induced charge is, except in one special case ( 316), less than the inducing charge. 311. Continued Production of Electricity. The Electropliorus. In last section we saw how it is possible to obtain a positive or a negative charge at will by means of a single insulated charged FIG. 168. body and two conductors. The instrument, based on this principle, which is generally used for the purpose, is the electropliorus. It consists of a flat, circular, cake of resin, contained in a shallow metal vessel db (Fig. 169). The resin is slightly warmed, and is electrified negatively by friction with cat's skin. A metal disc cd, insulated by means of a glass handle, is used instead of the con- ELECTROSTATICS. 388 ductor B of last section, while the earth takes the place of the conductor C. As the metal disc is brought near to the electrified resin, positive electricity is induced on its near side and negative electricity is repelled to its far side ; and the more closely the disc is approached to the resin, the greater is the resultant separation of electricity. When the disc is laid upon the resin, contact is made between it and the ground by means of a metal pin which passes through the centre of the cake of resin and is connected with the metal vessel enclosing it, and therefore with the ground upon which the vessel rests. The negative electricity escapes to the ground, and the disc is left charged with positive electricity, the charge being practically equal to that on the resin. The disc may now be lifted away, and positive electricity can be communicated by contact from it to any conductor ; and the process may be repeated from the beginning. Two points in this explanation may present some difficulty. It may appear that the negative electricity of the resin should be destroyed whenever the disc is placed upon the surface of the cake ; for the induced positive electricity in the disc would combine with it, and then the remaining negative charge in the disc would pass to the ground through the metallic connecter. This would really happen if the two surfaces came in contact throughout their whole extent ; but, because of inequalities, they only touch over a com- paratively small area. Again, the negative electricity in the disc is repelled by the electricity on the resin. How, then, can it pass to the ground by a connection which passes through the resin ? We cannot be content with the reply that the disc and the earth are then both parts of one conductor, so that a road is opened up by which the electricity can get to a greater distance from that which repels it. This statement seems very like a statement to the effect that a stone would roll a short distance up one side of a hill, in order that it might get a longer roll down the other side. The fact that electricity flows like an incompressible fluid ( 335) makes an explanation easy. There are three parallel layers of electricity in the apparatus two negative layers with one intermediate positive layer. The lower negative layer tends to draw positive electricity from the ground ; the intermediate positive layer repels it ; and we are left with the upper negative layer which attracts it. We may suppose, therefore, that the negative layer on the upper side of the disc draws positive electricity from the ground and unites with it. The earth is consequently left with an equal negative charge, and the effect cannot be distinguished from 384 A MANUAL OF PHYSICS. that which would have ensued npon an actual passage of the negative electricity of the disc to the ground. We have no means of dis- tinguishing between the two cases, and therefore we are justified in saying that the electricity does pass from the disc to the ground. Returning from this digression, we remark that the production of electricity by means of the electrophorus becomes more and more continuous the more rapidly the various motions of the disc are performed. The principle of all machines used for the produc- tion of a statical charge is the same as that of the electrophorus, but they are so constructed as to give a strictly continuous production. 312. Law of Electric Attraction and Repulsion. We have hitherto explained the various facts which have come under our consideration by the assumption of attractive or repulsive force, which diminishes in intensity as the distance between the attracting or repelling quantities increases ; and we found that the assumption enabled us to give a consistent account of the facts. Therefore, adopting this assumption as a working hypothesis, we must now consider more minutely the exact law of force. The law was elaborately investigated by Coulomb by means of his torsion balance. In this instrument a vertical wire attached to the FIG. 169. torsion-head h (Fig. 169) carries a horizontal insulating arm, at the end of which a small metal disc d is fastened. A scale fastened to the glass cover which surrounds the instru- ment enables us to determine the angular position of the arm ; and the position of the torsion-head, when there is no torsion on the wire, is also noted. Now let a positive charge be given to the disc d, and let another positive charge, contained in a metal ball fixed to ELECTROSTATICS. 385 an insulating handle b, be introduced into the interior through the aperture a in the glass cover of the instrument, the length of the handle being such that the ball and the disc are in one horizontal plane. The mutual repulsion of the two quantities of electricity will cause the arm to twist round through a certain angle. Additional torsion is then put on the wire, by turning the head round, until the disc is brought back to its former position. Now increase the charge in the ball in any ratio and repeat the same series of operations, having previously turned back the torsion- head into its old position. It will be found that the torsion, which must now be put on the wire in order to turn the disc back to its first position, is increased in the same ratio. This proves that the force is proportional to the quantity of electricity in the ball, and therefore, also, that it is proportional to the quantity in the disc. Next, perform a series of experiments in which the charges of both bodies are kept constant, while their mutual distance is varied. The amount of torsion which is requisite at the different distances will show that the force varies inversely as the square of the distance. The same law will be found to hold when the charges are nega- tive, and also when one is negative and the other positive. Of course, in the latter case, the angular rotations of the arm and the torsion-head are necessarily reversed in direction. Let q, q', be the quantities, and let s be the distance between them. The law of force is expressed by where the positive sign corresponds to repulsion and so indicates that the force is attractive (i.e., is in the direction of decreasing dis- tance) when q and q' are of opposite sign, and that it is repulsive (i.e., is positive outwards) when they are of like sign. This is the well-known law of gravitational force, and, therefore all results which we have deduced (Chap. VIII.) regarding that force will at once apply to the case of electric force, provided that we take account of a possible reversal of sign. 813. Electric Potential. Electromotive Force. If we attempt to increase the charge of an insulated conductor by any stated means in which the same conditions are maintained, as in the electro- phorus, we find that it is more and more difficult to do so the farther the process is carried out, and that the charge cannot be increased beyond a certain limit. To make the reason for this clear we must make an apparent digression. 25 386 A MANUAL OF PHYSICS. Electrified systems obviously possess energy in virtue of their electrification, for the mutual attraction or repulsion between their various parts may be used for the production of mechanical work. We define the Mutual Potential Energy of two systems as the amount of work which may be obtained from their mutual repul- sion until they are at an infinite distance apart ; and we define the Potential at any point, due to a given electrical system, as the mutual potential energy between the system and unit quantity of positive electricity placed at that point. This definition makes the sign of the potential coincide with the sign of the electrification of the system to which it is due, and it makes the potential represent potential energy and not exhaustion of potential energy, as in the case of gravitation ( 95). Let V and V be the potentials at two points which are at a dis- tance s apart. The average force which acts so as to transfer the unit of positive electricity from the point which is at potential V to the point which is at potential V is (V - V) /s ; and, if dV is the change of potential in the small distance ds measured from any point, the actual force at that point is _dV ds for our definition makes V decrease as s increases. This quantity is the rate of variation of potential per unit length, and is called the Electromotive Force at the given point, for it is the force which acts so as to transfer electricity. We see therefore that no transference of electricity can occur between two conductors which are at the same potential. Now we have seen that two like quantities of electricity, q and q', situated at a distance s apart, repel each other with a force q'g, ~* and the force with which q acts on a unit of electricity of like sign is ds s' 2 00 This gives (see 35). Now, the potential of a conductor must be uniform throughout if the electricity which it contains is at rest, for otherwise an electro- ELECTROSTATICS. 387 motive force would act from one part of it to another. Hence we are justified in speaking of the potential of a conductor ; and the above expression shows that the potential of a conductor becomes larger and larger in strict proportion to the quantity of electricity which it contains, being positive when the charge is positive, and negative when the charge is negative. And it follows that the work, which must be expended in order to bring up to a conductor a given charge, becomes greater and greater as the charge already contained in the conductor increases. In particular, if the potential of the body which carries up the new charge (the disc of the electrophorus) does not exceed a certain fixed value, the potential of the conductor to which the charge is given cannot be made, by this process, to exceed that fixed limit. Therefore the charge of that conductor cannot be increased above a fixed limit, which is the statement made at the commencement of this section. The potential is zero only at an infinite distance ; but, in order to practically carry a charge to an infinite distance, it is necessary merely to connect the conductor, which contains it, to the ground. For the earth is a conductor which is practically at an infinite dis- tance, and any ordinary charge which is communicated to it pro- duces no sensible variation of its potential. (See next section.) 314. Capacity. Condensers. The quantity of electricity which must be given to a conductor in order that its potential may be raised by unity is called the Capacity of that conductor. It follows at once, from the result of last section, that the capaci- ties of two spherical conductors are in strict proportion to their respective linear dimensions. For, since electrostatic force obeys the same law as gravitational force ( 312) from which ( 88) we know that the action of each quantity is the same as if it were con- densed at the centre so far as points outside the sphere are concerned the potential of a sphere, of radius a, is Hence, when V = l, q = a; that is, the capacity is measured by the radius. The capacity of a compound conductor which consists of two concentric spherical conductors may be made extremely great. Let A and A' (Fig. 170) represent the two surfaces, and let their radii be respectively a and a+r. Let A be charged with a quantity q of positive electricity, and let A' be charged with an equal quantity 252 A MANUAL OF PHYSICS. of negative electricity. These two charges will spread uniformly over the respective surfaces ( 316). Now the potential at any point of A', due to its own charge, is ; and its potential, due to the charge of A, is ql(a + T). FIG. 170. Its total potential is therefore zero ; and so no flow of electricity will occur if it be connected to the ground by a conductor. This proves that if the sphere A be charged in any way, an equal and opposite charge will be induced on A' if it be connected to the ground. Hence, if we add a positive unit of electricity to A, a negative unit will necessarily appear on A' when it is * put to earth ' to use the technical expression. But the work done in increasing the charge q of A by unity is q/a, and the work done in increasing the negative charge q of A' by a negative unit is -qKa+r). Hence the whole 'work done is a a-\-r -* a(a-\-T) When T is very small, this gives q = a 2 wjr ; and, when w is unity, q represents the capacity of the arrangement, which is therefore a quantity which may be made extremely great by sufficiently decreasing T. Any arrangement of this sort is called a Condenser, since it enables us, without much expenditure of work, to store up a large quantity of electricity. The only essential feature in any such arrangement is that the two conducting, and mutually insulated, surfaces shall be extremely close together in comparison with their own dimensions. A common form of the condenser is known as the Ley den jar. This consists (Fig. 171) of a thin glass jar coated externally and internally with tin-foil. The neck and upper portion of the jar are not coated ELECTROSTATICS. 389 with the foil, so that the insulation between the two conducting sheets may be as complete as possible. The mouth of the jar is usually closed with a cork, through which passes a metallic rod, terminating externally in a knob, and making communication in- FIG. 171. ternally with the inner coating of the jar. The external coating can be readily connected to the ground, while the internal coating is charged, through the agency of the rod, by means of any electric machine. Discharge of the jar is effected by means of the discharging-rod, (Fig. 172), which consists of two jointed metal rods connected to a glass handle. The two knobs a and 6 are placed in connection with the outer and inner coatings of the jar ; and the discharge, resulting FIG. 172. in the combination of the two equal and opposite quantities of elec- tricity, takes place through the metallic circuit acb. Great care must be taken in the use of jars charged to a high potential, as very serious, if not fatal, effects might ensue upon their discharge through the human body. Another common form of condenser consists of a pile of sheets of tin-foil separated by paper soaked in paraffin. All the odd sheets in the pile (counting from one end) are connected together. So also are the even sheets, but the odd and the even elements are carefully kept separate. This arrangement constitutes a condenser of great capacity. 390 A MANUAL OF PHYSICS. A practically identical arrangement may be made by joining together all the internal coatings and, independently, all the ex- ternal coatings of a number of Ley den jars. The capacity of the whole is equal to the sum of the capacities of the separate jars. If the jars be connected together in series that is, with the outer coating of one joined to the inner coating of the next in order, and so on the resultant capacity is only equal to the capacity of each jar (all being supposed to be equal in this respect), and the whole charge is equal to the charge of one jar when its coatings are raised to the same (total) difference of potential. When the capacities are not alike, we may let V and V x indicate the potentials of the outside and inside coatings of the first jar, whose capacity in C a , while Vi and V 2 are the similar quantities for the second jar, the capacity of which is C 2 , and so on, V being the potential of the inner coat- ing of the last jar. If C be the total capacity, C(V n -V ) is the total charge. But this is equal to the sum of the separate charges. Hence But the charge of any one jar is necessarily equal to that of any other ; for the outside coating of each has a charge which is equal and opposite to that of the inner coating of the jar to which it is joined, since the two form a single insulated conductor ; and the charges in the two coatings of any one jar are also necessarily equal. Hence we have the n 1 equations In all there are n equations connecting the 2n + l quantities C, d, . . ., C a , V , ____ , V,. If V =0, while the quantities d ---- C n , are known, we can eliminate the Vs and calculate C. 315. Specific Inductive Capacity. In last section we obtained the expression a 2 /r for the capacity of an arrangement consisting of two concentric conducting spherical surfaces of mean radius a and separated by an insulating interval of thickness r. It is customary to consider air as the standard insulating material. If the air be replaced by some other insulating material, such as glass, resin, etc., it is found that the capacity of the same arrange- ment becomes where K is a constant for that particular medium and is called its Specific Inductive Capacity. ELECTROSTATICS. 391 The Specific Inductive Capacity of a substance may therefore be determined by means of measurements of the quantity of electricity which is required to charge a jar, of which the given substance forms the insulating medium, up to a given potential. The ratio of this quantity to the quantity which is required in order to raise the potential of a precisely similar jar to the same extent, when air is the insulator, is the value of the required constant. Faraday determined the Specific Inductive Capacity of various substances by measurements of potential. He used two precisely similar and equal Leyden jars which were so constructed that the insulating medium could be changed when desired. The coatings of one of the two jars were insulated by air ; those of the other were insulated by a substance whose (unknown) inductive capacity K, was to be found. He charged, the air jar to potential V and then divided its charge between itself and the other jar by making connection between their outer coatings and their inner coatings respectively. V being the resultant common potential of the jars, while C is the electrostatic capacity of the air jar, the equation V'C+V'KC=VC expresses the condition that the total quantity of electricity in tha two jars is equal to that originally possessed by the air jar. Hence V-V is found in terms of the known potentials. The value of this constant (sometimes called the Dielectric Con* stant) might also be found by observations of the reduction of the potential of a condensing arrangement, with a given charge, when the layer of air between the oppositely electrified surfaces is partially displaced by a layer of another insulating material. If t be the thickness of the new layer, the effective thickness of air displaced is tjK. If vacuum be taken as the standard insulating medium, the specific inductive capacity of air is 1-00059, according to the deter- minations of Boltzmann. Faraday, with the apparatus at his dis- posal, was unable to observe any difference between the inductive capacities of different gases ; but Boltzmann has shown that small differences really occur. The dielectric constants of solids and liquids are greater than those of gases and differ considerably among themselves. 316. Distribution of Electricity on Conductors. Electric Density. A static charge of electricity, which is communicated to 392 A MANUAL OF PHYSICS. a conductor, is necessarily confined to the surface of that body. For, otherwise, the mutual repulsion between like quantities of electricity would produce continuous currents of electricity in the interior of the conductor. The fact that the electricity is at rest on the surface of the con- ductor, also shows that the distribution must be such as to produce a uniform potential all over the conductor. If the conductor be spherical, we see, from the principle of symmetry, that the surface distribution of electricity is uniform; but if the conductor be not symmetrical, we would infer that the quantity of electricity distributed, per unit of area, over the surface cannot be uniform. This quantity is called the density of the surface distribution. It is large where the curvature of the surface is large, and is small where the curvature is small. In order to determine the density at various parts of any surface (whether conducting or not), we might place a small plane metallic disc, attached to an insulating handle, in contact with the surface. If the curvature of the surface be not too great, and if the disc be sufficiently thin and small, the electricity on the part of the surface covered by the disc will be transferred to the outside surface of the disc, for it is then practically a portion of the electrified body. The disc may then be removed, and the magnitude of its charge may be tested. If the charge be q, while the area is a, the density at that part of the surface is qja. The process may be repeated as often as we please, at different portions of the surface, BO long as the total charge of the conductor is not sensibly diminished ; and this objection might be entirely avoided by taking precautions to have the potential of the conductor maintained con- stant. As this method was first used by Coulomb, the disc is called Coulomb's Proof Plane. In a number of cases the law of distribution of density can be calculated from the known electrostatic laws. Some of these cases will be discussed in the next two sections. In particular, it is known experimentally that a closed conductor, which has an insulated charge placed inside it, and is connected to the ground, will become charged in such a way that no force is exerted at any external point by the internal and superficial charges ; and the reason is that its potential is zero, so that the potential is uniformly zero in all external space. Hence the surface distribu- tion, with its sign changed, acts at all outside points precisely as the internal charge does. Similarly the conductor screens in- ternal bodies from the influence of external charges. 817. Electric Images. We have already seen that a uniform ELECTROSTATICS. 393 distribution of electricity over a spherical surface produces the same effect at external points as an equal quantity of electricity con- densed at its centre would produce. This imaginary quantity is called the electric image of the uniform spherical distribution. [There is a reason for calling this quantity imaginary beyond the mere fact that it does not actually exist, and this is that the volume density of a finite quantity of electricity condensed at a point would necessarily be infinite, and such a condition cannot exist ( 321). J We proceed now to the general discussion of the method of electric images, which is due to Sir W. Thomson, and which gives in many cases simple solutions of very formidable problems. Draw a sphere (Fig. 173) with radius CM = &, and divide CM ex- ternally and internally in the points A, A' repectively, so that FIG. 173. CA . CA' = a 2 . Take any point P in the circumference of the sphere, and join PA, PA', PC. Let PA = r, PAW, CA = d. We know by geometry that the triangles CAT, CPA are similar. Therefore CA' Now put and we get CA' e'= -e a 394 A MANUAL OF PHYSICS. But if we draw A'Q perpendicular to CA, we have QA perpen- dicular to CQ, and therefore CA'/a =CA'/CQ = CQ/CA=a/d. Hence a e'= e-. d If a quantity e of electricity be placed at A, while a quantity e' is placed at A', the condition e'/r' -\-e\r =0 asserts that the potential at the point P, due to these charges, is zero ; and therefore the sphere is a surface of zero potential. Consequently, if the sphere be an infinitely thin conductor, and be connected to the ground, the action of the charge e placed at A will induce upon it a charge, the effect of which at external points will be the same as that of e' placed at A' ; and the charge e' placed at A' will induce on the sphere a charge, the effect of which at in- ternal points is the same as that of e placed at A. And further, these two induced charges are precisely equal and similarly dis- tributed, but of opposite sign ; for. when the sphere is uncharged and insulated, and the charges e and e' are placed at A and A' respect- ively, the potential of the sphere is zero, and no resultant charge is induced upon it, neither can electricity flow from any one part of it to any other part. [To find the law of distribution of density which produces this effect, produce AP to K, describe a circle round the points E, A', and A, and let A'P meet this circle in S. The triangles EPS and A'PA are similar. Also PR and PS are proportional to the forces, / and /', at P due to e and e' respectively. For these forces are e/r 2 and e'/r' 2 respectively, and e\r -e'/r'. Hence the forces are inversely proportional to r and r', and therefore are directly pro- portional to PE and PS, by construction. Consequently, the result- ant force F at P is proportional to SE, and so SE/SP = F//' = AA'/r. AA' AA'e 2 1 AA'e' 2 1 This gives F = /' = --s = -7,- r J e ' r* e r' A Hence a sphere, the density of which is inversely as the cube of the distance from an external (or internal) point, attracts internal (or external) matter, as if it were condensed at that point.] The point A' is the image of A with respect to the given sphere, and by means of the relation CA . CA' a 2 , we can find the image of any distribution of electricity. For example, let there be a given distribution of electricity along the straight line Aj A^ (Fig. 174). The image of the point Aj in a sphere, of radius a, the centre of which is at C, is A'j, which is such ELECTROSTATICS. 395 that CAi . CA f ! = a 2 . Similarly, CA' 2 =A' 2 =a 2 , and so on. It is easy to prove that the line A'iA'a is a portion of circle. But if FIG. 174. AiA 2 is very small A^A'g is practically straight, and is inclined to AiC at an angle equal to the angle of inclination between CjA and AjAo. Let CAj = r, CA'^r', A 1 A a =Z, A'jA'^Z', and we get, as the ratio of an infinitely small length in the image to an infinitely small length in the direct system, Similarly, ' and a representing areas, v' and v representing volumes, ^_^!_ r ' 4 . !_^L_?!l! r 4 a, 4 ' v r 6 a 6 Next let X, which gives V - V = t A / F . V a [We might have deduced this result from the expression for the energy contained in unit volume of the dielectric ( 320), which is STT 8?r t 2 Hence the total energy contained in the volume at between the discs is 1 (V-V) 2 - , Ct* STT t And the rate at which this varies per unit of thickness gives the force In Thomson's absolute electrometer, in which this principle is used, a concentric circular portion of the upper disc is alone moveable, and so the difficulty of the non-uniformity of the force at the edge of the ELECTROSTATICS. 409 discs is avoided. The moveable part as nearly as possible fills the aperture without touching its sides, and it is suspended by means of a delicate (spring) balance, which has a fiducial mark by means of which the lower face of the movable disc can always be placed in one plane with the lower face of the surrounding portion of the upper disc called the guard-ring. The lower disc can be accurately moved, perpendicular to its own plane, through known distances, by means of a screw. The balance and disc are surrounded by a metal case for the purpose of preventing any disturbance which might arise from external electrification. When the two discs are connected to bodies of different potential, the balance is depressed, and the screw is turned until it returns to its standard position. In this way the distance t is determined. Also, by previous experi- ments, it is known what weight must be placed on the disc in order to bring it to its standard position, and this gives the (constant) value of F. In using the instrument it is preferable to keep its lower disc at a constant potential by means of a charged condenser the constancy being determined by means of a secondary electrometer. The value of t is then found, first, when the upper disc is connected to the ground, and again, when it is connected to the body whose potential is to be determined. If 9 be the difference of the distances, the above equation gives us the value of the potential - /87TF -v -^-' a being now the area of the attracted disc. In Thomson's quadrant electrometer an aluminium needle swings inside a hollow metal cylinder, which is divided into four quadrants. FIG. 175. The two opposite pairs of quadrants (Fig. 175) are connected by wires. The needle in its normal position is suspended with 410 A MANUAL OF PHYSICS. length directed along one of the lines of division between the quadrants, and it is charged to a high positive potential. One pair of quadrants, say those connected to the wire a, may be connected to the ground, while the other pair is connected by the wire 6 to a body at positive potential V. The quadrants connected with b become positively charged, and the quadrants connected with a become negatively charged. The needle is therefore deflected towards the negatively charged quadrants, and the deflecting couple is proportional to V, if the potential of the needle be sufficiently high. Modifications of this instrument may be used for the measure- ment of extremely small, and of extremely large, differences of potential. 326. Electric Machines. The electrophorus, which is the simplest form of electrical machine, has been already described. As an example of the older machines, used for the continuous production of electricity, we shall take the cylinder machine. This machine consists of a glass cylinder C (Fig. 176), which is turned FIG. 176. round in the direction AmB. An insulated leather rubber A, coated with zinc amalgam, presses against the rotating cylinder, and causes the development of positive electricity on the glass, while it becomes itself negatively electrified. The positive electricity is carried round on the surface of the glass until it reaches the sharp metal points p, which project from the insulated metallic conductor B. It induces in the points negative electricity which is discharged on the surface of the glass, destroying its electrification, and leaving B positively electrified. A silk flap m, connected to the rubber and resting in contact with the upper portion of the cylinder, prevents the electri- fication from slipping back along the surface of the glass. The potential of the positive electricity rises rapidly as it is carried from A to B, and the resulting electromotive force may cause the ELECTROSTATICS. 411 electricity to slip back, so that the potential of B cannot rise very high. The silk flap becomes negatively electrified and so prevents the slipping, or stops it if it does occur by using the slipping elec- tricity to destroy its own negative electrification, which tends to equalise the potential. In the plate machine the glass cylinder is replaced by a circular glass plate, which is rubbed on both sides. The Holtz machine is one of the best modern forms of electric machines. A fixed glass disc D (Fig. 177) has two paper armatures (a, a'} fixed on it near the opposite extremities of a diameter. Near each armature an opening (shown by dotted lines) is cut in the glass, through which a paper point attached to the armature pro- jects so as nearly to come in contact with a revolving glass disc C, JL FIG. 177. which is mounted upon an axis passing through the centre of D. A metal conductor ra, fitted with a row of sharp points, faces the revolving disc on the other side opposite the armature a. A similar conductor n faces the armature a', and can be placed in communi- cation with m by means of the rod Z, which slides through the knob m. In order to work the machine, the knobs n and m are placed in connection, and a charge (positive, say) is given to the armature a by means of the electrophorus or otherwise, while the disc is rotated in the direction aCa'. After a short time a rustling sound is heard, and the machine becomes difficult to drive. It is producing elec- 412 A MANUAL OF PHYSICS. tricity, and the extra work which has to be performed is the equiva- lent of the electrical energy which is developed. We may explain the process of charging in the following manner : The positive charge given to a induces negative electrification in the points of the conductor m. This is discharged upon the glass sur- face, and the glass carries it round to the opposite side of the machine, leaving m positively charged. Here it induces positive electricity in the sharp point of the armature a', and this is dis- charged upon the inside of the glass disc, leaving a' negatively charged. The armature a' now draws positive electricity to the points of the conductor n. This electricity is discharged upon the outer surface of the disc, leaving the conductor n negatively charged. Thus we may regard the glass disc as constantly carrying positive electricity from n to m in the one half of its revolution, and as constantly carrying negative electricity from m to n in the other half of its revolution. It is possible that the negative charge is given to a' by way of the conductor n. The positive electricity, which is produced in m by the first motion of the machine, and flows to n, may be supposed to act inductively on the armature a', drawing negative electricity to the body of it, and repelling positive electricity to the sharp point to be discharged upon the glass, leaving a' negatively charged. When the conductors n and m are slightly separated, a brush discharge passes through the air space. This brush discharge may bo changed into a violent spark discharge by connecting the inner coatings of two Leyden jars to the conductors n and m, the outer coatings of the jars being joined together. The jars have to be charged up to the potential required for discharge through the air space, and, as their capacities are large, a great quantity of elec- tricity passes at each discharge. CHAPTEE XXVIII. THERMO-ELECTRICITY. 327. Thermo-electric Phenomena. Though the principle of con- servation of energy shows ( 324) that the sum of the electromotive force in a closed metallic circuit must be zero, provided that there be no difference of temperature between the various parts of the circuit, we cannot assert that their sum will be zero if the tempera- ture be not uniform. For, in the process of equalisation of tempera- ture, it is possible that there may be transformation of thermal energy into electric energy ; and this transformation will occur if the electromotive forces of contact between the metals which form the circuit depend upon the temperature. Now, Seebeck discovered in 1822 that, in general, a current of electricity flows around a circuit, which is composed of two different metals, if there be a difference of temperature between the two junctions of the metals ; and this shows that the equilibrium of the contact forces has been destroyed because of the variation of tem- perature. And we cannot assert, without experimental evidence, that there will be no resultant electromotive force in a closed circuit composed of a single metal which varies in temperature from point to point. But the experiments of Magnus showed that no such force exists. Still, in order that Magnus's result should hold, it is necessary that every part of the circuit should be physically similar and not merely chemically similar. For example, two portions of the same metallic substance, which are in a different state of strain, are physically different substances, and are also thermo- electrically distinct. And two portions of the same substance which are at finitely different temperatures are in different physical states, and might, therefore, exhibit thermo-electric phenomena. Such phe- nomena were observed by Le Roux and others at the instant when 414 A MANUAL OF PHYSICS. contact was made between two portions of the same metal which differed abruptly in temperature. 328. Laws of Thermo-electric Circuits. It is found by experi- ment that the introduction of a piece of metal into a thermo- electric circuit does not contribute to the electromotive force of that circuit, provided that the extremities of the metal are at one and the same temperature. We 'may, therefore, use solder to connect together the various parts of the circuit. Let the lines A (Fig. 178) represent two pieces of the same metal. Let two of their ends have a common temperature 1} while the other two have a common temperature t a ; and let the ends which are at the temperature ^ be joined by a metal C, while the ends which are at the temperature t a are joined by a metal B. As the whole arrangement is symmetrical with respect to the pieces A, FIG. 178. it is obvious that there can be no resultant electromotive force in the circuit. And if the temperature of one of the junctions between C and A be changed from ^ to t. 2 , the metal B will still, from its symmetrical position, contribute nothing to the electromotive force, although there may now be a resultant electromotive force in the circuit. Next let us arrange pieces of two metals alternately, as in Fig. 179, and let the temperatures of their extremities be as indi- cated. The pieces B, which are at temperatures ^ and t. 2 respect- ively, contribute nothing to the total effect, so that the whole arrangement really consists of two metals (A and B), the two junctions of which are t 3 and t respectively. Now we may join the pieces ^B^ and 2 B 2 respectively to those points of the piece t.jBt 01 which are at the temperatures t and t. 2 , by means of con- nections made of the metal B ; and these new connections contribute nothing to the total electromotive force in the circuit. But the electromotive force in the part BB^A^B is due to the metals B and A when the temperatures of the junctions are ^ and t . Similarly, the force in the part BB^A&jB is due to A and B, with temperatures t : and t. 2 at the junctions ; and that in the part BB 2 A 3 is due to A THERMO-ELECTRICITY. 415 and B with temperature t. 2 and t. A at the junctions. Hence the elec- tromotive force, due to temperatures t. A and t , is equal to the alge- braic sum of the electromotive forces due to temperatures t 3 and t , t. 2 and ti, 3 and t. 2 , respectively. fc B FIG. 179. This result is~quite general, and, therefore, the algebraic sum of the various electromotive forces in a compound circuit, which is composed'^of a number of pieces of two metals ivith their junctions at various temperatures, t and 1? t and &>, . . . . t n -\ and tm is equal to the electromotive force in a tivo functioned circuit of the same metals with its junctions at the extreme temperatures t and t n . We can thus obtain a comparatively large electromotive force by means of a small' difference of temperature. This is the essential principle of the Thermopile, an instrument which, in conjunction with a galvanometer, is used for the measurement of small differ- ences of temperature. Lastly, arrange four metal wires, A, B, C, and B in the manner FIG. 180. indicated in Fig. 180, and let the one wire B be raised to tempera- ture t while the other is kept at temperature t . The wires B merely serve as junctions, and so the total electromotive force is due to a circuit composed of A and C with the junctions at tempera- tures h and t respectively. But we may join the two wires B by third wire of the same metal without altering the distribution of electromotive force in the circuit. And in the circuit BjA BB the 416 A MANUAL OF PHYSICS. electromotive force is due to A and B with junctions at temperatures 1 and t ; while in the circuit B^C^BB the electromotive force is due to B and C, with junctions at the same temperatures. If we define the thermo-electric power of a circuit of two metals as the rate at which the electromotive force in that circuit varies per unit of difference in temperature between the junctions, we see that the result which we have just obtained shows that at every temperature the thermo-electric power of A and C is the algebraic sum of the thermo-electric powers of A. and B, and B and C. 329. Variation of the Electromotive Force with Temperature. Soon after Seebeck's discovery Gumming observed that in certain circuits (such as that of iron and copper), while one junction is main- tained at a constant ordinary temperature and the other is gradually raised in temperature, the electromotive force gradually increases to a maximum, then diminishes, vanishes, and finally is reversed. The law of variation of the electromotive force has been very fully investigated by Gaugain and others. They found that, with most pairs of metals, the curve which is obtained by plotting difference of temperature along the (horizontal) axis of a?, and electromotive force along the (vertical) axis of y (Fig. 181) is in general a parabola with its axis vertical. Therefore, if we denote the electromotive force by e, and the temperature by t and let E and T respectively represent the electromotive force and the temperature which cor- respond co the vertex of the parabola, we obtain E-e = &(T-) 2 (1) where 6 is a constant ; for this equation merely expresses the well- known property of the parabola, that the square of the distance of THERMO-ELECTRICITY. 417 any point on it from the axis is proportional to the distance of that point from the tangent at the vertex. In particular cases the curve is a straight line ; in others, it is made up of portions of parabolas with parallel (vertical) axes, but with their vertices alternately turned in opposite directions. 330. The Thermo-electric Diagram. Now another well-known property of the parabola is that the rate of increase of the ordinate at any point per unit of increase of the abscissa is proportional to the value of the abscissa measured from the axis. The proof of this is simple, for (1) gives (2), which is the direct expression of the above statement. If, therefore, we plot the value of dejdt (which is the thermo- electric power) against difference of temperature, we shall obtain, instead of a parabola, a straight line. And we inay form a self- consistent diagram of such lines for any number of metals by means of observations on circuits consisting of each of these, in turn, with some standard metal whose line is made to coincide with the axis of temperature. (In the true diagram the lines might, of course, be T T! T 2 FIG. 182. curves obtained from these straight lines by a process of shearing.) We see, by (2), that these various lines will intersect the axis of temperature at points which correspond to the temperatures of the maximum ordinates in the original diagram (Fig. 181). And, further, the point of intersection of any pair of lines in the diagram, indicates the temperature, T, at which the electromotive force in a circuit of the two corresponding metals attains a maximum value. This diagram is called the thermo-electric diagram. Sir W. Thomson first suggested the construction of such a diagram. The actual construction of it, upon the assumption (suggested by 27 418 A MANUAL OF PHYSICS. theoretical considerations) that the curves which represent the thermo- electric powers of the metals are, in general, straight lines, is due to Tait. The diagram on page 422 is reduced from his results. The area included between two temperature lines and the lines which represent the thermo-electric position of any pair of metals represents the electromotive force in a circuit of these metals when the junctions are kept at the two given temperatures. This follows (see 34) from the way in which the diagram of lines has been deduced from the diagram of parabolas. 331. The Peltier Effect. A thermo-electric circuit forms a system which is in stable equilibrium. For, if it were in unstable equi- librium, increase of temperature of one of the junctions would produce effects which would still further increase the temperature. But we know that the application of heat to one junction produces a current of electricity which flows in a certain direction across that junction. Therefore by the principle of stable equilibrium ( 15), we can assert that the passage of electricity in the given direction will cool the junction. The current at the hot junction always flows from the metal which has the lower thermo-electric power to the metal which has the higher thermo-electric power. Conversely, heat is absorbed at a junction where a current flows in this direction, or is evolved at a junction where the current flows in the reverse direction. Peltier discovered this by direct experiments made without reference to any theoretical considerations ; and so the phenomenon of the absorption or disengagement of heat at a junction across which electricity flows is known as the Peltier Effect at that junction. The total Peltier effect in any circuit vanishes when the two junctions are at the same temperature, for the absorption of heat at one of the junctions is equal to the disengagement of heat at the other. S32. The Thomson Effect. In order to explain the fact that, in auch circuits as iron-copper, the direction of the electromotive force changes when the hot junction is sufficiently raised in temperature, Thomson assumed that the Peltier effect vanishes at that tempera- ture at which the electromotive force reaches its maximum value, that is, at the temperature at which the lines of the metals intersect in the thermo-electric diagram. The metals are then said to be neutral to each other, and so this temperature is called the Neutral Temperature. Now no heat is being absorbed or developed at the junction which is at the neutral temperature, and heat is being developed at the cold junction, for there the current is flowing from the metal of higher thermo-electric power to the metal of lower thermo-electric THERMO-ELECTRICITY. 419 power. It would seem, therefore, that there is no source of thermal energy in the circuit by the transformation of which we can account for the development of electrical energy. But there is no other possible source of the electrical energy, and hence Thomson was led to predict that heat is absorbed at parts of the circuit other than the junctions, either in that metal in which the current flows from hot parts to cold parts, or in that metal in which it flows from cold parts to hot parts, or in both metals. And he subsequently verified his prediction by direct experiment. In copper, heat is absorbed when the current, is passing from cold parts to hot parts ; in iron, it is absorbed when the current is passing from hot parts to cold parts. [It is assumed here that we know the direction in which a current is flowing. The convention by which this is determined will be stated in next chapter.] Now, in a tube through which a liquid is flowing, heat is absorbed by the liquid when it passes from cold parts to hot parts. Hence, in copper and similar metals, electricity acts as an ordinary fluid would do ; and so Thomson speaks of the specific heat of electricity. It is positive in copper and similar metals, and is negative (at ordinary temperatures, at least) in iron and similar metals. 333. Further Discussion of the Thermo-electric Diagram. The Peltier and the Thomson effects can be readily represented on the thermo-electric diagram. Let e l represent the electromotive force in a certain circuit gom- posed of some metal together with the standard one. T, being the neutral temperature, we get, at temperatures t and t' respectively, while the temperature t of the cold junction remains constant whence Similarly, if we use any other metal giving the electromotive force e. 2 with the standard metal under the same conditions . . (4). Hence the electromotive force in a circuit of the two given metals under the same conditions as to temperature is 272 420 A MANUAL OF PHYSICS. Now (2) shows us that 26 X and 26 2 are the tangents of the angles at which the lines of the metals meet the line of the standard metal in the thermo-electric diagram. So, if we let t represent the absolute zero of temperature, while all the other temperatures are given in absolute units, we get qt = 26^! and pt = 26 2 T 2 and so (Fig. 183) qp = 2(6 1 T 1 - 6 2 T 2 ). But qp = 2T(6 X - & 2 ), whence b^ - 6 2 T 2 = Tfa -6 2 ), - -~, .... (5), and therefore tf -=2(6 1 - which is the general expression for the electromotive force in a circuit of two metals in terms of the inclinations of their lines to the line of the standard metal, of their neutral temperature, and of the temperatures of the junctions. Now (5) may be put in the form ^=2(6 1 -6 2 )(T-e')(^-^) + (&i-& 2 )(^'-e) 2 .... (6) The first term on the right-hand side of (6) vanishes when t' = t and also when the temperature of the hot junction is at the neutral point. It therefore represents the part of the electromotive force which corresponds to the Peltier effect. Therefore, if there is no other effect than the Peltier effect and the Thomson effect, the second term must represent the part of the electromotive force which corresponds to the Thomson effect. If we suppose that unit quantity of electricity is transferred along the circuit under the electromotive force +9 where q lt etc., are the strengths of the several parts, we have which proves the statement for longitudinal division. But it is obvious that the statement is true whatever be the forms of the parts into which the magnet is divided, for each part may be supposed to be built up of an infinitely great number of infinitely small rectangular portions. And it follows from this consideration also that the proposition is true whatever be the form of the original magnet. The quantity I, the pole-strength per unit of area, is called the Intensity of Magnetisation of the given magnet. It is evident that we may regard it as being the magnetic moment per unit of volume. Let us imagine a cylindrical crevasse to be cut out in the interior of a uniformly magnetised body. Let it be bounded by plane surfaces perpendicular to the direction of magnetisation ; and, while all its dimensions are infinitely small, let the perpendicular distance between these planes be infinitely smaller than the transverse dimensions. The surface density of magnetism on the plane faces of the cavity is 3E, north magnetism being distributed on the plane face next the south pole of the magnet, while south magnetism is distributed on the other ; and hence the force in the space between the planes is 47rH ( 99). We may therefore suppose that 47r$ lines of force are drawn per unit of area across this cavity in the direc- tion of magnetisation (that is, from the south pole to the north pole within the substance of the magnet). It is customary to call these lines the Lines of Magnetisation. The lines of magnetisation do not constitute all the lines of force in the interior of the magnet. There may be lines of force due to external magnetisation. This distribution of force must be investigated in precisely the same manner as that in which we investigate the distribution of force outside a magnet. Let us suppose that the cylindrical cavity is infinitely long in comparison with its cross dimensions. The magnetisation at the ends of this cavity exerts no effect upon a point at its centre, and hence any force found at the centre must be due to external magne- tisation. This quantity is denoted by the symbol jfy. and is called the magnetic force at the point. The total force, IB, is called the 448 A MANUAL OF PHYSItS. Magnetic Induction at the given point in the magnet, and is equal to It is usual to call the total lines of force inside a magnet, Lines of Induction. They consist, therefore, partly of lines of magnetisa- tion, and partly of the lines of force within the uncut magnet. They are continuous with the lines of force external to the magnet. [It must be remembered that the three quantities 13, fi, and 1|, are vector quantities, and are therefore subject to the laws of vector addition ( 40). In most practical cases, however, IE and 1$ are either similarly or oppositely directed.] The force due to the magnetism at the surface of the magnetised body is included in the quantity H?, and is obviously directed oppositely to E since it acts in the direction of a line drawn from the north pole to the south pole through the material of the body. It therefore acts so as to demagnetise the body, and has its greatest value 27rl at points close to the ends of the magnet (Compare 99). [To obviate demagnetisation, bar magnets, when not in use, are placed parallel to each other with their like poles oppositely directed, and ' keepers ' made of a magnetic metal (soft iron preferably) are placed in contact with their ends (Fig. 195). A closed magnetic S N S N S NS N FIG. 195. circuit is thus formed, for the effect of the magnetism at the poles is annulled by that of the magnetism which is induced in the keepers.] It appears, therefore, that the form of a magnet must have an effect upon the distribution of magnetisation throughout its interior. Thus, while in a long rod placed parallel to the direction of the force in a uniform field, the magnetisation is sensibly uniform except near the ends, in a short rod the magnetisation is very far from being uniform. The introduction of a para-magnetic body into a previously uniform field of force disturbs the uniformity of the field. The lines of force, which were originally parallel and equidistant, close in upon the body and become continuous with the lines of induction in its interior. MAGNETISM. 449 352. Permeability and Susceptibility. That property of a sub- stance in virtue of which the lines of induction are more or less closely arranged than are the lines of force in the originally undis- turbed field is called the Permeability .of the substance. We have ^E+l i **** which may be written in the form In this equation, the quantity ft represents the permeability, and & represents the Susceptibility. The permeability is therefore the ratio of the induction to the force within the substance of the magnetic body, while the susceptibility is the ratio of the magnetisa- tion to the magnetising force, and is a measure of the readiness of the body to acquire magnetisation. In a paramagnetic substance, as we have already seen, the lines of induction are more closely arranged than are the lines of force. That is to say, the permeability of such a substance is greater than unity ; and therefore the susceptibility is positive. On the other hand, in a diamagnetic substance, the lines of induction are less closely arranged than are the lines of force ; and so the permeability FIG. 196. is less than unity, and the susceptibility is negative. In a para- magnetic body, north magnetism is manifested at that extremity which faces in the direction in which the external lines of force are drawn; in a diamagnetic body, north magnetism appears at the opposite end. Consequently, while a paramagnetic substance is attracted towards the pole of a magnet, a diamagnetic substance is repelled from it. In Fig. 196, the body marked p is paramagnetic, the body marked d is diamagnetic. More generally ; a paramagnetic substance moves from weak parts to strong parts of a field of force, while a diamagnetic sub- stance moves from strong parts to weak parts. 29 450 A MANUAL OF PHYSICS. 353. Residual Magnetism. Betentiveness. Coercive Force. We have already seen that some substances, such as steel, retain to a considerable extent their state of magnetisation after the magne- tising force is removed. The property in virtue of which this occurs is termed Betentiveness. The magnetism which remains, because of retentiveness, is called Besidual Magnetism. From its great retentiveness, hard steel is employed in the construction of so-called permanent magnets. The residual magnetism of a long bar of steel is more permanent than is that of a short bar, for the self -demagnetising force ( 351) has less influence in the former case than it has in the latter. Betentiveness has very different values in different materials. It is relatively small in good specimens of soft iron. In order to get rid of residual magnetism in any substance, we must either heat the substance to redness, or employ a reverse magnetising force. It is usual, therefore, to speak of a Coercive Force as existent in the material in virtue of which residual magnetism is retained. 354. Belation connecting Magnetisation and Magnetising Force. If the magnitudes of any two of the quantities IS, $, and ^), are determined in any particular case, the value of the remaining quantity can be calculated from the relation 13 = 47r!-}-cf)' Methods for the determination of each of the three quantities will be de- scribed subsequently. FIG. 197. Fig. 197 represents the usual course of the variation of intensity with magnetising force. The force is measured along the axis O;ij, while the intensity of magnetisation is measured along the axis O1E. MAGNETISM. 451 At first, while the force is small, the magnetisation increases very slowly, and at a sensibly uniform rate. Then, as the force is increased, the law becomes $. = 0$ -\-bffi, a and 6 being constants. After this, a very slight increase of the magnetising force produces a great change in the magnetisation. With still larger forces, the rate of variation becomes rapidly smaller, and ultimately the magnetisation becomes sensibly constant. These various stages in the process of magnetisation are represented by the parts OA, AB, and BC, of the curve. If the force be now gradually removed, the magnetisation will diminish at a relatively slow rate, until, when the force is entirely removed, a considerable amount of residual magnetisation remains. This is represented by OD. If a reverse force be now applied, the magnetisation will fall off rapidly in magnitude, and will disappear entirely when the reverse force has a definite value OE. This may be supposed to represent, as Hopkinson suggests, the coercive force. If the reverse force be now increased until it reaches a value equal to the maximum value of the direct force, if it be then diminished to zero, and if, finally, positive force be reapplied until the original maximum value is attained, the magnetisation will pass through successive values represented by the part EC'D'E'C. The curve OR represents the residual magnetisation which is left after various magnetising forces have been applied and removed. The dotted curve represents the change which takes place in the magnetisation when the same substance (say a soft iron wire) is hardened by being stretched beyond its limits of elasticity. The maximum magnetisation is lessened. The residual magnetisation is also lessened, but the coercive force is increased. Since the magnetisation practically reaches a maximum value when the force is sufficiently great, the substance is then said to be saturated. However much the force may be further increased, the intensity remains appreciably constant. The susceptibility increases from a small value to a maximum which is indicated by the tangent drawn from to the curve OBC. Thereafter it diminishes to zero as the force increases without limit. The relation ju = 47r&+l shows that the permeability also increases from a small value to a maximum (attained at a somewhat greater value of 35 than that at which the maximum susceptibility is reached), after which it gradually diminishes to unity as the force is indefinitely increased. When soft iron is magnetised (more especially when the force is feeble and, the specimen of iron is large) it is found that the magnetisation takes some time to attain its full value corresponding 292 452 A MANUAL OF PHYSICS. to the force which is acting. This effect is, by analogy, said to be due to Magnetic Viscosity. 355. Hysteresis. We see from Fig. 197, that the changes of magnetisation tend to lag behind the changes of force which give rise to them. Thus, when the stage C has been reached, a much greater change in the value of the force is requisite in order to effect a given diminution in the magnetisation than was requisite for the production of an equal increase of magnetisation just before the stage C was reached. A similar effect is observable at the point C'. Ewing has called this tendency Magnetic Hysteresis. As the result of hysteresis, different values of the magnetisation may correspond to one given value of the magnetic force, and we must therefore limit our definitions of permeability and suscepti- bility to the case of a substance which is originally unmagnetised, and which is subjected to a force which increases in magnitude con- tinuously from zero upwards. If E represent the magnetic energy of the magnetised body, the increase of energy which accompanies an increase of intensity of magnetisation d$. is - -Ps- -L II H-M <-ss +P <- -L where P may be translated ' increase of tension.' 456 A MANUAL OF PHYSICS. [It is important to observe that, when we regard only changes of magnetisation and of pressure (or tension), we are dealing with energy flowing from an external system into the iron ; and that, when we regard only changes of pressure (or tension) and of length, we are dealing with energy flowing from the iron rod to another external system ; while, when we regard changes of magnetisation and of length alone, we are dealing with energy flowing through the rod from one external system to another, and being transformed through its agency from one form to another.] Joule proved that no observable change of volume takes place when a rod of iron is magnetised, and therefore that longitudinal magnetisation in weak fields must cause a diminution of the sec- tional area of the rod. Hence he concluded that if a rod of iron be magnetised circularly, that is, if the lines of magnetisation be circles surrounding the axis of the rod, longitudinal contraction will ensue. He verified this conclusion by experiment. Torsional strain, also, is accompanied by variations in the magnetic qualities of iron, nickel, and cobalt rods. These effects can, as Sir W. Thomson has shown, be deduced from the known effects of longitudinal stress upon the magnetic qualities. Thus it is known that the susceptibility of iron in weak fields is increased along lines FIG. 199. of traction, and is decreased along lines of compression. But, when a circular rod of iron is twisted in the manner which is indicated by the arrows in Fig. 199, all lines such as a a 1 suffer traction, while all lines such as b b' suffer compression. Hence the susceptibility is increased along a a' and is diminished along b b'. The effect of this is practically to produce two components of magnetisation one longitudinal, the other circular when the twist is sufficiently great. MAGNETISM. 457 Hence torsion in weak fields diminishes the longitudinal susceptibility of iron. Conversely, a circularly magnetised iron rod, when twisted, becomes longitudinally magnetised. It is easy to deduce the cor- responding effects in nickel and cobalt. [No reversal of the direc- tion of longitudinal magnetisation takes place in iron, however strong the circular magnetisation may be. Ewing explains this by the fact that the intensity of magnetisation in the direction of the line of traction, or of compression, never reaches the point at which the Villari reversal occurs.] Since torsional stress produces circular magnetisation in a longi- tudinally magnetised rod, and since it also gives rise to longi- tudinal magnetisation in a circularly magnetised rod, we might expect that the superposition of longitudinal and circular magneti- sations would cause torsional strain. This effect was discovered experimentally by Wiedemann in the case of iron. The twist in weak fields takes place in such a direction as to be completely explainable by the increase of length which occurs in the direc- tion of resultant magnetisation. Knott has shown that the twist occurs in the reverse way in nickel a result which he expected to find, since nickel contracts in the direction of magnetisation. We may observe here that the twisting of a magnetised' rod, or the magnetisation of a twisted rod, gives rise to a transient electric current in the magnetised material. This effect will be considered in next chapter. 358. Magnetometric Measurements. The magnetometer consists essentially of a small magnet, which is suspended by a long fine fibre, whose co-efficient of torsion is negligeable in'most cases, and which is free to turn about that fibre as an axis. A small mirror is usually attached to the magnet, so that, by means of a reflected beam of light, very small angular motions of the magnet may be made evident. This apparatus is placed in a uniform field of force of known intensity, say the earth's field ( 359). The magnet then places its length in the direction of the force in the given field. Let the magnet be placed at P (Fig. 200), and let the direction of the controlling force be PQ. Let AB be a bar magnet, the intensity of magnetisation of which we have to determine, and let the points A and B represent the position of its poles. Place it symmetrically with regard to PQ in the position which is indicated in the figure. Let I be its (unknown) intensity of magnetisation, while a is its sectional area. Then la is the strength of its pole. The effect of the north pole, A, at P is in the direction AP, and is equal to 458 A MANUAL OF PHYSICS. I&/AP 2 . Similarly, the directive force of the south pole, B, at P is in the direction PB, and is equal to Ia/PB 2 . If we represent these forces by AP and PB respectively, it is obvious that the re- sultant effect of the two is represented on the same scale by AB. The magnitude of the resultant is therefore IaAB/AP 3 , and acts so as to place the little magnet at P parallel to AB, with its poles facing oppositely to those of AB. B F G. 200. Now let PR represent this force on the same scale that PQ repre- sents the force of the external field. PS is the resultant of these forces, and the little magnet sets itself in the direction PS, making an angle 9 with PQ, such that tan = PR/PQ. If F be the intensity of the external "force, this gives (1) from which we can calculate I. In order to find the value of the force F, if it be unknown, we may set the magnet AB oscillating under the action of F alone. The time, T, of a small oscillation is given by the equation T2~~ ~ K ' where Kis the moment of inertia of the magnet about its axis of suspension, and AB is its length. For if the direction of the force F be denoted by the arrow (Fig. 201), and if ns represent the magnet inclined at an angle 9 to the direction of the force, FS in 9 is the force which is acting perpendicularly to the length of the magnet, and which tends to turn it around its axis of suspension. This force acts at each end of the magnet so as to produce rotation MAGNETISM. 459 in the positive direction, and the turning moment is therefore F sin 9 laAB. When the angle is small, this becomes F0I&AB. The angular acceleration is ( 42, 45), and the momentum which is porduced per unit of time is m0K, where m is the mass of the magnet / s n> F FIG. 201. and E is its radius of gyration ( 70). Hence the moment of momentum which is produced per unit of time is ra0B 2 =K0, where K is the moment of inertia ( 70). We therefore have the minus sign being used, since the angular acceleration is nega- tive. Now every quantity in this equation is constant, with the exception of ; and so the equation expresses the fact that the angular acceleration is negative and is proportional to the displace- ment. The small oscillations of the magnet, therefore, obey the simple harmonic law, and the angular position of the magnet is given ( 51) by the equation Pcos where" P and Q are constants and t is the time ; whence T being the periodic time, we get ( 51) By elimination between the equations (1) and (2), we can find the values of F and of laAB (which is the magnetic moment of the magnet). Also, a and AB being known, we can obtain the value of I ; and hence, if we know the intensity of the magnetising force, we can calculate the susceptibility and the permeability of the substance. Equation (2) shows that the intensities of two fields are inversely 460 A MANUAL OF PHYSICS. proportional to the squares of the periods of the small oscillations of a magnet of known magnetic moment, which is suspended first in one field and then in the other. The ballistic method of making magnetic measurements will be discussed in 369. 359. Terrestrial Magnetism. The earth exerts a magnetic action in virtue of which compass needles point in a northerly direction. The angular distance between the line along which a compass needle points and the geographical meridian is called the magnetic declination or variation. The magnetic needle, if it were carefully supported on an axis which passes through its centre of inertia, and which is perpendicular to the magnetic meridian, would, in Britain, place its magnetic axis in a direction which is inclined to the horizon the north end pointing downwards at a considerable angle. This angle is called the Magnetic Dip. The declination and the dip vary considerably from one part of the earth's surface to another. In some regions the declination is easterly, in others it is westerly. The line on the earth's surface, at all points of which the declination is zero, is called the Magnetic Equator. It does not coincide with the geographical equator, and is not a great circle. That point on the surface at which the north pole of a magnet points vertically downwards is called the North Magnetic Pole of the earth, and that point at which the south pole of a magnet points vertically downwards is called the South Magnetic Poleoi the earth. These poles do not coincide with the geographical poles of the earth, neither do they lie at opposite extremities of a diameter. [Observe that the magnetism which we may suppose to be collected at the north pole of the earth must be south magnetism, i.e., it must be of the same kind as that which appears at the south- pointing pole of a magnet. Similarly, the magnetism which is manifested at the south magnetic pole of the earth must be north magnetism.] The earth's magnetic force is in a constant state of variation. It changes with the hour of the day and the time of the year ; and it depends also upon the position of the moon. Yet these variations do not appear to be due to any direct action of the sun or the moon. Sudden disturbances sometimes take place in addition to these more regular variations. A period of maximum disturbance occurs every eleven years, and coincides with the period of maxi- mum sun-spot disturbance. A slow secular change of the position of the magnetic poles is also in progress. MAGNETISM. 461 360. Theories of Magnetism. At one time magnetic phenomena were explained by the assumption of the existence of two imponder- able fluids, one of which constituted north magnetism, while the other constituted south magnetism. In Poisson's elaboration of this theory a magnetic body was supposed to be made up of spheres of infinite'' permeability, uniformly distributed in an absolutely non- permeable fluid. This made the problem of magnetic induction identical with that of electric induction in a non-conducting medium, throughout which perfectly conducting insulated spheres were uniformly distributed. Among other objections to this theory is the fatal one pointed out by Maxwell, that the permeability of iron is too great to be accounted for even if the spheres were packed in the closest possible arrangement. In modern theories the molecules are supposed to be little magnets. In an unmagnetised body the magnetic molecules have their axes distributed, on the whole, uniformly in all directions ; and the substance becomes magnetised when the axes of its molecules get, on the whole, a definite set in one direction. Saturation will take place when all the molecules have set their axes in the direc- tion of the magnetising force. The fact that the slightest force does not produce saturation shows that displacement of the molecules must be resisted by some force. Weber assumed that each molecule is acted upon by a constant force in the original direction of its axis, which tends to prevent its orientation from that direction, and tends to make it resume its original direction when it is displaced from it. It follows from this assumption that the curve of magnetisation (Fig. 197) should at first be a straight line, that it should afterwards become concave to the axis along which the force is measured, and that it should ultimately approach an asymptote parallel to that axis. This does not agree with experiment. Maxwell improved this hypothesis by the additional assumption that a molecule could return to its original position if it were turned through an angle of less than a certain finite magnitude, and that if it were displaced through an angle greater than this, it would retain, after removal of the force, a displacement equal to the excess of its total displacement over this quantity. This form of the theory leads to a magnetisation curve similar to that given by Weber's unmodified theory, and it indicates that the curve of residual magnetisation starts from a point on the force-axis at a finite distance from the origin, is always concave to that axis, and approaches a parallel asymptote. These results also are incorrect. Ewing, following a limit by Maxwell, regards each molecule as 462 A MANUAL OF PHYSICS. subject only to the mutual action of the entire system of surround- ing molecules. He has constructed a model of such a system by means of a number of pivoted magnets, which are arranged in parallel rows. So long as no external magnetic force acts, the magnets arrange themselves in positions of stable equilibrium under their mutual forces, some of them pointing in one direction, some in another. This illustrates the condition of non-magnetised steel. If only a feeble uniform magnetic force acts, each magnet is slightly turned from its first position, which it reassumes when the force is removed. This illustrates the first stage in the process of magneti- sation. A somewhat stronger force causes instability in the origin- ally less stable groups of the magnets, and the magnets which compose these groups swing round into a new stable position. As the external force is increased still further, more and more groups break up, until all have taken the new position of equilibrium under their own mutual forces and the external directive force. This illustrates the second stage of magnetisation, in which the ratio of magnetisation to magnetising force increases with great rapidity. The third stage, in which this ratio is practically constant, is exemplified by the fact that an infinite force is now needed to make the magnets point exactly in the direction of the external lines of force. If the external force be now removed, a considerable pro- portion of the magnets retain their final positions of equilibrium in other words, magnetic retentiveness is exhibited. This model can also show the effects of strain on the magnetic properties. For this purpose the magnets are placed on a sheet of indiarubber. If the indiarubber is stretched the magnets are separated out from one another in one direction, and are brought nearer to each other' in a direction at right angles to the former. The magnetic susceptibility is increased or is diminished, according as the stability of the magnets is diminished or is increased by the alteration of relative position. Similarly, the increase of the susceptibility of iron with rise of temperature is explained by the diminution of mutual magnetic influence which results from increased distance. Professor Ewing suggests that the total loss of magnetisation which occurs at a high temperature is due to a con- tinuous whirling motion of the magnetic molecules. He suggests, also, that the dissipation of energy, which occurs when hysteresis is exhibited, is due to the induced electric currents ( 342), which are caused by angular motions of the magnetic molecules. CHAPTER XXXI. ELECTROMAGNETISM, ETC. 361. Oersted's Discovery. Oersted found that a magnetic needle always tends to place its length in a direction at right angles to a plane which, passing through its centre, contains a linear circuit, through which an electric current is flowing. The direction in which the north pole points depends upon the direction in which the current flows. The north pole always tends to move round the linear circuit in a direction which is related to that of the current in the way in which the rotation of a right-handed screw is related to its linear motion. After this fact was discovered, it was surmised that the converse phenomenon might also be found to exist that motion of the linear circuit, through which the current flows, would take place if the magnet were fixed while the circuit was free to move. This was verified experimentally by Ampere. And, further, since a magnet can act thus upon two* neighbouring circuits, through which electric currents flow, it was supposed that mutual action might be found to exist between these circuits them- selves if the magnet were removed. Ampere proved the existence of this action also. 362. Magnetic Action of Closed Electric Circuits. The above statement regarding Oersted's discovery shows that a linear electric current is surrounded by circular magnetic lines of force. The researches of Ampere and of Weber have enabled us to find the distribution of electric force in the neighbourhood of any conducting circuit. It is found that a small plane closed circuit produces the same magnetic action as a small magnet, placed at some point inside the circuit with its length perpendicular to the plane, and having the direction of its axis (measured from south pole to north pole) related to the direction of the circulation of the current according to the rule of right-handed screwing motion, while its magnetic 464 A MANUAL OF PHYSICS. moment is equal to the area of the circuit multiplied by the strength of the current. [The word small means that the point at which the action is determined is very far off in comparison with the dimen- sions of the circuit.] It is of no consequence in what part of the circuit the equivalent magnet is placed ; and so we may assume that it is an extremely thin magnetic shell, which fills the entire circuit, and is possessed of a magnetic intensity which is numerically equal to the strength of the electric current divided by the thickness of the shell. For, if I, a, and t represent respectively the magnetic intensity, the area, and the thickness of the shell, lat is the moment of the shell, and, therefore, ia=Iat, or i = It, where i is the strength of the cur- rent, and It is called the strength of the shell. Now, let any finite circuit PQES (Fig. 202) be filled with a net- work of infinitely small conducting meshes of any shape. Let a current i circulate in the circuit in the positive direction. We may assume that an equal current flows similarly in each of the meshes, such &s ziqrs, f r the currents in each common side of two adjacent FIG. 202. meshes exactly neutralise each other. But, as above, the magnetic effect of the current in each mesh may be supposed to be due to a magnetic shell of strength i. And hence we see that the magnetic action of the circuit PQES at external points is equivalent to that of a magnetic shell of strength i, and of any form, which completely fills the circuit. It is easy to find a simple expression for the magnetic potential of the shell. For, if d$ be an element of its surface, we may replace the part of the shell which corresponds to dS by a small magnet, ELECTROMAGNETISM, ETC. 465 the strength of whose, pole is idSjt. The force with which the north pole n (Fig. 203) acts on a unit north pole, placed at a point P, is where r is the distance from P to n. The potential at P FIG. 203. due to n is, therefore, idS/tr. Similarly, the potential of s at P is - idS[tr' 9 where r' is the distance from P to s. The total potential is, therefore, _ : ~~ since r' is practically equal to r. But r' rt cos 0, where 9 is the angle between the axis of the magnet and the line joining P to its centre, and t is the length of the magnet (equal to the thickness of the shell). Hence idS cos Now, since the element of the surface of the shell is at right angles to the axis of the magnet, dS cos 9 represents the resolved part of the element normal to r, and dS cos 0/r 2 is the elementary solid angle which the surface subtends at P. We may, therefore, write wnere di represents this elementary angle. And, in order to find the total potential V at P, due to the whole shell, we have merely to sum all the quantities, such as V, for the whole surface. Hence that is, the potential at any point external to the magnetic shell is equal to the product of the intensity of the current into the solid angle which the shell subtends at the given point. It follows that the work which is done upon a unit north pole by the magnetic forces due to the current is ^i-wo), if the pole passes from a place where w has the value w to a place where it has the value w. 2 - I n particular, if the pole completely describes a closed path which does not pass through the interior of the circuit in which the current flows, the work is zero. If the pole passes 30 466 A MANUAL OF PHYSICS. by an external path from a point P, which is infinitely close to one side of the shell, to a point P', which is infinitely near to P on the opposite side of the shell, w changes by an amount which is infi- nitely nearly equal to 47r, and the work which is done by the magnetic forces is practically 4-n-i. But any shell of the proper moment, which is bounded by the circuit, produces the same magnetic effect at distant points as the current produces; and so we may now replace the shell, whose action we are considering, by another shell of the same moment, which is everywhere finitely distant from P and P', and which is bounded by the same circuit. The forces due to this shell do infinitely little work upon the unit north pole when it passes from P' to P, and hence the work which is done upon the unit pole in a single complete passage round a closed path which passes through the circuit is 4?ri. In n such passages the work is 4-Trin. 363. Electrodynamic Action on an Electric Circuit. Let the intensity of magnetisation of a shell, which produces the same magnetic action as a given circuit, be I ; and let the shell be placed in a field of force the intensity of which is F. We may choose the shell so that each element of its surface is at right angles to the direction of the force in its immediate neighbourhood. Consider a small portion ds of the north face of the surface. The total force acting on this part is -\-FIds, the plus sign being used since we consider it to be positive when it acts in the direction of the outward normal to the north face. Similarly the force which acts on the corresponding part of the south face is FIcZS ; and therefore the potential energy of this portion of the shell in the given field is - FIdSt, where t is the thickness of the shell. Now it, the strength of the shell is equal to t, the strength of the current ; and so the potential energy is -FidS. Consequently, the total potential energy of the shell is -N where N is the whole force acting on the shell, that is, the number of lines of force which pass through the circuit whose action is represented by that of the shell. When a force F produces a change df in one of the quantities which determine the position of the circuit, the work which is expended is Fdf, and the corresponding change in the potential energy of the circuit is idN. Hence And we see that the force F tends to produce or to oppose the change df according as that change is accompanied by an increase, ELECTROMAGNETISM, ETC. 467 or by a decrease, of the number of lines of force which pass through the circuit in the positive direction. In particular, if a part of the circuit be moveable, the electro- magnetic forces which act upon the circuit will tend to produce such a displacement of the moveable part as will cause an increase of the number of lines of force which pass through the circuit. 864. Case of Linear Circuits. We have already seen that a linear circuit carrying an electric current is surrounded by circular lines of force, the direction of which is related to that of the current in the same way as the rotation of a right-handed screw is related to its linear motion. Let A B (Fig. 204) represent part of a fixed linear circuit through which a current flows from A to B, and let ab be a portion of a moveable parallel circuit through which a current flows from a to b. We may assume that the circuit ab is completed by way of p. The lines of force due to AB pass through abp in the positive direction, and a displacement of ab towards AB would increase the area abp, B (! J f A ""*** FIG. 204. and so would cause an increase in the number of positively drawn lines of force. Hence the electrodynamic action between the circuits is such as to cause their mutual approach. [We would arrive at the same result by the supposition that the circuit ab is completed by way of q. For the lines of force due to AB pass through the electric circuit abq in the negative direction. Hence ab will move so as to diminish the area of abq, that is, so as to diminish the number of negatively drawn lines of force which pass through it.] Similarly, mutual repulsion will ensue if the currents are oppositely directed. Next let the circuits AB and ab be inclined to each other, and let 00' (Fig. 205) be the shortest line between them. Let us suppose that, while AB is fixed, ab is free to turn around 00' as an axis. Complete the circuit ab by way of p. No lines of force due to AB will pass through abp when AB and ab are mutually perpendicular. On the other hand, the number of lines which pass through it in the posi- CO -2 468 A MANUAL OF PHYSICS. tive direction is a maximum when the currents in AB and ab are parallel and similarly directed. The electrodynamic action on ab is therefore such as to cause it to place itself parallel to AB. If either current be reversed the moveable circuit will turn so that ba is co -directional with AB. 365. Circular Circuits. Solenoids. 'Ampere's Hypothesis re- garding Magnetism. A circular circuit, of radius r, which is traversed by a current i, may be replaced by its equivalent magnetic shell of intensity i[t ( 362). So also, a second circular circuit, which is traversed by a current i', may be replaced by a shell of intensity i'jt r . These shells will turn so as to place their oppositely magnetised forces parallel, and will then exhibit mutual attraction. Hence the electric circuits will tend to turn so that the currents in each are parallel, and will then exhibit mutual attrac- tion. But if, while the circuits are in this position, one of the currents be reversed, mutual repulsion will be exhibited. [It is an easy matter to calculate the force which such a circuit carrying a current i, exerts at its centre. Let the centre be at o (Fig. 206) and let a, c, be the points in which the circuit cuts the plane of the paper the plane of the circuit being supposed to be perpendicular to that plane. Imagine the circuit to be replaced by an equivalent hemispherical magnetic shell ( 362) abc. The work ELECTROMAGNETISM, ETC. 469 which is done in displacing a unit pole from through a small distance Od=r is i(w Boyle's Law, 103, 105, 149, 150 BREWSTER, 203, 208, 217, 223, 238, 243, 251, 263 Brittleness, 83 CALORIC, 254 Capillarity, 119-127 Carnot's cycle, 290, 291 CAUCHY, 146, 209, 258 Caustics, 184, 189 Cavendish experiment, 89 Characteristic function, 201 Charles' Law, 150, 265 Chemical combination, 280 Cohesion, 118, 137 Colour, 180 body, 205 surface, 206 Colours of crystalline plates, 248, 249 mixed plates. 222 thick plates, 223 thin plates, 218-221 Compressibility, 79 of gases, 102 of a perfect gas, 104 of vapours, 106 of liquids, 113 of solids, 128 Conductivity, thermal, 151 Conservation, 4, 12 of energy, 4, 10 of matter, 4, 5 Consonance, 175 Contact, angle of, 121 Coronse, 232 Coulomb's torsion balance, 312 Couple, 70 Critical angle of reflection, 188 Crucial test, 15 Crystalline structure, 143, 144 Curvature, 43 DENSITY, 64 of earth, 89, 90 Deviation, iirioimum, 192 Dew, 277 Dialysis, 117 Diamagnetism, 346 Dichroism, 205 Dielectrics, 306 Diffraction gratings, 233 Diffusion of gases, 109, 151 of liquids, 116 Diffusivity, 109 Dilatancy, 382 32 498 INDEX. Dilatation of gases, 265 of liquids, 264 of solids, 263 Dimensions, 18, 67 Discharging rod, 314 Dispersion, irrationality of, 197 theories of, 209 Displacement, 40 Disruptive discharge, 321 Dissociation, 152, 280 Dissonanc^, 175, 176 Divisibility, 79, 139 Doppler's principle, 178 Dulong and Petit, law of, 270 Dynamical similarity, 67, 73, 76, '127, 168, 170, 171 Dynamics, 63 Dynamo, 367 EFFICIENCY, thermo-dynamic, 293 Effusion, 110 Elasticity, 79 of gases, 107 of liquids, 114 of solids, 134, 156 .Electric absorption, 321 charge, 310 current, 335, 343 density, 316 displacement, 319 force, lines of, 318 images, 317 induction, 310, 319 machines, 326 polarisation, 338 potential, 313 quantity, 310 v resistance, 335, 343 Electrodynamometer, 369 Electrokinetic energy, 368 Electrolysis, 337 Electromagnetic induction, 367 medium, 92 units, 370 waves, 376 Electrometer, 325 Electromotive force, 313, 343 Electrophorus, 311 Electroscope, gold-leaf, 309 Electrostatic capacity, 314 energy, 320 units, 370 Emissivity, 203, 256 Empirical formulae, 17 laws, 17 Energy, 4, 7-11, 62 conservation of, 10 dissipation of, 11, 297, 298 forms of, 8 kinetic, 8 potential, 8, 1 1 transformation of, 9, 11 Entropy, 296, 298 Epoch, 51 Equation, personal, 16 Equilibrium, of fluids, 75 of particles, 65 of rigid bodies, 69 stable, 15, 127, 331, 357 Equipotential surfaces, 96 Eriometer, 232 Errors, instrumental, 16 observational, 16 probable, 16 Ether, 179, 202 Evaporation, 152 Exchanges, theory of, 255, 256 Extension, 79 FARADAY, 276, 315, 319, 337, 367, 371 Fluid, equilibrium of, 75 motion of, 61, 74 motion of perfect, 74 Fluorescence, 208, 253 Focal lines, 184, 189 Focus, principal, 183, 193 Force, 62 lines and tubes of, 96 moment of, 70 measurement of, 64 Forces, composition bf, 65 molecular, 83, 125 range of, 145 Form, 79 Freedom, degrees of, 41, 54, 57 Freezing mixtures, 279 FRESNEL, 186, 212, 214, 237, 239, 240, 245 FresnePd rhomb, 252 Friction, 62 GALVANOMETER, 369 Gas battery, 341 Gaseous pressure, 149 Geysers, 275 Glaciers, motion of, 273 Graphical method, 17 j Gravitation, 8, 62, 85-101 INDEX. 499 Gravitation, law of, 87 Gravity, centre of, 88 hypotheses in explanation of, 91, 92 GREEN, 245, 380 Gyration, radius of, 70 Gyrostats, 58, 156 HAIDINGER'S brushes, 249 Hall's effects, 373 Halos, 198 HAMILTON, Sir W. B., 201, 246 Harmonic motion, simple, 51, 52 Harton experiment, 90 Heat, conduction of, 282, 287 convection of, 282, 288 latent, 272, 274, 276 mechanical equivalent of, 289 radiant, 253 radiation of, 257-259, 261 specific, 268, 269, 271, 301 total, 276 HELMHOLTZ, Von, 173, 209, 249, 338 Hoarfrost, 277 Hodograph, 48 HOOKE, 135, 147, 237 Hooke's Law, 135 Hope's experiment, 264 HUYGHENS, 186, 234, 235, 237 Hygrometer, 277 Hypothesis, 15 Hysteresis, 355, 356, 357 IMAGE, 182 real, 183, 195 virtual, 183, 195 Impact, 68 , Impenetrability, 79 Impulse, 68 Indicator diagram, 294-297 Inductive capacity, specific, 315 Inertia, 6, 79 centre of, 69, 88 moment of, 70, 71 Isothermals, 24, 295 JOULE, 147, 148, 289, 303, 357 Joule's Law, 342 KEPLER'S Laws, 86 Kerr's effects, 374 Kinetics, 63 KIRCHHOFP, 203, 204, 256 Kirchhoff 8 Laws, 336 LEAST action, 200, 201 confusion, circle of, 184 time, 200, 201 . Lenses, 193, 194 achromatic. 197 LE Boux, 207, 327, 333 LE SAGE, 91, 147 Ley den jar, 314 Light, absorption of, 202-209 colour of, 180 diffraction of, 224-233 dispersion of, 196, 197, 207, 209 intensity of, 177 interference of, 210-223, 247 nature of, 179, 242 polarisation of, 237-252, 371 propagation of, 177, 186, 225 radiation of, 202-204 reflection of, 181-186, 240 refraction of, 187-200, 240 refraction (double), 234-236, 244, 245, 250 scattering of, 181 speed of, 178 theories of, 179, 185, 186, 199, 200, 375 Lightning rod, 322 LLOYD, 213, 246 MACCULLAGH, 239, 240, 243, 245 Magnetic axis, 349 force, lines of, 350 induction, 351 intensity, 351 moment, 349 permeability, 352, 354 pole, 349 potential", 350 retentiveness, 353 shell, 362 susceptibility, 352, 354, 356, 357 viscosity, 354, 355 Magnetism, residual, 353 terrestrial, 359 theories of, 360, 365 Magnetometer, 358 Magnets, coercive force in, 353, 356 permanent, 344 temporary, 344 Malleability, 83 MALUS, 201, 237, 238, 243 ' Mass, 7 measurement of, 64 500 INDEX. Matter, 4, 5 definitions of, 77 general properties of, 79 heterogeneity of, 142, 145, 146 special properties of, 80, 83 specific properties of, 81 statts of, 78 MAXWELL, 24, 108, 145, 147, 152, 153, 154, 250, 319, 320, 360, 372, 375, 376 Metallic reflection, 206, 243 Microscope, 195 Mirage, 191 Molecules, 82, 154 size of, 146 Moment, 49 Momentum, 64 Motion, laws of, 63, 64, 65, 68 of centre of inertia, 69 of connected particles, 68 of non-rigid solids, 73 of particles, 64, 65, 66 of rigid body, 56, 57 Motor, electric, 366 Musical intervals, 167 NEBULAE hypothesis, 93 NEUMANN, 239, 240, 245 NEWTON, 63, 67, 68, 87, 88, 92, 186, 223, 237, 259 Newton's rings, 221 Note, 157 OHM'S Law, 336, 338 Osmose, 117 PARAMAGNETISM, 346 Paraselenes, 198 Parhelia, 198 Path, mean free, 148, 153 Pendulum, 9 ballistic, 68 Peltier effect, 331 Period, 51, 53 Phase, 51 Phosphorescence, 208, 253 Photometer, 177 Piezometer, 113 Polarising prisms, 252 Porosity, 79 I'osition, 39 Potential, 94 gravitational, 95 Precession, 58 PREVOST, 147, 203, 255 Primary cells, 340 Prisms, 192 Projectiles, 47 Pyrheliometer, 260 Pyro-electricity, 232 Pyrometers, 267 QUARTZ fibres, 89 RADIOMETER, 153 Rainbows, 198 RANKIXE, 254, 295 Reflection, total, 188 Refraction, conical, 246 index of, 187, 223 REGNAULT, 264, 265, 269, 271, 275, 276, 277 Residual discharge, 321 Resonance, 173 Restitution, co-efficient of, 68 Rigidity, 79, 83, 128, 131, 156 flexural, 133 torsional, 131 Ripples, 76 Rotation, 54, ,55 Rotational equilibrium, 72 Ruhmkorff coil, 367 SCALAR quantity, 40 Schehallion experiment, 90 Secondary cells, 341 Shear, 59 Solar radiation, 260 Solenoid, 365 Solution, 279 Sound, diffraction of, 164 intensity of, 161 interference of, 165, 175 pitch of, 166 propagation of, 158 quality of, 174 reflection of, 162 refraction of, 163 speed of, 159, 160, 172 wave-length of, 157 Spectrometer, 204 Spectrum analysis, 204 diffraction, 2?3 heat, 258 refraction, 196 Speed, 41 average, 45 mean square, 148 Specific heat of electricity, 332 INDEX. 501 Statics, 63 STEWART, BALFOUR, 203, 256, 259, 261 STOKES, 208, 217, 224, 225, 234, 239, 242, 245, 379 Strain, 59, 60 Stress, 68 Surface-tension, 120, 125, 126, 127 Syren, 166 TAIT, 153, 278, 284, 330, 333 Telephone, 9 Telescope, astronomical, 195 Temperature, 262 absolute, 293 absolute zero of, 266, 303 critical, 24, 278 measurement of, 267 triple point, 24 Tenacity, 83, 138 Tensor, .42 Theory, 15 mathematical, 15 Thermal capacity, 268 conductivity, 283, 284 Therm< 'dynamic motivity, 298 Thermodynamics, fist law of, 289 second law of, 292 Thermo-electric diagram, 330, 333 power, 328 Thermometer, 267 hypsometric, 275 wet and dry bulb, 277 Thermometric conductivity, 284 of crystals, 286 of earth's crust, 285 Thermopile, 328 THOMSON, 15, 92, 101, 126, 137, 141, 146, 148, 155, 209, 260, 273, 292, 293 Thomson effect, 332 Tone, 157 combination, 176 partial, 173 Transpiration, 111 Twist, 57 VECTOR, 40 Velocity, 41, 46 angular, 43, 58 average, 45 Vibrations of air-columns, 172 plates, 170 rods, 168, 169 strings, 17l_ Viscidity, 115 Viscosity, 79, 83 of gases, 108, 151 of liquids, 115 of solids, 136 Vortex atoms, 141, 156 motion, 61 Vortices, molecular, 254, 372 WATER equivalent, 269 Wave-length, 53 motion, 53 Waves, long or solitary, 76 on stretched cords, 73 oscillatory or free, 76 Weight, 64, 79, 85 Wheatstone's bridge, 343 Work, 7, 62 YOUNG, 211, 221, 222, 232, 237 Young's modulus, 130 ZONE plates, 230 THE END. Bailliere, Tindall