THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES DEC 11 1976 THEORETICAL PHYSICS ELEMENTS OF THEORETICAL PHYSICS BY DR. C. CHRISTIANSEN PROFESSOR OF PHYSICS IN THE UNIVERSITY OF COPENHAGEN TRANSLATED INTO ENGLISH BY W. F. MAGIE, PH.D. PROFESSOR OF PHYSICS IN PRINCETON UNIVERSITY MACMILLAN AND CO, LIMITED NEW YORK : THE MACMILLAN COMPANY 1897 All rights resei~eed GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. Physii TRANSLATOR'S PREFACE. THE treatise of Professor Christiansen, of which a translation is here given, presents the fundamental principles of Theoretical Physics, and develops them so far as to bring the reader in touch with much of the new work that is being done in that subject. It is not in every respect exhaustive, but it is stimulating and informing, and furnishes a view of the whole field, which will facilitate the reader's subsequent progress in special parts of it. The need of such a book, in which the various branches of the subject are developed in connection with one another and in a consistent notation, has been long felt by both teachers and students. The thanks of the translator are due to Professor Christiansen for his courtesy in permitting the use of his book. The translation was made from the German of Miiller. The first draft of it was prepared by the translator's wife, without whose aid the task might never have been accomplished. W. F. MAGIE. PRINCETON UNIVERSITY, September, 1896. TABLE OF CONTENTS. PAGE INTRODUCTION, - 1 CHAPTER I. GENERAL THEORY OF MOTION. SECTION I. Freely Falling Bodies, - 5 II. The Motion of Projectiles, 7 III. Equations of Motion for a Material Point, - 8 IV. The Tangential and Normal Forces, - 13 V. Work and Kinetic Energy, - 14 VI. The Work Done on a Body during its Motion in a Closed Path, - 16 VII. The Potential, .... - - 21 VIII. Constrained Motion, 24 IX. Kepler's Laws, - - 27 X. Universal Attraction, - 30 XI. Universal Attraction (continued}, - 31 XII. The Potential of a System of Masses, - ... 34 XIII. Examples. Calculation of Potentials, - - 36 XIV. Gauss's Theorem. The Equations of Laplace and Poisson, - 41 XV. Examples of the Application of Laplace's and Poisson's Equations, ------- ---46 XVI. Action and Reaction. On the Molecular and Atomic Structure of Bodies,. - - - 48 XVII. The Centre of Gravity, - -.- * . - - - 50 CONTENTS. SECTION PAGE XVIII. A Material System, 53 XIX. Moment of Momentum, - - - - - - 55 XX. The Energy of a System of Masses, 56 XXI. Conditions of Equilibrium. Rigid Bodies, - - - 58 XXII. Rotation of a Rigid Body. The Pendulum, 60 CHAPTER II. THE THEORY OF ELASTICITY. XXIII. 62 XXIV. Components of Stress, - - - 64 XXV. Relations among the Components of Stress, 67 XXVI. The Principal Stresses, - 69 XXVII. Faraday's Views on the Nature of Forces Acting at a Distance, - 72 XXVIII. Deformation, --------- 74 XXIX. Relations between Stresses and Deformations, 79 XXX. Conditions of Equilibrium of an Elastic Body, 82 XXXI. Stresses in a Spherical Shell, 83 XXXII. Torsion, 85 XXXIII. Flexure, ---------- 87 XXXIV. Equations of Motion of an Elastic Body, - 89 XXXV. Plane Waves in an Infinitely Extended Body, 90 XXXVI. Other Wave Motions, - - - - 93 XXXVII. Vibrating Strings, - - - .... 95 XXXVIII. Potential Energy of an Elastic Body, - - - - 96 CHAPTER III. EQUILIBRIUM OF FLUIDS. XXXIX. Conditions of Equilibrium, ... XL. Examples of the Equilibrium of Fluids, 101 CHAPTER IV. MOTION OF FLUIDS. XLI. Euler's Equations of Motion, XLII. Transformation of Euler's Equations, 103 106 CONTENTS. IX SECTION XLIII. XLIV. XLV. XL VI. Vortex Motions and Currents in a Fluid, Steady Motion with Velocity-Potential, Lagrange's Equations of Motion, - Wave Motions, PAGE 107 109 111 112 CHAPTER V. INTERNAL FRICTION. XL VII. Internal Forces, XLVIII. Equations of Motion of a Viscous Fluid, XLIX. Flow through a Tube of Circular Cross Section, 115 118 119 CHAPTER VI. CAPILLARITY. L. Surface Energy, - LI. Conditions of Equilibrium, - LII. Capillary Tubes, - 121 123 125 CHAPTER VII. ELECTROSTATICS. LIIL Fundamental Phenomena of Electricity, - - - 127 LIV. Electrical Potential, 128 LV. The Distribution of Electricity on a Good Conductor. - 130 LVI. The Distribution of Electricity on a Sphere and on an Ellipsoid, - - - 132 LVII. Electrical Distribution, - ..... 135 LVI 1 1. Complete Distribution, - - 139 LIX. Mechanical Force Acting on a Charged Body, - - 141 LX. Lines of Electrical Force, 143 LXI. Electrical Energy, - - - - - - - - 145 LXII. A System of Conductors, 147 LXIII. Mechanical Forces, 150 LXIV. The Condenser and Electrometer, 151 LXV. The Dielectric, 155- CONTENTS. SECTION PAGE LXVI. Conditions of Equilibrium, 157 LXVII. Mechanical Force and Electrical Energy in the Dielectric, 158 CHAPTER VIII. MAGNETISM. LXVIII. General Properties of Magnets, 163 LXIX. The Magnetic Potential, ------ 166 LXX. The Potential of a Magnetized Sphere, - - - 168 LXXL The Forces which Act on a Magnet, - - - - 169 LXXII. Potential Energy of a Magnet, - - - - - 171 . LXXIII. Magnetic Distribution, - - . _ .' - 173 LXXIV. Lines of Magnetic Force, 174 LXXV. The Equation of Lines of Force, - - - - 178 LXXVI. Magnetic Induction, - 179 LXXVII. Magnetic Shells, - - . . fc . . 180 CHAPTEE IX. ELECTEO-MAGNETISM. LXX VIII. Biot and Savart's Law, - . _ . j 84 LXXIX. Systems of Currents, - - . , . . - 186 LXXX. The Fundamental Equations of Electro-Magnetism, - 188 LXXXI. Systems of Currents in General, - - . - 190 LXXXII. The Action of Electrical Currents on each other, - 192 LXXXIII. The Measurement of Current-Strength on the Quantity of Electricity, - .. . . . . _ '-194 LXXXI V. Ohm's Law and Joule's Law, . . . - 197 CHAPTER X. INDUCTION. LXXXV. Induction, - - .... LXXXVI. Coefficients of Induction, - ' . LXXXVII. Measurement of Resistance, LXXXVIII. Fundamental Equations of Induction, LXXXIX. Electro-Kinetic Energy, XC. Absolute Units, - 202 205 - 210 - 211 CONTENTS. CHAPTER XL ELECTRICAL OSCILLATIONS. SECTION XCI. Oscillations in a Conductor, ------ XCII. Calculation of the Period, ------ XCI II. The Fundamental Equations for Electrical Insulators or Dielectrics, -------- XCIV. Plane Waves in the Dielectric, - XCV. The Hertzian Oscillations, XCVI. Poynting's Theorem, 224 215 217 219 221 223 CHAPTER XII. REFRACTION OF LIGHT IN ISOTROPIC AND TRANSPARENT BODIES. XCVII. Introduction, - - XCVIII. Fresnel's Formulas, XCIX. The Electro-Magnetic Theory of Light, Equations of the Electro-Magnetic Theory of Light, Refraction in a Plate, 242 Double Refraction, ----.... 246 Discussion of the Velocities of Propagation, - - - 249 CIV. The Wave Surface, 251 The Wave Surface (continued), - - ... 254 The Direction of the Rays, 256 Uniaxial Crystals, - - - 259 C. or. GIL GUI. CV. CVI. CVII. 229 231 235 237 CVIII. Double Refraction at the Surface of a Crystal, - - 261 CIX. Double Refraction in Uniaxial Crystals, - - - 264 CHAPTER XIII. THERMODYNAMICS. CX. The State of a Body, ------- 266 CXI. Ideal Gases, . 270 CXII. Cyclic Processes, - - 272 CXIII. Carnot's and Clausius' Theorem, 274 CXIV. Application of the Second Law, 279 Xll CONTENTS. SECTION PAGE CXV. The Differential Coefficients, - 280 CXVI. Liquids and Solids, - - 281 CXVII. The Development of Heat by Change of Length, - 282 CXVITI. Van der Waal's Equation of State, - - 283 CXIX. Saturated Vapours, 290 CXX. The Entropy, - - 292 CXXI. Dissociation, --------- 295 CHAPTER XIV. CONDUCTION OF HEAT. CXXII. Fourier's Equation, - 298 CXXIII. Steady State, - - 300 CXXIV. The Periodic Flow of Heat in a given Direction, - 301 CXXV. A Heated Surface, - - 303 CXXVI. The Flow of Heat from a Point, 304 CXXVII. The Flow of Heat in an Infinitely Extended Body, - 305 CXXVIII. The Formation of Ice, - - 307 CXXIX. The Flow of Heat in a Plate whose Surface is kept at a Constant Temperature, 308 CXXX. The Development of Functions in Series of Sines and Cosines, - 312 CXXXI. The Application of Fourier's Theorem to the Conduction of Heat, - - 315 CXXXII. The Cooling of a Sphere, 318 CXXXIII. The Motion of Heat in an Infinitely Long Cylinder, - 322 CXXXIV. On the Conduction of Heat in Fluids, - - - - 325 CXX XV. The Influence of the Conduction of Heat on the In- tensity and Velocity of Sound in Gases, - 330 INTRODUCTION. IN the Science of Physics it is assumed that all phenomena are capable of ultimate representation by motions, that is, by changes of place considered with reference to the time required for their accomplishment. We therefore begin with a brief discussion of the theory of pure motion (Kinematics). We will treat first the motion of a point. The continuous line traced out by the successive positions which a moving point occupies in space is called its path. The symbol s represents the distance which the point traverses along its path in the time t. In measuring these quantities the second is used as the unit of time; the centimetre, as the unit of length. The measures of all the magnitudes which occur in the discussion of motions may be stated in terms of these two units. Motions are distinguished by the form of the path, as rectilinear, curvilinear, or periodic. Eectilinear and curvilinear motions are sufficiently defined by their names. A periodic motion is one in which the same condition of motion recurs after a definite interval of time; that is, one in which the moving point returns after a definite time to the same position with the same velocity and direction of motion. Rectilinear motion may be either uniform or variable. It is uniform if the moving point traverses equal distances in equal times. In this case the point traverses the same distance in each unit of time, and the distance traversed in the unit of time measures its velocity. If the point traverses the distance 5 in the time t with a uniform motion, the velocity c is the ratio of s to t, or (a) c = s/t. A velocity is therefore a length divided by a time. If a point moves on the circumference of a circle with a constant velocity, the radius vector drawn to this point sweeps out equal sectors in equal times. In this case the angle which is swept out by this radius vector in the unit of time measures the angular velocity. 4 INTRODUCTION. is x l + x 2 . The component of velocity in the direction of the a--axis is ac = sc 1 + 2 . Similar expressions hold for motions in the directions of the other axes. The resultant velocity is represented by the diagonal of the parallelepiped, whose edges are x, y, z, or by s = *]d? + y- + z 2 . Since an acceleration is the increment of a velocity, the resultant acceleration will be determined in a similar manner. Let ajj, a% represent the .r-components of the increments of velocity due to the two motions. We then have for the total acceleration in the direction of the .r-axis, x = x l + x. 2 , and for the acceleration of the point, 3 = ^/(iBj + 2 ) 2 + (y l + i/^f + (^ + z 2 ) 2 , by which s is expressed as the diagonal of the parallelepiped whose edges are a;, y, z. If the coordinates of the moving point are given as functions of the time, the equation of the path is obtained by determining the values of x and y which hold for the same time L If, for example, x=f l (t)a,ndy=f 2 (t), the relation between x and y is found by eliminating the variable t from the equations by any appropriate method. From this brief discussion of these purely kinematic questions we turn to the consideration of the causes of motion, taking as our starting point the researches of Galileo on freely falling bodies. CHAPTER I. GENERAL THEORY OF MOTION. SECTION I. FREELY FALLING BODIES. THE investigation by Galileo of the motion of freely falling bodies was the first step in the development of modern physics. It is advantageous to start from the same point in our study of the subject. Galileo concluded from his experiments that all bodies falling freely in vacuo will fall at the same rate. This is one of the most important discoveries in natural science, since it shows that all bodies, independent of their condition in other respects, have one property in common. No parallel to this has been found in Nature. It points to a unity in the constitution of matter, of which we certainly do not as yet appreciate the full significance. Galileo's conclusions have been confirmed by the careful experi- ments of Newton, Bessel, and others. Galileo concluded further, that the distance s traversed by a falling body in the time t is proportional to the square of the time, so that (a) s = ^gt 2 , where g is a constant. The constant g is called the acceleration of gravity. The falling body has a uniformly accelerated motion, since = s = t and Its acceleration is therefore constant. This second law of falling bodies is not to be considered a fundamental law in the sense in which the first is.* In the time r immediately following the time /, the body traverses the distance v , kinetic energy is produced during the motion, and will increase continuously if the motion is continued. On the other hand, supposing vZpy>-'dYj'dz)dydz. In general, the work done by a force during the movement of a body around a surface element dS^ which is parallel to the y^-plane, is (b) F.dS x In the same way we obtain G . dS, = (dX/'dz - SECT. VI.] MOTION IN A CLOSED PATH. 19 F, G, and H are the quantities of work done during the movement of the body around a unit area at the point 0, when perpendicular to the x-, y-, and s-axes respectively. If OABC (Fig. 8) is an infinitely small tetrahedron, whose three edges OA, OB, OC are parallel to the coordinate axes, and if the body moves on the boundary of the surface ABC in the direction given by the order of the letters, the work done is equal to that which is done by moving the body in succession about OAB, OBC, and OCA. By this set of motions, the distances AB, EC, CA will each be traversed once in the positive direction, while the distances FIG. 8. OA, OB, OC will each be traversed twice and in opposite directions, so that the work done in them is zero. The work done during the movement of the body about the surface ABC = ds is, therefore, (c) J .dS=F .dS.l + G.dS .m + H.dS.n, where /, m, n are the cosines of the angles which the normal to the surface dS drawn outward from the tetrahedron makes with the coordinate axes. Hence the work J done during the movement of the body around unit area is (d) J=Fl + Gm + Hn, where Z, m, and n determine the position of the unit area. 20 GENERAL THEORY OF MOTION. [CHAP. i. If the work done during the movement of a body about an infinitely small surface is zero, we must have .7=0 for all positions of the surface, or F=G = H=Q, ( -dZfdy - 'dY/'dz = 0, ax/a? - VZ/Vx = 0, \ 9F/aB-aX/^ The surface may be divided into '- -t -i L*~*il /\ surface-elements, as shown in Fig. 9. / / //, I I \ If the body moves about these ~7 7 7 7 7 7| elements one after another in the v 4- + 4- + -r -f 7 same direction, the total work done X^ / / / / I/ will equal zero. It is here assumed " x ^,/ 7 1^^^ ^ Dat ^ e f rces X, F, Z are con- ^~^^_ " J tinuous and single valued func- tions of the coordinates. Every line-element thus introduced will be traversed twice in opposite directions, with the exception of those which form the boundary of the finite surface. Those forces or systems of forces, which are such that the work done by them is independent of the path in which the body is trans- ferred from its initial to its final position, are called conservative forces. The most important examples of such forces are those which act from a fixed point, and have values which depend only on their distance from it. If the force acting at the point P depends only on the distance of that point from the origin of coordinates 0, that is, SECT, vi.] MOTION IN A CLOSED PATH. 21 if it equals /(r), then X=f(r) . x/r, since x/r is the cosine of the angle which the line OP makes with the #-axis. We have similarly X=f(r).x/r, r=/(r).y/r, Z=f(r).z/r. If we set f(r)/r = R, then X=Ex, Y=Ry, Z=Rz. We have then -dZfiy = dR/dr . yz/r, 3F/3* = dR/dr . yz/r. The equation of condition ~dZ/c)y -?>Y/'d2 = is therefore satisfied. The same is true of the other equations of condition (e). The work done during the movement of the body about a surface is given by the integral \(Xdx + Ydy + Zdz). This work is also done if the body moves in succession about all the surface-elements into which the finite surface is divided (Fig. 9). In this process the motion must be uniformly carried out in the same sense. From (c) this work is equal to \(Fl + Gm + Hn)dS. If we substitute the ex- pressions for F, Gf, H formerly obtained, we have, by the use of (a) and (b), {\(X . dx/ds + Y. dyjds + Z . dz/ds)ds = I j [(dZfdy - -dY/3z)l + (dX/Vz - -dZ/3x)m + (dY/3x-'dXI'dy)n]dS, where s is the perimeter of the surface S, and I, m, n are the direction cosines of the normal to each surface-element. Equation (f) shows that the line integral along a closed curve may be replaced by a surface- integral over a surface bounded by this curve. The only conditions which the surface S must fulfil are that it shall be bounded by the curve and have no singular points. The theorem contained in (f) was discovered by Stokes. SECTION VII. THE POTENTIAL. The only applications of the potential that we will discuss are those like the foregoing, in which the work done during the motion is completely determined by the initial and final points of the path. That this may be the case, we must have We exclude from the discussion all cases in which these equations do not hold. Let the components of the force in the field be X, Y, Z, Let there be a unit of mass at the point (Fig. 10), whose rectangular 22 GENERAL THEORY OF MOTION. [CHAP. i. coordinates are a, b, c, and let it move from to P along the path s. The work V done by the force during this motion is (a) V= f'(Xdx + Ydy + Zdz) = V P - it being assumed that A', Y, Z are the partial derivatives of a single function V, which itself is a function only of x, y, z. The work required to transfer the unit of mass from any point to P is equal to the difference of the potentials V P and V at those points, or is equal to the difference of potential. Such differences of potential are all that can be directly measured. The value of the potential itself FIG. 10. involves an unknown constant, and therefore cannot be completely determined. If we assume that the potential is zero at the point 0, then V P is the potential at P. Hence the potential at any point is the work required to transfer the unit of mass to that point from a point where the potential is zero. The potential V is a function of the coordinates. The equation (b) y(x, y, z) = C, when C is constant, represents a surface which is the locus of points, such that the amount of work required to SECT. VII.] THE POTENTIAL. 23 transfer the unit of mass from a point where the potential is zero to any one of them is the same. If different values of C are taken, we obtain a system of surfaces, which are called level or equipotential surfaces. Let PF and QQ' (Fig. 11) be two infinitely near surfaces of this system ; let the potential on PP be V, and on QQ' be V+dV. Let ds be the element of an arbitrary curve crossing these surfaces, which is cut off by them. If the force acting in the direction of ds is T, the quantity of work Tds will be done by the transfer of the unit of mass from P to Q ; this work is also equal to V^-V^dV. We have, there- fore, (c) T .ds = dV or T=dVjds. Hence the force in any direction at a point is determined from the potential ; the relation between force and potential being given by equation (c). Since the direction of the element ds is arbitrary, we may substitute for ds the elements dx, dy, dz, and obtain for the com- ponents of the force X='dV/'d.r, Y= dF/c)y, Z^'dF/^z. From equa- tion (c) the force is inversely pro- portional to the element ds drawn between the two equipotential sur- " faces V and V '+ dV. If the direction of ds is that of the normal to the surface PP', the force has its greatest value. If a series of lines is drawn which cut the equipotential surfaces orthogonally, their directions are the directions of the force at the points of intersection. Such lines are conse- quently called lines of force. The tangent to the line of force at a point gives the direction of the force at that point. If P l and P 2 are two infinitely near points in an equipotential surface, no work need be done to transfer a body from P l to P 2 , for V Pl - V P ^ = \ the force acting on the body is perpendicular to the direction of motion. 1. Example. Gravity. If at a place near the earth's surface we set up a system of rectangular coordinates, so that the .r^-plane is horizontal, and the positive ?y-axis directed vertically upward, then ^=0, Y= -mg, Z=0. Hence we have V= -mgy, that is, the equipotential surfaces are horizontal planes. FIG. 11. 24 GENERAL THEOEY OF MOTION. [CHAP. i. 2. Example. In the case discussed in V., Ex. 2, the work F=fjf(r)dr is needed to move the body from its position at the distance r from a fixed point to another position at the distance r. Hence we have F"= F(r) - jP(r ), and the equipotential surfaces are spheres whose centres are at the centre of attraction 0. SECTION VIII. CONSTRAINED MOTION. Galileo investigated not only freely falling bodies and the motion of projectiles, but also motion on an inclined plane and the motion of a pendulum, and thus made the first step in the investigation of constrained motion. If a body is compelled by any cause to move in a given path, which is not that which it would follow if free to yield to the action of the forces applied to it, its motion is said to be constrained. 1. Example. The Inclined Plane. Let the body D (Fig. 12), acted on by gravity, slide down an inclined plane AB, which makes the FIG. 12. angle a with the horizontal plane BC. We neglect any resistance which may arise from friction. The force or reaction exerted by the inclined plane acts perpendicularly to the plane AB, and does not affect the motion of the body. The expression sought may be best obtained by using the relation between kinetic energy and work. If in represents the mass of the body, v the velocity acquired at B, g the acceleration of gravity, and I the length AB of the inclined plane, we have fymv 2 = mg sin a . /, assuming that the motion begins at A, so that the initial velocity is zero. If a represents the height AC of the inclined plane, we have I sin a = a, and the work -SECT VIII.] CONSTEAINED MOTION. 25 -jft where ds is an element of the path AB. 2. Example. The Pendulum. If we suspend a body A (Fig: 14) at the end of a weightless rod which can swing freely about the point 0, it is compelled to move on the surface of a sphere whose radius is equal to the length I of the rod. We will treat only the simple case in which the departure of the pendulum from its position of equilibrium is small. If, at the time t = 0, the body starts from rest at A, it will move in the arc A BCD through the point C lying perpendicularly under 0. If we set OA = l, ^AOC = a, i_OC=0, and if A A' and BB' are drawn perpendicular to OC, then the velocity which the body gains in moving from A to B equals that which would be gained if it were to fall freely from A' to B'. The distance from A to B' is A'B' = l(cos 0-cosa), and therefore the velocity at B is v = *j2gl (cos - cos a). 26 GENERAL THEORY OF MOTION. [CHAP. i. For v = 0, or for 6= a, the pendulum bob will be at rest, and is then at A or D, if i.DOC = LAOC. The time t taken by the body to move from A to B is found by substituting this value of v in (b). We thus obtain (c) t= - I e id6/J2gl(cos 6 - cos a), 'a This expression is easily integrated if a, and therefore 0, are so small that we may set cos0=l-0 2 and cosa = l-a 2 . The ex- pansion of the cosine in a series is of the form and if x is very small we may neglect terms of higher orders than the second. Making this restriction, we have and by integration (d) 6 = acos(t>JgJl). If t^/g/l = ^Tr, we have = 0; the body moving from A reaches the lowest point of its path in the time t = ir . *Jl/g. The time T required for the movement of the body from A to D is twice this, or (e) T=irjl/g. T is called the period of oscillation.* The period of oscillation is directly proportional * In the case of the pendulum here treated, which swings in a plane, it must be clearly understood that by the period of oscillation T only one advancing or returning beat is meant. In other periodic motions, the period of oscilla- tion is the time between two instants, at which the motion of the body is precisely similar, that is, at which the body has the same velocity and direction of motion ; or, it is the time required for both the advancing and returning l>eats. SECT. VIIL] CONSTEAINED MOTION. 27 to the square root of the length of the pendulum, and' is inversely proportional to the square root of the acceleration of gravity. The equation (e) holds only for very small arcs. In the case of finite values of a, we use the formula (f) It is only for very small arcs that the oscillations of the pendulum are isochronous, that is, independent of the size of the arcs. If the arc is not infinitesimal, the period will increase rather rapidly with the length of the arc. The pendulum may also be studied in the following way. Let an oscillating body of mass m be at the point J5, and be acted on by the force mg. We may represent this force by the line OE (Fig. 14). Draw EF perpendicular to OB; then OF and FE are components of the force OE. The magnitude of the tangential force is mgsind. If we set BC = s, and reckon the tangential force posi- tive, when it tends to increase s t we have P= -mgsin(s/l), and if we assume s to be very small, P= -mgs/L Hence the equation -of motion is (g) ms = P or = -gs/l. By integration we obtain, by a suitable choice of constants, (h) s = a cos (t\/g/l). This equation corresponds to (d). If a body is compelled to move on a given surface, the deter- mination of its motion is in general very difficult. We will not enter into the discussion of the general case, but will consider only the motion of an infinitely small body on a spherical surface, when the body during the motion always remains near the lowest point C of this surface, and when gravity is the only force acting on it. We may then assume that the component of gravity which moves the body is directed toward the point C, and that it is equal to mgsjl, when I represents the radius of the sphere. This assumption gives the motion treated in III., Ex. 3. The path is an ellipse and the time of oscillation is T=2ir*Jl/g. The time of oscillation is therefore independent of the form and dimensions of the path. SECTION IX. KEPLER'S LAWS. In our deduction of the principal theorems of the general theory of motion, we proceeded from Galileo's laws of falling bodies. We turn now to that force of which gravity is a special example, and 28 GENERAL THEORY OF MOTION. [CHAP. i. from whose properties the laws of planetary motion may be deduced. Starting with the hypothesis of Copernicus, that the sun is stationary and that the earth rotates on its own axis and also revolves round the sun, Kepler announced the following laws : 1. A radius vector drawn from the sun to a planet describes equal sectors in equal times. 2. The orbits of the planets are ellipses with the sun at one of the foci. 3. The squares of the, periodic, times of two planets are proportional to the cubes of the semi-major axes of their orbits. These laws may be expressed" analytically in the following way. Let S be the centre of the sun (Fig. 15) and APQ a part of the orbit of a planet. Let the planet be at A at the time = 0, and at P at the time t. In the next time element dt the planet moves from P to Q, and its radius vector drawn from the sun describes the sector PSQ. Let -ASP = Q, LPSQ = dQ, and SP = r. The surface PSQ is equal to \r-dQ. Since by Kepler's first law the surface described by the radius vector increases pro- portionally to the time, we have r 2 dQ = Mt, where k is constant, or, writing the equa- tion in another form, (a) 7- 2 .0 = &. Kepler's first law is a special case of a general law which is FIG. 15. If the force which acts upon a the surface described by the called the law of areas. This law is moving body pi'oceeds from a fixt radius vector drawn from that point to the body increases at a constant rate. Hence Kepler's first law holds for all central forces. From (a) it appears that the angular velocity is inversely proportional to the square of the distance of the planet from the sun. We represent the velocity of the planet at P by v, and the perpendicular from S upon the tangent to the orbit at the point P by SN=p. SECT, ix.] KEPLER'S LAWS. 29 If we set PQ = ds, the area of the sector PSQ is equal ,to ^pds = fypvdt. But it is also equal to ^r 2 dQ = \k . dt. Hence we have pvdt = Mt, or pv = Tc, that is, the velocities of the planet at different points in its orbit are inversely proportional to the distances of the tangents at those points from the sun, the centre of attraction. Let BPC be the elliptical orbit of the planet (Fig. 16), with the sun situated at the focus S. Let the major axis be BC=2a, and let SA be a fixed radius vector which makes the angle a with the major axis. We set SP = r, _ASP = Q. If n i ci if- v. FIG. 16. F and S are the foci; we have PF+PS=2a, PF=2a-r, and hence (2a-?-) 2 --=4a 2 e 2 + r 2 + 4aercos(6-a), if e is the eccentricity, and if, therefore, FS=2ae. From this equation we obtain for the equation of the path in polar coordinates (b) 1/r = [1 + e cos (6 - o)]/a(l - e 2 ). From equation (a) we have J^r 2 . dO = |& . T, if the integration is taken over the whole orbit and, if T is the periodic time. The integral is equal to the area of the ellipse, or to a . b . TT, if b represents the minor axis. Hence we have %Trab = k.T. If we notice that a 2 = b 2 + a?e 2 , we obtain (c) 27r 2 V 1 - e 2 = JcT, and squaring, By Kepler's third law T 2 /a 3 is constant for all the planets. We must therefore have (d) /* = k 2 /a(l - e 2 ) = 47r 2 a 3 /T 2 , a constant. The velocity v may be determined in the following way. Let S (Fig. 16) be the origin of a system of rectangular coordinates, and let SA be the a-axis. We have x = rcosQ and y = rsin6, and v 2 = z? 4- if 2 . From the equations (e) x = r cos - r sin . ; y = r sin + r cos . 9, we obtain (f) v z = r 2 + r 2 6 2 . If we substitute for r6 its value given by equations (a) and (b), and for r the value got by differentiating equation (b), we have v 2 = (l + 2e cos (0 - a) + e 2 ) . & 2 /a 2 (l - e 2 ) 2 . Noticing that 1 +2ecos (Q - a) + e 2 = 2(l + e cos (6 - a)J - (1 -e 2 ), we obtain, by the help of equation (b), 2 = (2/r- I/a) . & 2 /a(l -e 2 ), or, introducing the quantity p. defined by (d), (g) v 2 = 2p,/r p/a. 30 GENERAL THEORY OF MOTION. [CHAP. i. SECTION X. UNIVERSAL ATTRACTION. We owe to Newton the determination of the law of the force which must act on a planet in order that its motions may conform to Kepler's laws. To determine this force, we use equation (g) IX. Let the centre of the sun be the origin of a system of rectangular coordinates, and let the planet be situated at the point (z, y) Represent the components of the unknown force by X and Y. From the law of kinetic energy (V.) we have |# 2 - t' 2 = { Xdx + Ydy. If v is the velocity at the distance r , we obtain, by the help of equation (g) IX., fJL/r-fjL/rQ = \Xdx + Ydy. If Xdx+Ydy is a complete differ- ential d, we will have X = 'tyfdx**'d(plrydx and or X = - /x/V 2 - 3r/ac = - /iz/r 8 , Y The force R with which the sun acts on the planet is R= - /*/r 2 , that is, is a force which is inversely proportional to the square of the distance of the planet from the sun. It is evident from equation (d) IX. that the quantity p. has the same value for all the planets. FIG. 17. We may also obtain these results from the general equations x = X and y=Y. The unknown components of force X and Y may be represented by the lines PA and PB (Fig. 17), and resolved into a component R in the direction SP = r and a component T perpendicular to SP. Setting -PSX = Q t we have T= - XsinQ+ Fcos6. SECT, x.] UNIVERSAL ATTRACTION. 31 By the help of equation (e) IX., this becomes (d) 72 = r-r6 2 and T=2fQ + re=l/r . d(r*6)ldt. But since r 2 6 = constant by Kepler's first law, we have T=Q. The attractive force is therefore directed toward the sun. Using equations (a) and (b) IX., we obtain (e) R= - 2 /a(l -e 2 >' 2 = - f*,'r 2 . We apply this result to the motion of the moon. By reference to (d) IX., where the value of p. is given, we find that R= -4irV/ZV. The orbit of the moon is approximately a circle with a radius 60,27 times as great as that of the earth. Setting r=a=4. 10 9 .60,27/27rcm, we have the acceleration 7 of the moon toward the earth, 7 = 47r 2 a/r 2 = 877 . 60,27 . 10 9 /2 360 600 2 cm, since the period of the moon's rotation is 27,322 days or 2 360 600 seconds. Hence we have 7 = 0,27183 cm. If the centre of the moon were situated at the distance of the radius of the earth from the earth's centre, it would have an acceleration equal to 0,27183.60,27 2 cm = 987cm, assuming that the force is inversely proportional to the square of the distance. This value accords so well with that of the accelera- tion at the surface of the earth, that we are justified in assuming that the motion of a falling body is an action of the same force as that which keeps the moon and the planets in their orbits. The final proof of the validity of Newton's law of mass attraction is obtained from the complete agreement of the theoretical conclusions drawn from it with the results of observations on the heavenly bodies. SECTION XI. UNIVERSAL ATTRACTION (continued). We will now use a method precisely the opposite of our former one. We will assume the law of attraction known, and determine the path of a planet whose position and velocity at the time t = are given. Let S be the centre of the sun (Fig. 18), let the attracted body be situated at A at the time = 0, and let AC represent the velocity r , whose direction makes the angle CAD = Avith SA = r produced. If the acceleration which the sun imparts to the planet is set equal to p./r 2 , then, using a system of polar coordinates whose 32 GENERAL THEORY OF MOTION. [CHAP. i_ origin is at S, and writing the force with the minus sign because it is directed toward the sun, we obtain (a) (b) f - r6 2 = - fM/r 2 and 1/r . d(r*Q)/dt = 0. It follows from (b) that (c) r 2 9 = &, where k is a constant. This for- mula was obtained from Kepler's first law, (a) IX. Since ?-6 is the component of velocity perpendicular to the direction of r, we obtain for t = Q, (d) k/r = v Q sm(f>. By the help of equation (c), (a) take& the form r - F/r 3 = - /*/r 2 . If this equation is multiplied by 2i-dt, we have d(f 3 )+d(k 2 /r 2 ) = 2d(p/r), and by integration r 2 + F/r 2 = 2/i/r + Const. FIG. 18. In the initial point A, we have v = v () and r = v cos, therefore for / = 0, we obtain ?' 2 cos 2 < + F/r 2 = 2fi/r + Const., from which by the help of (d) it follows that ?; 2 = 2/*/r + Const. Hence we have (e) r 2 = V-2f0o + 2/Vr-F/r>. Since the velocity v, from (f) IX., may be expressed generally by 2 j we obtain by the use of (c) and (e) This equation agrees with (g) IX. The same result may be derived from the theorem connectin kinetic energy and work. From (e) we have (g) r=x/V-2 ft where the upper sign is to be taken, if r and t increase or diminish together. It follows from equation (c) that Q = k/r 2 . Since r and depend only on t, we obtain from (c) and (g) ft . d(l/r)/dB = + VV-2/ SECT, xi.] UNIVERSAL ATTRACTION. 33 This is the differential equation of the orbit. By adding and sub- tracting fj. 2 /k 2 under the radical sign, and by noticing that p/k is a constant, and that therefore its differential is zero, this equation may be written (h) dQ = d(k/r - If we set M 2 = v z - 2/x/r + p?/k 2 , we get by integration = arc cos (k/ur - p/uk) + a, where a is a constant. Hence the equation of the path is (i) l/r=(l+Jb//tco8(e-o))/<^//*), . In this equation u may always be considered positive, since a is arbitrary. The polar equation of a conic is (k) l/r= (l +e cos (0 - a))/a(l -e 2 ), which represents an ellipse, a parabola, or a branch of an hyperbola respectively, according as el. If e = we have the equation of the circle. From the equation e = ku/p, by introducing the value of , we obtain (1) 1 -e 2 = (2//r -i> 2 ) . & 2 //* 2 . If a body approaches the sun from infinity to the distance r , its velocity will be v v determining by the following equation Therefore we have (m) e 2 = 1 - (v^ - v 2 ) . F//* 2 . Hence the path is either an ellipse, a parabola, or an hyperbola, according as o<*'i v o = v i or v o> v i'> that is, the path of the body is an ellipse, parabola, or hyperbola, according as the vis viva imparted to the planet at the first instant is too small to send it to infinity against the attraction of the sun, or exactly sufficient, or more than is sufficient, to accomplish that result. By comparison of the formulas (i) and (k), we obtain (n) o(l -#) = *//*. This corresponds to (d) IX. From (m) and (n) it follows, moreover, that (o) yu. = (flj 2 - v 2 ) . a. The upper sign is used when v 1 >v , the lower when v 1 and OB = u, we have ?= r-2-n-R sin < . Rd . d(j>, the integral takes the form V= faR/r . udu/u . . From equation (e) the force outside the sphere is inversely pro- portional to the square of the distance of the unit of mass from the centre of the sphere. We thus justify the assumption that the planets and the sun may be treated as points in which their re- spective masses are concentrated. In the interior of the sphere the force is proportional to the distance of the attracted point from the centre. If we transform equation (e) to dF t /dr = -^m^p.l/r 2 , we see that the force proceeds from that portion of the sphere whose distance from the centre is less than r. This only holds on the assumption made about p. The earth's density very probably increases toward the centre; hence the force of gravity will not have its greatest value at the surface, but at some point beneath it. This corresponds with the results of experiments on the time of vibration of a pendulum in a deep mine. 3. The Potential of a Circular Plate. Let AE (Fig. 21) be a circular plate of surface density a-; the centre of the plate is and the axis OP. The point P, for which the potential is to be determined, lies on the axis at the distance x from the plate. The potential V is then f* F= I 2irr) . drj . o-/0, V l = 2ir(r(p-x) If the radius of the plate is infinitely great in comparison with x, we may set P r l = C-2Tra-x and F 2 C+2iro-x, where C is an infinitely great constant, since p is infinitely great and o- remains finite. We have for x>Q, dV l jdx = -2ircr, and for x<0, d F 2 /dx = + 27nr. By passage through the surface, dF/dx, that is, the force, changes dis- continuously by 47rcr. 4. The Potential of an infinitely long straight line. Suppose each unit of length of the line AB (Fig. 22) to have the . Let G be a point at the distance a from AB, and CD the "B perpendicular let fall from C upon AB. The potential V, at the point (7, is V= 2^ log (z'/a Since z' is infinitely great in com- parison with a, we may neglect 1 under the radical and write (k) V= 2p log ( 2z'/a) = C.-p log a 2 , . where C is an infinitely great con- stant, if z' is infinitely great. Further, we obtain dF/da= -2/*/a, that is, the force is inversely pro- FlG ' 22> portioned to the distance of the point from the straight line. 5. The Potential of a Circular Represent the surface density of a circular cylinder by a-. Through the point P, for which the potential is to be determined, pass a plane 40 GENERAL THEOEY OF MOTION. [CHAP. i. perpendicular to the axis of the cylinder (Fig. 23). Let R be the radius of the cross section of the cylinder and r the distance of the point P from its centre. FIG. 23. We then have from (k) v= c-^JJme.a-. log a 2 . We now find the value of the integral in which a = *Jf 2 - 2rE cos 6 + If 2 . First consider the case in which r>R. The integral may then be written, if we set a = E/r, rir _ ^ = 2 J o Rd6(\og r + log Vl - 2a cos + a 2 ). Since cos 6 = \(& 9 + -**), we have 1 - 2a cos + a 2 = (1 - a.e i6 }( 1 - ae~ ie ), and jf Now developing the terms in this integral in series, and carrying out the integration, we find that the integral is equal to zero, and hence that A = Z-n-R log r. Thus the mean value of log a for all points of the circumference of the circle is equal to log r or to the logarithm of the mean distance from P to the circumference of the circle. If now V< R, that is, if P lies between and the circumference, and if we set a = rjR, we have = 2 Rd6(\0g R + log N/l-2aCOS0+a 2 ), and hence A = 2-rrE log R. In this case also the mean value of log a SECT, xiii.] CALCULATION OF POTENTIALS. 41 is equal to the logarithm of the mean distance from P to the circum- ference of the circle. Now setting V n and V i for the potentials of points outside and inside the cylinder respectively, we have from these values, r a =C- 47r^o- log r, V i = c- lirEo- log R. The potential is therefore constant, and the force zero within the cylinder. Outside the cylinder the force is given by (n) d7Jdr= - 4irftr/r, that is, the force is inversely proportional to the distance of the point from the axis of the cylinder. SECTION XIV. GAUSS'S THEOREM. THE EQUATIONS or LAPLACE AND POISSON. Let ABF (Fig. 24) be a closed surface, of which AB = dS is a surface-element, and at the point within the surface, let the mass m be concentrated. On the element ds at C, construct the normal CE. Let the length of the line connecting and C be r, and let the normal CE make the angle DCE = Q with 00 produced. If the potential at C due to m l is V v then P r l = m l /r, and the force ^ acting at the point C in the direction CE is N l = 'dV l [dn ) while the total force in the direction of CO is mjr 2 . We have (a) JV t = wij/r 2 . cos ( - 6) = - mjr 2 . cos 0. 42 GENEEAL THEORY OF MOTION. [CHAP. i. If a sphere of unit radius is described about the point as centre, the straight lines drawn to the contour of (IS mark out on this unit sphere a surface-element whose magnitude is equal to (b) du = dS.cose/r*. From (a) and (b) we obtain NjdS= - mjr 2 . cos QdS= - m^w and 'dPJ'dn . dS= - mfa. If there are still other masses, m^ m y etc., within the closed surface, we obtain similarly 3 fy3 . dS = - m^u, "d V^^n, . dS = - m s do>, .... V v Vy F s are the potentials at the point C due to the masses m v m 2 , m s respectively. For the total potential at the point C we ~ ..., and therefore .dS= - If we designate the mass enclosed by the surface by 2m, integration over the whole surface gives (c) foV/'dn. dS= - 47r2m. The force acting in the direction of the normal to the surface S is 'dF'/'dn ; we call 'dF'/'dn. dS the flux of force which passes through the element dS. Hence the total flux of force passing through a finite closed surface equals the sum of the acting masses contained within the surface multiplied by - 4:ir. Hence, if the entire acting mass is enclosed by the surface, and 'dVfdn is given for all points on the surface, the sum of the masses may be determined by the help of equation (c). SECT, xiv.] GAUSS'S THEOREM. 43 The theorem expressed in (c) also holds in case the acting mass lies outside the closed surface. Let the mass m' be situated at the point 0' (Fig. 25) outside the surface ABB' A. If the surface-element do> is taken on the surface of the unit sphere described about 0' as centre, the straight lines drawn from 0' through the boundary of this surface-element mark out on the closed surface the surface-elements AB = dS and A'B' = dS'. Let the normals to AB and A'ff directed outward from the closed surface be n and n' respectively, and let the forces 'dF'/'dn and 'dF'/'dn' act in the direction of these normals. The V in these expressions represents the potential due to m'. If the angles made by the normals directed outward and the straight line drawn from 0' are designated by 6 and 6' respectively, and if we set 0'A=r, 0'A' = r', we then obtain -dF'fdn = m'/r 2 . cos (a- - 9) ; 'dF'/^n' = m'/r' 2 . cos (a- - 6'), dS . cos (TT - 6) = r 2 du ; dS' cos 6' = r' 2 . du, and therefore Wpn.dS + W/dri .dS' = 0. We therefore have (d) \?)V'/'dn.dS=Q, if the integral is extended over the whole surface. The flux of force proceeding from a point outside a closed surface, and passing through the surface, is equal to zero. Therefore the value of the integral is independent of the mass outside the surface. We have then, generally, (e) foF'fdn.dS** -4-jrM, where S is a closed surface, V the potential, n the normal directed outward, and M the sum of all the masses within the surface. This theorem is due to Gauss. Equation (e) may be put into another form. We have 3 Ffdn = 3 Vfdx . dx/dn + 3 F/^ . dy/dn + 3 V\"bz . dz/dn, and dx/dn = A, dyjdn = p, dzldn = v, where A, /*, and v are the cosines of the angles which the normal to the surface makes with the axes. We have then 3F/3tt = \X + ^Y+ vZ. X, Y, and Z are the com- ponents of the force, and / is set equal to 1. We then obtain from Gauss's theorem (f) J(ZX + Yp + Zv)dS= - 1-rrM. Let x, y, z (Fig. 26) be the coordinates of the point 0, Ox, Oy, and Oz be parallel to the coordinate axes, and 00' be a parallelepiped whose edges are parallel to these axes. Let X, Y, Z be the components of the force acting at 0. The components of the force at the point A, whose coordinates are x + dx, y, z, will be X+VXfdx.dx, Y+'dY/'dx.dx, Z+-dZ/Vx.dx. We apply Gauss's theorem to the surface of this parallelepiped. The force acting normal to the surface OA' is - X, that acting normal to AO' is +X+~dX/c)x.dx. In the same way the force acting normal 44 GENERAL THEORY OF MOTION. [CHAP. i. to OB 1 is -Y, and that acting normal to BO' is Y+'dY/'dy.d similar statement holds for the z coordinate. We have therefore \-dVfdn, . dS = Jf J[ - Xdydz + (X+ VX/Vx . dx)dydz] + [ - Ydxdz + (Y+ VY/oy . dy)dzdx] We suppose the volume-element 00' to contain the mass M of density />, so that M=pdxdydz. We then have from (e) (g) or (h) On account of the frequent use made of this equation in mathe- matical physics, we use for the sum of the first derivatives of a function / with respect to the three coordinates the symbol and for the sum of the second derivatives with respect to the same variables the symbol V 2 /= 3 2 //a 2 + 3 2 //c)?/ 2 + 3 2 //3 2 . With this notation SECT, xiv.] GAUSS'S THEOREM. 45 equation (h) may be written (i) V 2 F+ 4ny> = 0. By the help of this equation, which was first used by Poisson, we can determine the density when the potential is known. If no matter is present in the region under consideration, that is, if p = 0, we have (k) 3 2 r/9a; 2 + 3 2 F/3y 2 + 3 2 r/o^ = V 2 r=0. This equation was first derived by Laplace. It may be obtained more simply in the following way. We start from where , i], and are constant, and obtain 3(l/r)/a* = - (x - )/r, ^(l/r)/^ = - l/r + 3(a? - Analogous expressions hold for 3 8 (l/r)/8jy 2 and 3 2 (l/r)<32 2 . Adding these equations, we have 3 2 = 0. Since the potential F=2w/r [(b) XII.], this is equivalent to V 2 F=0. Poisson's equation may also be obtained in the following way. Let the density at the point P be p. Describe a sphere of infinitely small radius R so as to contain the point P, and suppose the density in the interior of the sphere to be constant. The potential P at the point P consists of two parts, Fj and F~ a ; V a is due to the mass outside the sphere, and V i to the mass within the sphere. The potential at P is V=V a +V i . If P is at the distance r from the centre of the sphere, we have from (d) XIII. (k') F t = 2aX# - r 2 ) ; V= V a + **p(& - &*)> If , rj, and x, y, z are the coordinates of the centre of the sphere and of P respectively, we will have r 2 = (x - ) 2 + (y - r)) 2 + (z - ) 2 . By differentiation with respect to x, we obtain 3(r 2 )/ae = 2(x - ) and 9 2 (r 2 )/ae 2 = 2, therefore W = 6, and from equation (k') V*V=V*V a -lvp. Now V a is the potential due to the mass lying outside of the sphere, and therefore V 2 F" a = and V 2 F+47r/a = 0. This is Poisson's equation. In the parts of the region where p is infinitely great, Poisson's equation loses its meaning. In this case we return to the fundamental equation (e). For example, let a mass be distributed on a surface S with surface density a-. Draw the normals v t and v a to the element dS on both sides of the surface, and construct right cylinders on both sides of the surface on dS as base, and with the heights dv i and dv a ; the linear elements of these cylinders are lines of force. By applying equation (e) to the volume included in the cylinders, we 46 GENERAL THEORY OF MOTION. [CHAP. i. obtain ?>V^v i . dS+'dFJ'dv a . dS= -4WS, where V t and V a represent the values of the potential on both sides of the surface. Hence (1) 3r/3v, + ^V a l^v a + lira- = 0. This equation finds an application in the theory of electricity. Comparing formulas (e) and (h), and noticing that M= ^^pdxdydz, we obtain the relation (m) \\\V z Vdxdydz = Jf3F/3. dS. The triple integral in (m) must be extended over the volume bounded by the surface S, and the double integral over the surface S. This theorem may also be proved by integration by parts. SECTION XV. EXAMPLES OF THE APPLICATION OF LAPLACE'S AND POISSON'S EQUATIONS. The potential V at the point x, y t z is a function of the three coordinates, and, from the previous discussion, has the form (a) V= Mpd&vdt/J(z - & + (y- ^ + (* - 2 > where the density p at the point (, rj, ) is a function of the coordinates. We may, however, use the differential equation (b) V 2 F+47r/> = as the starting point for the determination of the potential; we thus often obtain the desired result by a more convenient method. The density p must be given as a function of x, y, and z. The integral of (b) is always given by (a), but V may often be found more conveniently by direct integration of Poisson's equation. In the solution of problems in potential, special attention must be paid to the boundary conditions which serve to determine the functions which are obtained by integration. We shall investigate the equations of condition to which the potential V t within a closed surface S and the potential V a outside that surface must conform, if the surface S encloses all the masses which are present in the field, and if no mass is present outside the surface. Applying Poisson's equation to the region enclosed by #, we have (c) V 2 J^ + 47r/> = 0. Outside the surface S we have (d) V 2 F" a = 0. If is any point within S, P a, point outside S, and if we set OP = r, we have, when r is very great, (e) F a = M/r. M represents the whole mass enclosed by S. Hence, for r = , the potential F tt = and (f) lim(rF a ) r=00 =M, that is, the product rV a approaches the finite limit M if the point P moves off to infinity. SECT, xv.] LAPLACE'S AND POISSON'S EQUATIONS. 47 If P l and P 2 are two points which lie infinitely near each other on different sides of the surface S, the potentials at both points are equal, and we have for all points on the surface S, (g) V i = F a . The dash drawn over V is used to denote the value of V at the surface. From (1) XIV. it follows further that for the points on the surface where o- = 0, we have (h) 'dF i /'dv = 'dF a /?)v, where the normal to S is designated by v = - v ; = v a . The potential is therefore everywhere finite. It is here assumed that p is everywhere finite. For the places where p = so we obtain other equations of condition, which may readily be derived from those already given. For example, if o- is the surface density on a surface S, in which, therefore, p is infinitely great, and if p = Q for all other points in the region, then, in our former notation, we have V 2 F"j = and V 2 F" tt = 0, but (i) Vi=V W t l'dv l + Wjdv a +4:ira- = Q for all points on the surface S. By these equations we may determine the potential if the density p within a sphere of radius R is constant. Outside the sphere p is supposed to be zero. The potential within the sphere is F t , and outside of it F a . We have then W, + 4ir/o = 0, V 2 F ft = 0. Now we have 'dF/'dx = dFfdr . x/r, &Ffda? = d*Fjdr* . x 2 /r 2 + dFjdr . 1/r -dF/dr . x 2 /^. Similar equations hold for the derivatives of V with respect to y and z. We thus obtain V 2 F=d?F/dr 2 + 2/r . dFjdr. Since, however, d(r F){dr = rd F/dr + V and d 2 (rF)/dr 2 = rd 2 F/dr 2 + 2d F/dr, we have (1) V 2 F= 1/r . d 2 (rF)/dr 2 . The differential equations which V i and F a must satisfy are therefore (m) d 2 (rF t )ldr 2 + l-rpr = 0, d 2 (r F a )/dr* = 0. From these we obtain by integration For r = ao it is assumed that F a = 0, so that F a = C l '/r. Since F t cannot become infinite for r = 0, we have (7 = 0, and hence Since the force is a continuous function of the coordinates, arid since, therefore, for all points on the surface, dF i /dr = dFJdr, we will also have, when r = B, ^TrpR = C l '/R 2 . Hence C^^-n-pB 3 , and therefore (n) F a = irE s p/r. Since F" 4 =F a when r = R, we have C=27rR 2 p, and therefore (o) F^^^R 2 -^- 2 ). These formulas are the same as (c) and (d) XIII. 48 GENERAL THEORY OF MOTION. [CHAP. i. If the potential depends on the distance of the point under con- sideration from a straight line, we choose this line as the -axis of a system of rectangular coordinates. Let the distance from the 2-axis of the point for which the potential is to be determined be r. We have then r~ = x 2 + y-, and further x*/r 2 . dtF/dr* + l/r . dFfdr - x 2 /^ . dF/dr, etc. Therefore (p) ^Y^^YI^ + ij r . dp/fir = i I T t d(rdF/dr)/dr. If we are dealing with an infinitely long circular cylinder of radius E and surface-density a-, the axis of which is taken as the z-axis, we have (q) V s F i = and V 2 F" = 0, while for r = R we have (r) F t =F M d FJdr - dFt/dr = - 47ro-. It follows from equations (p) and (q) that d(rdFt/dr)/dr = Q and d(rdf r a /dr)ldr = 0. Hence dF t /dr=C l /r and dF a /dr = C 2 /r, r i = (7 1 logr+(7 1 ' and F a =C 2 \ogr+C 2 . GI must be equal to zero, since no force acts at points in the axis of the cylinder. Further, for r = R, we have C^ = C. 2 log E + C 2 . From equation (r) we have C 2 = -47rKo-, and therefore V. = C 2 - lirRo- log R; F a = C 2 - iirRo- log r. These equations are the same as those given in (m) XIII. SECTION XVI. ACTION AND REACTION. ON THE MOLECULAR AND ATOMIC STRUCTURE OF BODIES. In our discussions up to this point we have considered the motion of a body under the action of given forces ; but nothing has yet been said as to the origin of these forces. A body upon which no forces act moves forward, by the principle of inertia, in a straight line with a uniform velocity. A change in the motion can arise only from outside causes. We learn from experience that the motion of one body in the presence of another undergoes a change, and we are therefore led to assume that in the mutual action of these bodies is to be found the reason for the change of motion. We will first consider the mutual action of two bodies. We thus obtain the means of investigating the more general case in which three or more bodies SECT, xvi.] ACTION AND REACTION. 49 act on one another. The mutual action may be of different kinds. If two bodies collide their motion changes. A similar change occurs when the bodies slide over each other. In both cases the bodies are at least momentarily in contact. Bodies also act on each other without contact; thus, for example, a magnet attracts a piece of iron, or a piece of amber when rubbed attracts a feather. The first serious eifort to explain these mutual actions or so-called actions at a distance was made by Descartes. His explanation was based on the assumption that all space is filled with very small particles in motion, and that all observed motions of bodies are due to collisions between them and these invisible particles. Hence the discovery of the laws of collision became one of the most important tasks in the study of physics. Descartes investigated this question, but without success. It was not until the close of the 17th century that Huygens, Wallis, and Wren contemporaneously succeeded in solving it. A sphere in motion can set in motion a sphere at rest; the moving sphere, therefore, possesses energy of itself. Let the collision be central, that is, let the direction of motion coincide with the line joining the centres of the spheres. An iron sphere produces a greater effect on the sphere at rest than a wooden sphere of equal size moving with the same velocity. Of two equally large spheres whose mass is the same, the one produces the greater effect which has the greater velocity. Hence the force which the moving sphere possesses increases with its mass and with its velocity jointly. The product of the mass and the velocity gives a measure for the force residing in the body, and is called its momentum, or quantity of motion. The principal result which Huygens, Wallis, and others obtained was the following : If two bodies collide, they undergo changes of momentum which are equally great and in opposite directions, or, they act on each other with equal but oppositely directed forces. The action and reaction are therefore equal and oppositely directed. This is one of the most important laws of natural philosophy, and we will discuss the grounds upon which it is founded. It was first derived from observations on collision, without its thereby becoming apparent how far it holds for other interactions between bodies. Xewton first recognized in this law a universal law of nature, which always applies when bodies act on one another. By careful investigation of the collisions of different bodies (steel, glass, wool, cork) he found that the action and reaction are equal, if allowance is made for the resistance of the air. In order to examine D 50 GENEKAL THEORY OF MOTION. [CHAP. i. whether the same law holds for actions at a distance, he mounted a magnet and a piece of iron on corks and floated them on water. The iron and the magnet approached each other and remained at rest after they had come in contact, so that the forces by which the iron and the magnet were mutually attracted were oppositely directed, and equal. He showed further by the following argument that action and reaction are equal in the case of attraction or repulsion : If two bodies acting on each other are rigidly connected, they should both move in the direction of the greater force if action and reaction were not equal ; this would contradict the principle of inertia. Since Newton's time this law has been established in many ways, and many discoveries in physics have furnished proofs of its correct- ness. It has led in many cases to new discoveries, and there is no longer any doubt of its universal applicability. The simplest conception of the structure of bodies is that, according to which bodies are composed of discrete particles, for whose mutual action the law of action and reaction holds. Starting from this view, Newton calculated the action of gravity. Gravity is a function of distance alone ; its value is therefore the same so long as the distance is unchanged. This conception of the structure of bodies has led to important results in other branches of physics. There are, however, many cases in which it seems inadequate. Chemistry teaches that bodies are composed of molecules, which themselves may be groups of smaller particles or atoms. These molecules have certainly a very complex structure, and the mutual actions among them, especially if the distances between them are great in comparison with their size, must therefore be of a very complicated nature. As yet we have little knowledge on this subject. In what follows we will confine ourselves to the treatment of the motions of particles acting on each other with forces which are functions only of the distances between them. SECTION XVII. THE CENTRE OF GRAVITY. Gravity acts on all parts of a body; the forces thus arising may be considered parallel for all parts of the same body. The action of gravity on all the particles of a body may be combined in a resultant whose point of application is at the centre of gravity. If the centre of gravity is rigidly connected with the body and rests SECT. XVII.] THE CENTRE OF GRAVITY. 51 on a support, the body is in equilibrium in any position. Since gravity is proportional to the mass, the centre of gravity coincides with the centre of mass. The resultant applied at the centre of gravity is the weight of the body. The straight lever is in equilibrium if there is applied to its centre of gravity a force equal to its weight and acting in the opposite direction ; the particles on the one side of the centre of gravity tend by their weight to produce rotation in one sense which is equal to that produced in the opposite sense by the particles on the other side. Let the masses ra x and m 2 , whose velocities are represented by A A' and BB' (Fig. 27), be situated at the points A and B. Let the point C be so determined on the line joining A and B that m l AC = m. 2 BC. The point C is then .called the centre of gravity of the two masses m l and m 2 . If the point C' is so determined on the line joining m 2 B'C', we may consider CC' the velocity of FIG. 28. tJie centre of gravity. If AD and BE are equal and parallel to CO', the velocity AA of the mass m l may be resolved into the components AD and DA', and similarly the velocity BB' may be resolved into the components BE and EB'. Now, since m l /m. 2 = BC/AC=B'C'/A'C', the triangles A'CfD and B'C'E are similar, the sides A'D and B'E are parallel, and hence m l /m 2 = B'EIA'D^ The velocities of the masses may be considered as compounded of the velocity of the common centre of gravity and two velocities v l and v 2 , which are parallel to each other and inversely proportional to the masses ; so that If, therefore, oa and ob (Fig. 28) represent the velocities of the masses m-^ and m 2 , and if ab is divided by the point c into the parts ac and be, which are inversely proportional to the masses, then oc represents the velocity of the centre of gravity, and ca and cb 52 GENERAL THEORY OF MOTION. [CHAP. i. represent the velocities of the masses m l and m. 2 relative to the centre of gravity. It is convenient to resolve the velocity in this way, because the velocity of the centre of gravity is changed by external forces only. If momenta are resolved and compounded like forces, then from Fig. 28 the momentum of the centre of gravity, in which we may consider both masses united, equals the resultant of the momenta of the separate masses m l and m 2 . The momentum m l .oa may be resolved into m 1 oc + m 1 c, the momentum m 2 ob into m 2 oc + m 2 cb. Now, m^ca and m.ycb are equal but opposite in direction. Hence we have for the resultant momentum m^c + m. 2 oc = (m l + m 2 )oc. The velocity of the centre of gravity remains unchanged if the bodies m^ and m 2 act on each other according to the law of action and reaction. In this case both bodies receive momenta which are equal but oppositely directed, and which annul each other. This result may be derived analytically in the following way. If x l and x 2 are the coordinates of the particles m^ and m. 2 , the line joining which is taken for the a-axis, and if the ,r-components of the forces with which the masses act on each other are X^ and X>, the equations of motion are (a) m 1 x l = X l and m^c z = X 2 . Adding these equations, we have (b) d*(m l x l +m

) 2 situated at that point. SECTION XVIII. A MATERIAL SYSTEM. We will now consider a system of separate masses in vacuo, which act on each other with forces which are functions of the distances of the masses from each other, and obey the law of action and reaction. The forces which act in such a way within the system are called internal forces. External forces, proceeding from bodies which do not belong to the system, may also act on it. The masses are designated by %, m 2 , m 3 , etc., and the positions of the masses are determined by the coordinates x, y, z, with appropriate indices. We may determine the position of the system by supposing each mass to be made up of different numbers of units of mass ; the mean values , y, of the x-, y-, ^-coordinates will then be (a) = (mft + m^ + m y x 3 + ...)/(m l + m 2 + ...). etc. , -i], are the coordinates of the centre of gravity of the system of masses. If the equation (a) is differentiated with respect to the time t, it appears that the velocity of the centre of gravity depends on the velocities of the particles. That is, (b) = (m^ + m^c 2 + m^b s + . . . )/(/, + m 2 + m 3 + ... ), etc. The internal forces cannot change the motion of the centre of gravity, since by the law of action and reaction two masses impart to each a, other equal and opposite momenta, the sum of whose projections on any axis is equal to zero. This result may be represented geometrically in the following way. From any point (Fig. 29) draw the lines Oa, Ob, Oc, etc., which repre- sent the velocities of the masses m v m 2 , m 3 , etc. Then if the masses m v m v m s , etc., are placed at the points a, b, c, etc., respectively, and if p is the centre of gravity of the masses, Op is the velocity of the centre of gravity. 54 GENERAL THEORY OF MOTION. [CHAP. i. In order to determine the motion of the separate particles we must know the separate forces which act on them. If the com- ponents of the external forces acting on the mass m a are designated by X M Y a , Z a , if F ab is the force with which m a is attracted by m M and if r,^ is the distance between rn a and m b , the ,r-component of the forces acting on m a is X a + F ab .(x a -x b )/r ab + F ac (x , so that b coincides with B and c is brought to c'. By a second rotation about the axis AB, c' may be made to coincide with C. The motion of the body is thus reduced to a translation and two rotations. A rigid body, therefore, cannot be moved, if it can neither be displaced nor rotated. In order that a body acted on by external forces shall be in equilibrium, its centre of gravity must remain at rest. The necessary condition for this is that the resultant of the external forces is zero. It must also have no rotation about any axis. If such rotations exist, its particles receive a certain momentum, which has a moment with respect to the axis. Since each of the elements in this moment is positive, because the parts of the body all move in the same sense, and, therefore, the momenta of the separate particles have the same sign, the sum of the momenta can vanish only when each one of them is separately zero. Now the sum of the moments of momentum is [XIX.] equal to the product of the moment of force and the time during which the force acts. Hence it is required for equilibrium that the forces which act on the body have no moment with respect to the axis. This must hold for each axis about which the body can turn. Now, if the moment of the forces equals zero, the sum of the moments of momentum equals zero, therefore the moment of momentum of each particle equals zero; that is, each particle is in equilibrium. Furthermore, since moments can be compounded like forces, equilibrium will exist if the moments with respect to three arbitrary axes are zero. FIG. 34. 60 GENERAL THEORY OF MOTION. [CHAP. i. SECTION XXII. ROTATION OF A RIGID BODY. THE PENDULUM. Let a solid body revolve around an invariable axis, which is chosen as the 0-axis of a system of rectangular coordinates. Let the angular velocity of the body be w. If r represents the distance of any particle m from the 2-axis, the velocity of this particle is ro> and its kinetic energy ^mr-wr. Since tu has the same value for all particles, the kinetic energy T equals T=W 2 2mr 2 . The factor 2mr 2 is called the moment of inertia J of the body with respect to the 3-axis ; the moment of inertia is equal to the sum of the products of the particles into the squares of their respective distances from the 0-axis. Hence we have T=\(-J, that is, the kinetic energy of a rotating body is equal to its moment of inertia multiplied by half tJie square of its angular velocity. A length K may always be found, such that 2w 2 = K' 2 -m. This length is called the radius of gyration of the body. It is the distance from the axis at which a mass equal to the mass of the body would have the same moment of inertia with respect to the axis as that of the body. If the only external forces which act on the body pass through the axis, the work done by them is zero, since the axis does not move. Since the internal forces also do no Avork, the kinetic energy and therefore also the angular velocity o> must remain constant. Since [XVIII.] the centre of gravity moves as if the resultant of all the forces acted on the mass of the body concentrated at the centre of gravity, this resultant R can be determined. If we represent the distance OP (Fig. 35) of the centre of gravity from the r-axis by a, we have [IV. (b)] E = ^ma-(a-/a = ^mao)-. E is the resultant of the forces with which the body acts on the axis of rotation. In general, the forces applied to the body so act that they have no resultant; they tend only to produce rotation about the axis. In order to determine them, the theorem in XIX. concerning moments of momentum must be used. If external forces act on the body, its angular velocity changes. The amount of this change is determined from XIX. The momentum of a particle m is represented by mro>, and its moment of momentum by mrtar. Hence the moment for all particles of the body is by dS = + ^\ since the terms 2w.?', rj^my' and 2mtf vanish. Now, if we set /= a 2 2w + & 2 2m, where k is the radius of gyration, we obtain from (e) (g) t = irj(a 2 + k*)/ga. We call l = the reduced length of the pendulum or the length of the equivalent simple pendulum. The point S which is at the extremity of the line OS = l (Fig. 35), drawn through and P, is called the centre of oscillation. If an axis is passed through S parallel to the -axis, and the body oscillates about it, the reduced length of the pendulum I' is Since, however, l-a = k 2 /a, we have I' = (a 2 + k 2 )/a = I. The reduced length of the pendulum and therefore the time of oscillation are the same for this new axis as for the former one. CHAPTER II. THE THEOEY OF ELASTICITY. SECTION XXIII. INTERNAL FORCES. IF all parts of a body are in equilibrium and if no tensions or pressures act on them, yet internal forces must be present acting between the separate parts of the body. Every action produces changes of form in the body, and thus develops forces in its interior, which act in a sense opposite to the external forces. These internal forces con- dition the nature of the body, determining, for example, the difference between solids and fluids. No sharp distinction can be drawn, how- ever, between these two classes of bodies. Viscous fluids and jelly- like solids are bodies which seem to be transition forms between true solids and fluids. If a pressure acts on the surface of a fluid, it must be equally great on equal areas of the surface at all points, and it must be perpendicular to the surface, if the fluid is to be in equilibrium. This pressure is exerted throughout the whole mass ; all equal surface-elements at a point are subjected to equal pressures, which are always perpendicular to the surface-elements. We call such a pressure hydrostatic pressure. A similar pressure may also be present in solids. If a solid, a piece of glass, for example, which fills the volume enclosed by its external surface, is immersed in a fluid on which a pressure is exerted, the same pressure exists at every point in the surface of the glass as in the fluid. The pressure is everywhere the same, and perpendicular to the surface-elements. We may there- fore speak of hydrostatic pressure in solids also. Yet, in general, internal forces in solids are very different from those in fluids. Let a cylindrical rod be fastened at one end, and let the force V be applied at the other end so as to lengthen the 62 CH. II. SECT. XXIII.] INTERNAL FORCES. 63 rod. In a cross section perpendicular to the axis of the cylinder the internal forces are everywhere equal. Let the area of the cross section be A (Fig. 36), then the force V\A acts on unit of area in A. This quotient represents the stress S in the rod. If another plane cross section B is taken in the rod, which makes the angle < with A, the force & acts on each unit of area of B, so that & . AJ ^ *b . ^4 = o . Jj cos . The stress S' is no longer perpendicular to the surface B on which it acts ; its magnitude decreases with cos and vanishes for < = |TT. A surface-element within the cylinder and . parallel to its axis is therefore subjected neither to pressure nor to tension ; this conclusion holds for an element of the surface of the cylinder. We may resolve S' into two components, one of which, T, is tangent, and the other, N, normal to B, and have (b) N=Scos- sin . If internal forces of this type exist within a body, we call the stresses axial. In the direction of the axis the stress is S; a unit of surface whose normal makes the angle < with the axis is acted on by a force S cos < in the direction of the axis. We will consider a rectangular parallelepiped (Fig. 37), of which the lines OA, OB, and OC are adjacent edges. The stresses which act on each unit of area of the faces which are perpendicular to OA, OB, and OC are S a , S b , S c respectively. If the normal to an arbitrarily situated unit of surface / makes the angles a, (3, y, with the edges OA, OB, OC respectively, the force acting on / is the resultant of the forces S a cos a, S b cos /3, S c cos y, which are parallel to OA, OB, OC re- / spectively. If the stresses S a , S b , S c have the same value S, this resultant is &/ cos 2 a + cos L '/3 + cos'-'y = S. Hence three equal stresses which are perpendicular to each other cause a hydrostatic stress, since their resultant has the same value whatever may be the position of the surface. Since the components of this stress are S cos a, S cos ft, and S cos y, it is perpendicular to the unit area /. y B ,/ ' A x o / / J3' FIG. 37. 64 THE THEOEY OF ELASTICITY. [CHAP n. On the other hand, if S c = and S a = S b = S, that is, if two stresses act at right angles to each other, while the stress perpendicular to them both is zero, the components in the directions OA, OB, OC respectively are S cos a, S cos /3, 0. Hence the force acting on / is Sj cos-a + cos 2 /? = &Jl - cos-y = S sin y, and is perpendicular to OC. Such a state of stress in a body may be called equatorial. The plane which contains OA and OB, or rather, every plane parallel to both these lines, may be called an equatorial plane. The same stress S acts on each unit area perpendicular to the equatorial plane. If the normal to the surface / makes the angle (f) with the equatorial plane, the stress on it is proportional to cos <. SECTION XXIV. COMPONENTS OF STRESS. Let the surface F (Fig. 38) divide a body into two parts, A and B. If the portion of A which touches the element dF of the surface F is removed, a force must act on dF to keep B in equilibrium. This force SdF is not, as a rule, perpendicular to the element dF. The forces acting at the various points of F are, in general, different. If the force tends to move the element dF into the space occu- pied by B, it is called a pressure on the surface dF; if it tends to move the element dF into the space occupied by A, it is called a tension. In all cases we call the force S a stress; if this acts as a tension, it is a positive stress, if as a pressure, it is a negative stress. If the part of B which touches dF is removed, then to maintain equilibrium in A a force SdF must act on dF, since action and reaction are equal Hence both forces which act on an element of surface within a body are equal, but oppositely directed. It is characteristic of a stress that it may be looked on as made up of two equal and opposite forces. FIG. 38. SECT, xxiv.] COMPONENTS OF STRESS. 65 If the surface-element dF remains in its original place in the body, but is turned about one of its points, a particular value of the stress corresponds to every one of its positions; for special positions the stress may be zero. When the body in which the surface is drawn is a fluid, the stress is independent of the position of the surface. We assume in the body a system of rectangular coordinates. The stresses in the surface-elements, which are perpendicular to the directions of the axes, are determined by their components. Let the surface-element dF be perpendicular to the z-axis, and let dF = dy . dz. If that part of the body is removed which lies on the positive side of the surface-element dydz, the positive side being determined by the positive direction of the z-axis, then, to maintain equilibrium, a force Sdydz must act on the surface dydz. The force AS' is resolved into the components X a Y a Z^ which are respectively parallel to the coordinate axes. The index indicates that the forces act on an element which is perpendicular to the a--axis. X x is perpendicular to the surface-element; it is therefore called the normal force ; Y x and Z x are tangential forces. Xow, let the element dF remain in the same place, but be turned so that it is perpendicular to the y-axis. We may then set dF=dzdx. As before, there are three components of force X^ Y y , Z y acting on the surface-element dzdx, of which Y y is the normal force, X y and Z y are the tangential forces. If the surface-element dF is turned so as to be perpendicular to the z-axis, we have as components X a Y a Z,, of which Z t is the normal force and X z and Y z are the tangential forces. There are therefore, in all, nine components, X a Y^Z*; X0Y,,Z,; X a Y z , Z z . By these components the stress on any surface is determined. Let OA, OB, 00 (Fig. 39) represent line-elements, parallel respectively to the x-, y-, 2-axes. Let a plane be passed through A, B, and (7, so as to form the tetrahedron OABC. Let P, Q, and R be the components of the stress in the direc- tions of the coordinate axes at a point in the base ABC of the tetrahedron. We now form the equation of condition, which must hold that the tetrahedron shall not ' FIG. move in the direction of the z-axis. The forces which tend to move the tetrahedron in that direction are P. ABC acting on its base, and -X X .OBC, -X y .OAC, -X Z .OAB acting on its faces. Hence the force which urges the tetrahedron in 66 THE THEORY OF ELASTICITY. [CHAP. n. the direction of the o-axis is P. ABC - X x . OBC-X y .OAC- X : . OAB. Designate by a, /3, y the angles made with the axes by the normal to the surface ABC drawn outward from the tetrahedron; then the expression for the force in the direction of the ,r-axis becomes (P - X x cos a - X y cos (3 - X z cos y) . ABC. Now, if no external attractions or repulsions act on any part of the body, the conditions of equilibrium, obtained by setting this, and the two similar expressions which hold for the other axes, equal to zero, are f P = X x cos a + Xy cos ft + X z cos y, (a) I Q=Y x cosa+ F y cosp+Y s cosy, [^=^cosa+ Z y cos /3 + Z t cos y. If other forces besides the stresses act on the parts of the body, these must be taken into account in equations (a). If the force X acts on the unit of mass in the direction of the a;-axis, the force acting in that direction on the tetrahedron is Xpdv, if dv represents its volume, and p its density. The condition of equilibrium in the direction of the z-axis then becomes (P - X x cos a - X, cos j3 - X, cos y) . ABC + Xpdv = 0. Now, since dv = %h . ABC, where h is the height of the tetrahedron, this equation is equivalent to P - X x cos a - X y cos (3 - X z cos y + $hpX = 0. Since the height h of the tetrahedron is infinitely small, we may neglect the term containing it, and again obtain the first of equations (a), which hold generally. In order to exhibit the meaning of equations (a), we will consider the following case. Suppose a tension S to act in the direction of the re-axis, and a pressure of the same value to act in the direction of the y-axis. Then X X = S, Y y = - S, and all other components of stress are equal to zero. Hence P = S cos a, Q = -Scosfl, E = 0. The resultant A of these components is ^=$siny. If A, //,, v are the angles between A and the axes, we have cos A B cos M = -. . cos v = 0. The angle e between A and the normal sm T o . , >/} to the surface-element considered is determined by cos e = ' . - -. smy If the surface - element is parallel to the ^axis, y = a A=S, COS e = COS 2a, = 2a. SECT. XXIV.] COMPONENTS OF STEESS. 67 If a= then e = ^, the resultant is a tangential force. Thus the surface of a prism whose axis is parallel to the s-axis, and whose o 1 FIG. 39 a. make angles of 45 with the xz- and y^-planes, is acted on only by tangential forces, each equal to S. SECTION XXV. RELATIONS AMONG THE COMPONENTS OF STRESS. The force which acts on the volume-element dxdydz (Fig. 40) is determined from the components of stress. Let the components acting at the point be given, and let the force which acts on OA' in the direction of the x-axis be equal to - Xjlydz. By development by Mac- laurin's theorem we obtain for the force acting on AO' the expression (X x + 'dX x /'dx.dx)dydz. The resultant of these two forces is 'dXJ'dx . dxdydz. The forces - X y dxdz and ( X u + 'dXJ'dy . dy}dxdz, whose re- sultant is 'dXj'dy . dxdydz, act on the surfaces OB' and O'B respectively in the direction of the z-axis. The resultant of the forces acting in the same direction on the surfaces O'C and OC' is 'dXJ'dz . dxdydz. Hence the total force acting on the parallelepiped dxdydz in the direction of the -axis is 68 THE THEORY OF ELASTICITY. [CHAP. n. If (X), (Y), and (Z) represent the components of the force with which the stresses act on unit of volume, we have (X) = 'dXJ (Y) = VYJ-dx + 3 YJ-dy + 'd YJ-dz, (Z) = c)ZJ3x + 'dZ y fdy + 'dZJ'dz. If the body is acted on only by stresses, equilibrium will exist if the three components (X), (Y), (Z) are each equal to zero. The equations (b) in this case are three differential equations which the components of stress must satisfy. If a force whose components are X, Y, Z acts on each unit of mass, and if the density of the body is p, we obtain the conditions of equilibrium, [ VXJ'dx + VXJdy + 'dXJ^z + P X= 0, (c) I 'dYJ'dx + 'dYJ-dy + 'dYJ-dz + pY^O, ( -d Z x /*dx + 3 ZJdy + ~d ZJ-dz + P Z=0. Internal forces produce both translations and rotations in the The tangential components tend to rotate the parallelepiped 00' about the -axis. The tangential force X y acts on the surface OB' in the .negative direction, while the tangential force X y + 3 XJ'dy . dy acts on the opposite surface O'B in the positive direction. These two forces form a couple acting on the parallelepiped with a moment X y . dxdz . dy, if terms of an order higher than the third are neglected. This moment tends to turn the parallelepiped about the 2-axis in the negative direction. The tangential forces acting on the surfaces OA' and O'A have the moment Y z . dydz . dx, which tends to turn the parallelepiped in the positive direction. The total moment which tends to rotate the parallelepiped about the z-axis is (Y x - X y )dxdydz. If the body is in equilibrium under the action of the stresses con- sidered, this moment must be zero, that is, (d) Y x = X y , and similarly Z y = Y a X,= Z^ The last two equations are derived in the same way as the first. If attractive forces, such as gravity, or in general, if any forces acting at a distance act on the body, equations (d) will still be applicable. The point of application of such forces, in infinitely small bodies, coincides with the centre of gravity ; such forces, therefore, cannot produce rotations, and, therefore, cannot make equilibrium with the forces which tend to rotate the body. It appears from equations (d) that six quantities are sufficient to determine the stress at a point in a body, namely, X x , I 7 ,,, Z z ; Z y =Y a X 2 = Z X , Y x = X y . The first three are normal forces, the other three tangential farces. It is possible to express these forces by a SECT, xxv.] COMPONENTS OF STEESS. 69 simpler notation, but we will retain the above, which has the advantage that it exhibits more clearly than any other the true significance of the quantities involved. It must be borne in mind that the value of a component of stress remains unchanged if the direction of the force and the direction of the normal to the surface-element, on which the stress acts, are interchanged. SECTION XXVI. THE PRINCIPAL STRESSES. In order to obtain a better understanding of the nature of internal forces, we will examine if it is possible to pass a surface through a given point in a body in such a position that no tangential force acts on it. We may anticipate our conclusion by the statement that three such surfaces may be drawn through any point and that they are perpendicular to each other. To show this, we proceed from the equations [XXIV. (a)] ( P = X x cos a + X y cos (3 + X z cos y, (a) \ Q= Y x cosa+Y y cos/3 + F.cosy, [ R = Z x cos a + Z y cos (3 + Z t cos 7, in which a, /3, y are the angles between the normal to the surface and the axes, and determine the position of the surface on which the components of stress P. Q, R act. It is to be shown that this surface may have such a position in the body that the stress acts perpendicularly to it ; we will call the stress in this case the principal stress S. The angles which the direction of S makes with the axes are as before, a, /3, y, and (b) P = Scosa, Q = Scos(3, Introducing these values in (a), we have f (X x - S) cos a + X y cos ft + X, cos y = 0, J (c) If cos a, cos (3, cosy are eliminated from these equations, we obtain (S*-(X t +Y,, + Z t )S* + (X x Y,+ YJ t + Z t X,-Z*-X*-Y 1 ?)S \ - (X x YJ t + 2Z y X z Y x - XJ? - Y y X* - Z Z Y X 2 ) = 0. This equation has always one real root A, and we can find the cor- responding values of a, /?, y from equations (c) and the relation 70 THE THEOEY OF ELASTICITY. [CHAP. n. cos 2 a + cos 2 /? + cos 2 y = 1. Therefore, through any point in the body there may be passed at least one plane having the property that no tangential forces act on it. We call such a plane a principal plant. Let the system of coordinates be so rotated that this principal plane is parallel to the 7/2-plane. On this supposition, we have X X = A, Y x = 0, Z X ^Q. The equations (c) then become (A - S) cos a = ; (Y,,- S) cos p+Y z cosy = Q; These equations are satisfied when we set S=A, cosa = l, cos [3 = cos y = 0. We thus return to the principal plane already found, with its appro- priate normal stress A. The same equations are also satisfied if we set cos a = ; cos /3/cos y = - Yj(Y y - S) = - (Z t - S)/Z, r Since cos a = 0, and a = |TT, the new principal planes are perpen- dicular to the first one. We have further, and cos /3/cos y=\(Y !l -Z 2 J( Y y - Z These equations present two values of S and two values each of (3 and y. If we represent the values of P and y by /5' and /3", y' and y" respectively, we have cos (3' cos P" / cos y' cos y" = -1, and hence cos ft' cos /?" + cos y' cos y" = 0. Since the corresponding values of a are equal to JTT, it follows that the two new principal planes are perpendicular to each other. It is thus proved that, in general, through any point in a body, there may be drawn three surface-elements, and only three, on which only normal forces act, and that they are perpendicular to one another. The normal stresses corresponding to the three planes may be designated by A, B, and C. From (d) the following relations hold among these normal stresses and the components of stress, (e) ( ABC= XJ^i + 1Z V X.Y X - XtZ* - YJt? - ZY?. The first of these equations should be especially noticed; it shows, that the sum of the normal forces for three planes perpendicular to each other is constant. SECT. XXVI.] THE PRINCIPAL STRESSES. 71 If the axes of the system of coordinates are parallel to the directions of the principal stresses A, B, and C, equations (a) become P = Aco$a, Q = Bcos[3, E=Ccosj. IfA>E>C, and we set A = B+S V C=B-S 2 , the principal stresses can be replaced by a hydrostatic stress B and two axial stresses S l and S. 2 , the first of which is a tension, the second a pressure. This investigation shows that through any point in a body three planes can always be passed which are acted on only by normal stresses, equal to the principal stresses A, B, and C. A, B, and C are the three roots of equation (d) ; their directions may be determined by the help of equations (c). A makes the angles a, (3, y, with the coordinate axes. We write cos a x = / 1} cos f3 l = m v and cos y l = n v The corresponding notation for B and C is exhibited in the following table : (g) From equations (c) the following relations hold among these quantities : Am 1 = YJ 1 + C1 3 Em. = Y x l z + Y y m., + Y z n 2 : Bn z = ZJ Z + Z u m. 2 + Zp z A> 3 , J 3 + Z y m 3 + Z t n a . These equations can be solved for the components of stress X x , Y y , etc. These quantities may, however, be determined more easily in the following way. Through a point P draw the lines PA', PB', and PC' parallel to the directions of the principal stresses A, B, and C. These three lines, together with a plane F parallel to the p-plane, deter- mine a tetrahedron. The plane F is so placed that the tetrahedron is infinitely small ; its base is dF. The areas of the faces which meet at P are I^dF, l 2 dF, and l s dF. The force acting on unit area in 1-^dF in the direction of the o>axis is Al^ ; the forces acting on unit area in the two other faces are B1 2 , Cl s , respectively, and the force 72 THE THEORY OF ELASTICITY. [CHAP. n. acting on unit area in dF is X x . That the tetrahedron shall not move in the direction of the ar-axis we must have 2 B . l 2 dF+ 1 3 C . l 3 dF= X x dF or X, = Al* + Bl. 2 2 + C7 3 2 . By a similar process we obtain for the other components the following equations : 2 + Cl 3 n 5 , Cl 3 m 3 . It may easily be seen that these values of the components of stress satisfy equations (h), if the known relations among the quantities given in (g) are taken into account. SECTION XXVII. FARADAY'S VIEWS ON THE NATURE OF FORCES ACTING AT A DISTANCE. Newton considered the action between two masses as an action at a distance which is not propagated from particle to particle of the medium surrounding the masses. Faraday, on the other hand, in discussing electrical action, held that the intervening medium is the seat of the action between two charged bodies, and that the action is transferred from particle to particle. In each of these particles electricity is displaced in the direction of a line of force, one end of which becomes positively and the other negatively electrified. In a body thus polarized the particles are so arranged that poles of opposite name are contiguous. Hence the lines of force tend to contract, and a state of stress arises in the medium. This stress is similar to the elastic stress, and was called by Maxwell electrical elasticity. In Chapter V. of his Treatise on Electricity, Maxwell, using Faraday's hypothesis, developed a theory which we will now proceed to discuss. Since electrical and magnetic forces conform to the same law as that of universal attraction, the discussion may be made perfectly general, and applicable to all forces between bodies which are inversely proportional to the squares of the distances separating the bodies. Let the potential $ be given for all points of the region. The density p is determined from the potential by Poisson's equation (a) 3 2 ^/?.r 2 + 3-Y/3y 2 + ^/dz 2 + TT P = 0. SECT, xxvii.] FORCES ACTING AT A DISTANCE. 73 The mass pdv contained in the volume-element d-v is acted on by a force whose components are The upper sign holds for magnetic or electrical attractions, the lower for mass attractions. Introducing the value of p given in (a) the component acting in the direction of the .r-axis becomes This quantity must be capable of representation as the sum of three differential coefficients with respect to x, y, and z. We have Hence the force which acts in the direction of the a-axis on the volume-element dv is If we designate the components of force which act on the unit of volume by (X), (Y), and (Z) [XXV.], and if, for brevity, we set X = - -d^px, Y=- we obtain f (X) = (b) | (F) ( (Z) = I/STT . [Zd(XZ)/-dx + Zd(YZ)l*dy + 3(Z 2 -X 2 - Since these equations are perfectly analogous to those which deter- mine the force with which stresses act on the unit of volume, we may consider forces acting at a distance as arising from stresses in the medium. If we are dealing with universal mass attraction, the ether may be assumed to be the intervening medium; if we are discussing electrical actions, the dependence of the stress in the ether on the matter which fills the region, air, water, etc., must be taken into account. It is not necessary to enter upon this question in our treatment of the subject. A comparison of equation (b) with equation XXV. (b) shows that X x = (X 2 - Y 2 - Z 2 )/Sir, Y (c) -I Y y =(Y 2 -X 2 -Z 2 )/87r, Z x Z z = (Z 2 -X 2 - FO/STT, X, 74 THE THEORY OF ELASTICITY. [CHAP. 11. To determine the principal stresses in the medium, we use equations XXVI. (e), Avhich give A + B + C = + (X- + F- + Z 2 )/8ir, BC+AC+AB= - ((A' 2 + F 2 + Z 2 )/87r) 2 , ABC = ((X' 2 + Y 1 + Z-')/8ir) 8 , If we set (d) (X' 2 +Y* + Z' 2 )/8ir = S, A, B, and C are the roots of the equation D 3 SD' 2 - SW + S s = or (D + S) (D S)' 2 = 0. We have therefore either (e) A = + S, B=C = -S or A =- S, B=C = + S. Hence two principal stresses are always equal. In order to deter- mine their directions, a, (3, and y must be calculated from XXVI. (c). It is easiest to determine the directions of the equal stresses B and C. If the values of S, given in (d), are substituted for S in the equations referred to [XXVI. (c)], using the negative value of S in combination with the positive value of X x , etc., and vice versa, we obtain (f) A" cos a + Ycos (3 + Zcosy = Q. Hence both of the equal principal stresses are perpendicular to the direction of the force ; the third principal stress is in the direction of the force, and is equal to the square of the force divided by STT. It has thus been shown that all forces acting at a distance may be explained by a state of stress in an intervening medium. From this point of view universal mass attraction is replaced by a negative stress, that is, a pressure, in the direction of the lines of force, and a positive stress, that is, a tension in all directions perpendicular to the force. A surface-element which lies perpendicular to the direction of the force is acted on by a tension which is equal to the force. In the case of magnetic and electrical attractions the opposite holds true. There is no independent evidence for the existence of such stresses in the case of gravity ; but several phenomena in electricity indicate that the medium between two electrified bodies is in a state of stress, and no facts are known that are inconsistent with the assumption that this stress is the cause of the forces acting on the bodies. SECTION XXVIII. DEFORMATION. If a body changes its shape or its position in space, one of its points, whose coordinates are originally x, y, .:, may be so displaced that its coordinates become x + , y + ??, z+ , ->;, are the pro- jections of the path which P has traversed or the components of SECT. XXVIII.] DEFORMATION. the displacement. If , 77, are given as functions of the time, the position of the point P at any instant is determined. The motions of the separate points of the body are in general different, that is, , ?/, are functions of x, ?/, z. We will first consider some simple motions of the body. If , 77, are equal for all points of the body, the points all move through equal distances and in the same direction ; the motion is a translation. In this motion all parts of the body remain at fixed distances from each other, and there are no internal forces developed. This holds also in the case of a rotation of the body about an axis. Let the axis of rotation be parallel to the o;-axis, and pass through the point P (Fig. 41), whose coordinates are x, ?/, s. Let a point Q, whose coordinates are x', y , z, traverse the path Qli = li x . r, where r=QS is the distance of the point Q from the axis, aud h x is the angle of rotation. By this rotation the ^/-coordinate is diminished by BB' = QR(z' -z}/r = h x (z'- z), and the .^-coordinate is increased by CO' = QR(y' -y}lr = h x (ij -y). If the body rotates at the same time about two other axes, which are parallel to the y- and z-axes, and if the angles of rotation are designated by li y and h z respectively, the coordinates of Q are increased by , 17, , which have the following values : * W --9)*.- &-*)*,- We may now proceed to the discussion of the general case, in which the points of the body change their relative positions. Let the point P, whose coordinates are .T, y, z, pass during this motion to the point P', whose coordinates are x + g, y + r,, z + ; let another 76 THE THEORY OF ELASTICITY. [CHAP. n. point Q, whose coordinates are originally x\ y', z 1 , pass to the point Q, whose coordinates are x' + ', y' + i/, z' + '. If is a known func- tion of x, y, z, we will have ' = + (x' - x^j-dx + (y - y)3/3// + (z' - *)3/3* + . .* . We may assume that P and Q are infinitely near, so that x' x = dx, y' y = dy, z' z = dz. Neglecting terms of the second order we obtain the following relations, ' = + 3/d.r . dx + ^I'dy . dy + V/3z . dz, 77' = ?; + 'drf/'dx . dx + 'drj/'dy . dy + 'dq/'dz . dz, (' = + 3/3.r . dx + 'dtfdy . dy + *dtfdz . dz. By introducing the following notation, y x = x y we obtain (' = 4- x,dx + x v dy + x.dz - h z dy + hjlz, rj' = r] + yjlx + y y dy + yjlz - hjz + h z dx, T = C + M* + z^y + z z dz - h y dx + h,dy. These equations determine the motion of a point in the neighbour- hood of P. This motion is compounded of a translation, whose components are , ;, a rotation, whose components are h^ k y , h : , and two motions, determined by x^ y^ z z and z f x a y f If we confine our attention to the way in which the form of the body changes, we need only consider the motion whose components e?, d^, d are determined by the following equations : dr) = y r d d = z t dz + z,dx + z y dy. To interpret the coefficients x z , y^ s z and z^ a;,, y^ we assume that all except x x are equal to zero. Then dg = x x .dx and dr) = d=Q. The change of form corresponding to this is a dilatation of the body in the direction of the ar-axis, by which dx increases by d. The co- efficient x x therefore represents the dilatation of a unit of length parallel to the x-axis, or is the dilatation in the direction of the ;e-axis. Hence y v and z t are the dilatations in the directions of the y- and 2-axis respectively. SECT, xxvin.] DEFORMATION. 77 If, on the other hand, all the coefficients vanish with the exception of z y , we have dg = Q, d^=z y .dz, d=z y .dy. The particles are dis- placed in a plane parallel to the yz-plane, and their distances from the yz-plane remain unchanged. Let the original coordinates of the point P (Fig. 42) be x, y, z; let ABCD be a square, the length of whose sides is 2a. The point A, whose original coordinates were x, y + a, z + a, referred to the axes PY and PZ, is displaced to A\ whose coordinates are a + z y a, a + z y a. A therefore lies on PA produced. The points B and D are displaced to B' and Z>', which lie on BD ; C is displaced along AC produced to C". The square ABCD becomes the rhombus A'B'C'D. This change of form is called a shear; the quantities z y , x a y x are called components of shear. In the theory of elasticity we consider ^ FIG 49 only very small deformations of the body ; the components x x , y y , etc., are consequently small quantities, whose second and higher powers may be neglected. The volume of the body is not changed by a shear; the square whose area is 4a 2 will become a rhombus A'B'C'D' whose area is 2PA' . PB' = 2(a 4 z y a)j2(a - z y a)j2 = 4a 2 (l - z?). If we neglect z y -, the area of the square is equal to that of the rhombus; hence the volume will not be changed by the shear. From Fig. 42 it is evident that the infinitely small angle between AB and A'B' is equal to az y /a = z y ; hence the right angle DAB is diminished by the shear by 2z y , so that As the result of a dilatation determined by x a y y , z z , the volume of the parallelepiped dxdydz becomes dxdydz(\ + x x )(l +y y )(l+z z ). If the components of dilatation are supposed infinitely small, we may neglect their second and higher powers. Hence the increase in unit volume is = x x + y y + z z . is called the volume dilatation. Sub- stituting the values of x^ y^ z a we have also (e) Q = 'dfdx + 'fr)l'dy + 'dtrdz. Let dr be an element of a straight line which makes the angles a, /3, y, with the coordinate axes ; then dx = dr cos a, dy = dr cos (3, dz = dr cos y. 78 THE THEORY OF ELASTICITY. [CHAP. n. By the deformation dr becomes dr', and makes the angles a, (3', y' with the axis, so that dx + dg = dr'cosa ; dy + djj = dr' cos (B' ; dz + d =dr' cosy', from which dg, dt], d may be determined by equations (d). If the direction of the line dr remains unchanged, we have a = a, /3 = /3', and 7 = 7', and hence dg = dpcosa, drj = dp cos (3, d=dpcosy, where dp = d(r'-r). The length dp is the elongation of dr, and dp/dr is the dilatation s in the direction of the line dr. Hence we have .9 = dp/dr. Equations (d) then assume the following form: {(x x - s)cos a + x y cos (3 + x z cos 7 = 0, y x cos a- + (y y - s) cos (3 + y z cos y = 0, 2^ cos a + z y cos (3 + (z z - s) cos 7 = 0. A comparison of these relations with those of XXVI. (c) shows that they both may be interpreted in a similar way. There are therefore three directions perpendicular to each other, called the principal axes of dilatation, in which only dilatations occur ; every line-element which is parallel to one of these three directions contains after deformation the particles which were in it before the deformation. This conclusion holds only on the supposition that the body does not rotate, a supposition which has been made in deducing equations (d). If the principal dilatations thus deter- mined are called a, b, c, we have, as in XXVI. (e), f a + b + c = x x + y y + z z , (g) j be + ac + ab = z$, + x^ + yp x - z* - x* - y*, { abc = xji^ t + 2ZfCjf f - x^f - yp* - zjy*. The first of these equations shows that the volume dilatation does not depend on the position of the system of coordinates. In the same way as that in which the components of stress are ex- pressed in terms of the principal stresses [XXVI. (i)] x a x y ,... may be expressed in terms of the principal dilatations a, b, and c. Denoting the cosines of the angles which the direction of a makes with the axes by l v m v n v and the cosines of the angles which b and c make with the axes by 1 2 , m. 2 , n 2 ; 1 3 , m 3 , %, we obtain f x x = al l 2 +bl. 2 2 +c/ 3 2 ; 2,om 1 n (h) | y y = a, l 2 + bm 2 2 + cffl./ ; x t -=al l n 1 I z t = an^ + bn. 2 - + en./ ; y x = al-pi^ SECT. XXIX. STRESSES AND DEFORMATIONS. SECTION XXIX. RELATIONS BETWEEN STRESSES AND DEFORMATIONS. The study of the deformations of an elastic body has shown that a parallelepiped which is stretched by forces applied to its ends, is increased in length and diminished in cross section. If we only consider forces which are so small that the limits of elasticity are not exceeded, the elongation s per unit of length is s = S/E, where E is the coefficient of elasticity and S the force acting on the unit of surface. The contraction s' per unit of length parallel to the end surfaces, is given by s' = k . S/E, where k is a constant. It is assumed that the body is isotropic, that is, equally elastic in all directions and at all points. We will first consider a rectangular parallelepiped, whose edges are parallel with the coordinate axes. The normal forces are denoted by X x , Y y , Z a and a unit of length which is parallel to the a-axis increases by x x ; the units of length which are parallel to the y- and 2-axes respectively increase by y y and z z . We then have -k(X x +Y y )/E. From XXVIII. (e) the volume dilatation is From these equations we deduce X x = kEQ/(l + k)(l- 2k) + Ex x /(l + k). Setting A. = kE/( I + k) (1 - 2k), p. = i/(l + k), we obtain (a) X, = XO + 2/tr, ; Y y = XQ + 2p.y y ; and by addition (b) X x + Y y + Z z = (3X + 2^)9. To investigate the relation between the shears and the tangential forces we may use the following method, due to V. v. Lang.* If the prism ABCD (Fig. 43) is stretched by the tension S applied to each unit of surface of its ends AB and CD, it takes the form AB'C'D'. Four plane sections EF, FG, GH, and HE are passed through the prism, which mark out the rectangle EFGH on a plane parallel to the axis; the rectangle EFGH becomes by deformation * V. v. Lang, Theoretisc.he Physik; 411. 80 THE THEOEY OF ELASTICITY. [CHAP. n. the parallelogram E'F'G'H'. The angle AFE is represented by . The tangential stress T, which acts on the surface EF in the direction EF, is given [XXIII. (b)] by T = S sin cos <. Since < BFG = |?r - <, the same tangential stress T acts on GF in the direction GF. On deformation the angle AFE becomes AF'E' = + d, and we have Now since s=--SjE and s' = kS/E are infinitely small, we have ( 1 + s)/(l - s') = 1 + s + s' = 1 + (1 + k)S/E. Further, we have and tg (< + dfy = tg + d(f>/co$ 2 , so that d = (l+k)S sin cos , the angle EFG diminishes by 2d(f>, and the angle FGH increases by 2d=2(l+k)T/E. But 2d is the quantity 2z y pre- viously introduced, when the rectangle EFGH is parallel to the 2/3-plane; and hence T=Z y and z y = (l+k)ZJE. If we set we have (c) ^ = 2^, A>2/^ z , Y x = 2^y x . The equations (a) and (c) are the solution of the problem, to find the components of stress, when the deformations are given, and conversely. They contain only two constants, A and /*, which involve the deformations caused by simple dilatation in the following way : f X = */(l +*)(!- 2*); ,* = t-<8V+4i*V(*+p); * Since A and /A are positive, & must be less than \. The relations between the elastic forces and the deformations may also be derived by another method. Let the principal stresses A, B, and C, at the point P, be known in magnitude and direction [cf. XXVI. (g)]. An infinitely small parallelepiped, whose edges are parallel to the directions of the stresses A, B, and C, is extended in those three directions. The increments a, b, and c of the unit of length are parallel to A, B, and (7, and as in (a), we have (e) A = \e + -2iM, B = XQ + 2pb, when = a + b + c, or [XXVIII. (g)], Q = SECT, xxix.] STRESSES AND DEFORMATIONS. 81 By applying the formula XXVI. (i), we obtain the equation X x = X0 + 2/x(a/ 1 2 + bl/ + c/3 2 ), which, from XXVIIL (h), becomes X x = X9 + 2/xa^ The expressions for Y y and Z t are obtained in a similar way. From XXVI. (i), we have and hence [XXVIIL (h)] Z y = 2fj.z y . We obtain the expressions for A", and Y x in a similar way. The coefficients E and k depend on the nature of the body. It was at one time believed that k had the same value for all bodies. This opinion was first expressed by Navier. He assumed that bodies are made up of material points which repel one another, and on this assumption concluded that k = . Poisson also had the same opinion. While k is a mere number, the coefficient of elasticity E is deter- mined by E = S/s; the fraction \\E is called the modulus of elasticity. Sis the force which acts on the unit of surface, and [III.] its dimensions are LT~ 2 M/L 2 = L~ 1 T~' 2 M. Since s is the ratio between the elongation and the original length, it is also a mere number. Hence the dimen- sions of E are L~ 1 T-' 2 M. In practical units E denotes the number of kilograms which would produce an elongation in a rod of one square millimetre cross section, such that its length is doubled. In order to transform it into absolute measure, we notice that the weight of one gram is about equal to 981 dynes, and that therefore the weight of one kilogram is equal to 981 000 dynes. The cross section must be taken equal to 1 sq. cm., and the number must therefore be multiplied by 100, so that the factor of transformation becomes 98,100,000. According to Wertheim, E equals 17278 in practical units for English steel ; therefore, in absolute units it equals 17278. 981 . 10 5 = 1,695 . 10 12 . In the case of fluids, the discussion is simplified by the condition that a fluid always yields to tangential forces, so that, when it is in equilibrium, there are no tangential forces acting in it. This condition, from (c), enables us to set /^ = 0. If the fluid is subjected to the pressure p, and if its volume v is thereby diminished by dv, we have from (b) (f) -{ or, since /^ = 0, dv =pv/X. 82 THE THEORY OF ELASTICITY. [CHAP. n. If, for example, the unit volume of water is diminished by 0,000 046, when the pressure is increased by 1 atmosphere, we have X =pv /dv = 7& . 13,596 . 981/0,000,046 = 2,204 . 10 10 . In the case of gases, if we represent the original pressure by P, and its increase by p, Mariotte's law gives the equation Pv = (P+p)(v-dv). Assuming that p is very small in comparison with P, we obtain dv=pv/P, and therefore, for gases, we have from (f), (g) P=X. SECTION XXX. CONDITIONS OF EQUILIBRIUM OF AN ELASTIC BODY. If a force whose components are X, Y, Z acts on the unit of mass of a body, we have from XXV. (c) (a) VX,J'dx + 'dX,l'dy + 'dXJ'dz + pX=O t etc. Further [XXIX. (a) and (c)] we have (b) X x = XQ + 2fj..^px, Z y =Y i = p.(dil'dy + 'd-ql^z\ etc. If the values for X M etc., are substituted in (a), it follows that c (X + p) . ae/a-e + /* V 2 + pX = 0, (c) I By the use of our former symbols for the components of rotation, viz., | 2A \ equations (c) become ? - r r Suppose a constant hydrostatic pressure p l applied to the inner surface, and a similar pressure p. 2 applied to the outer surface. The pressures p^ and p 2 are perpendicular to the surfaces. Let the centre of the sphere be the origin of coordinates, and let the distance from of any point in the shell be r. On the hypothesis that has been made with respect to the pressures, all points lying in the same spherical surface having the centre 0, receive equal displacements from the centre. Let the displacement of the point considered be er, where e is a very small quantity. We then have 84 THE THEORY OF ELASTICITY. [CHAP. n. Since e is a function of r only, we may set = fr . x/r = d/dr . "drfdx = 'd^j'dx, where is a new function of r. We may represent ij and in a similar way, so that (b) = 3./3., ^30/fy, t=3/32. Hence we have (c) 6 = V 2 <. The equations XXX. (c), if the action of gravity is neglected, become (A + 2fi).V 2 3/3y = 0, (A + 2/*) . V^fdz = 0, so that (d) = V-< = a, where a is a constant. From XXX. (b) the components of stress are c X x = Aa + 2p . (e) J Y y = Xa + 2 p. I Z, = Xa + 2/i.3 2 ^/?3 8 ; F, = 2/t . 3 2 /3o%. The stress in a surface-element perpendicular to r is given by XXIV. (a), if we set cos a = z/'r, cos (3 = y/r, cos y = z/r. If the components of stress are P, Q, and R, we have P = Xa . a-/r + 2/*(.r/r . 3 2 <^>/3a; 2 + y/r . ^fdxdy + zlr . &/'dzdz). Using the equations Pt/dr* - x*/i* . d/dr +l/r. dj/dr, xy/f* . d*/dr* - xy/r* . d/dr, xz/r* . d^/dr* - xzji* . d/dr, we have P = (\a + 2p. . d^jdr*) . xfr. Similar expressions may be obtained for Q and It. Hence a prin- cipal stress (f) A = Aa + 2/* . d*/dr 2 acts on the surface-element considered. For a surface-element which contains r, the components are obtained in the same way. If a, /?, y are the angles which the normal to the surface-element makes with the axes, we have P = Aa cos a + 2/*(9 2 /3x3^ . cos y). If we notice that in this case cos a . x/r + cos (3 . y/r + cos 7 . z/r = 0, and use the expressions given above for the differential coefficients, we have P = (\a + 2/*/r . d/dr) cos a. We may obtain Q and R by replacing a by ft and y respectively. SECT, xxxi.] TOESION. 85 Hence the principal stress B acting on the element is From (d) and XV. (1) we have and therefore (h) d/dr = *ar + b/r* ; d 2 /dr 2 = ^a- 26/r 3 . From (f) and (g) it follows that A = (X + |/i)a - 4/xJ/r 3 ; B = ( A. + |/z)a + fyb/r 3 . For r = r v A= -p v and for r = r z , A= -p 2 , therefore a = 3/(3X + 2/x) . (pfi and A- B = SECTION XXXII. TORSION. Let us consider a circular cylinder whose axis coincides with the -axis; and let the circle in which the xy -plane cuts the cylinder be the end of the cylinder and be fixed in position. If torsion is applied to the cylinder, a point at the distance r from the axis describes an arc r, parallel to the xy-plane, whose centre lies on the z-axis. This angle, in the case of pure torsion, is proportional to the distance of the point from the zy-plane, so that = kz, where k is a constant. The displacement of this point is krz, and its components , 77, are (a) =-kyz, r, = kxz, f=0. Using these values, we find that the volume-dilatation is zero, that is, pure torsion dots not cause a change of volume. We have further [XXX. (b)], X x = 0, Y y = 0, Z Z = Q, and hence no normal forces act on the surfaces which are parallel to the coordinate planes. On the other hand, we have Z^fjJcx, X t =-fdy, Y x = 0. A surface-element perpendicular to the -axis is acted on by the tangential forces Y,= + fj.kx and Jf z = - pJcy, whose resultant pkr is perpendicular to the radius r and to the z-axis. By XXIV. (a) we reach the same result. That is, we get f P = - ftky cos y, Q = pkx cos 7, I R = -nkycosa 86 THE THEORY OF ELASTICITY. [CHAP. n. For the stress on the surface of the cylinder we must set cos a = x/r, cos f3 = y/r, cos y = 0. We will then have P = 0, Q = 0, ^ = 0. Hence a surface-element perpendicular to the radius, or which is part of the surface of a circular cylinder whose axis is the z-axis, is not acted on- by a force. To find the surface-elements on which the only forces which act are normal forces, we use equation XXVI. (d), which, in the case before us, becomes If A, B, and C are the roots of this equation, we can set A = 0, B = pkr, C=-ph\ If the angles between the axes and the normal to one of these surface- elements are represented by a, /3, y, S cos a = p.ky cos y, S cos f3 = pJcx cosy, Scosy = //.?/ cos a + //Xvr cos /. If we substitute in these equations the particular values of S given by A, B, and (7, the values of a, J3, y thus obtained show that the stress A = acts on a surface-element perpendicular to r ; and that B and C act in directions perpendicular to the radius r, and making angles of 45 degrees with the 2-axis. B acts in the same direction as the torsion, C in the opposite direction. For example, considering a point which lies in the surface of the cylinder and in the .s-plane, and setting therefore Y=0 and A" = r, we have S cos a = 0, S cos /? = pkr cos y, S cos y = pier cos p. When S = Q we have y = /3 = ^7r; when S pkr we have a = ^7r, cos (3 = cosy. Since cos 2 a + cos 2 /? + cos 2 y = 1 , we have eos/? = s /|. The moment of force M to which the torsion of the cylinder is due is M= (f&r. Zirrdr.r. The upper limit R of the integral is the radius of the cylinder, Integrating, we have M=^TrfjJcR* = irii.R 4 j2l, where I is the length of the cylinder and the angle of torsion. The factor T = Trp.R*/2l is called the moment of torsion of the cylinder. It depends only on the dimensions of the cylinder and the constant of elasticity /x. For this reason //, is called the coefficient of torsion. SECT. XXXIII. FLEXURE. 87 0. a SECTION XXXIII. FLEXURE. It is not possible to give a rigorous discussion of the flexure of a prism. We will, therefore, confine ourselves to an approximate calculation in one very simple case*. Let ABCD (Fig. 44) be the prism considered. Its length is supposed horizontal and coincident with the axis Ox. The axis Oz is directed perpendicularly upward, and the axis Oy is therefore horizontal. After flexure, the cross section AB is displaced to A'', which may lie in the same plane as AB. Another plane cross section FG, also perpendicular to the axis, is displaced by the flexure to F'G' ; we assume that the section F'G' is also plane, and that the plane F'G' cuts the plane A'B' in a horizontal line passing y through P. This line of intersection is supposed to be common to the planes of all sections perpendicular to the axis. The parts of the prism which originally lay in OQ lie after the flexure in OQ', FlG 44 which we will consider as the arc of a circle whose centre is P. Such a flexure is called circular. All the lines in the prism which were originally parallel to the x-axis become circles, whose centres lie on the straight line passing through P. Represent the original coordinates of a point M in the section AB by 0, y, z, and its coordinates after flexure by 0, y + ^ , 2+f * The same changes occur in the other cross sections, for example in FG. If the coordinates of a point M' in FG are originally x, y, z, they will become by flexure x + g, y + t], z + We set L. OPQ' = , OP = pa,nd OQ = OQ'. This last assumption is admissible, since there is always one line whose length does not change by flexure, and since we have as yet made no assumption as to the position of the z-axis. We therefore obtain * + 1 = * + to -(/> + * + &)(! - cos <). If p is very great in comparison with x, z, and , we may set sin < = x/p ; 1 cos a^/2p 8 and obtain (a) = xz/p, n = ^ t=to- x2 / 2 P- * Barre de Saint- Venant, Mem. pres. par div. Savants. T. 14. Paris, 1856. 88 THE THEORY OF ELASTICITY. [CHAP. n. We may so determine % and that all the components of stress except X x vanish ; hence we may write (1 ) X, = A6 + 2^/p = S, (4) Z, = (2) y,=Ae+2/u3v^ =0 > ( 5 ) x z (3) Z t = \Q + 2iidf fdz = Q, (6) Y, Further, we have Q = z/p + 'dr] From (2) and (3) it follows that (b) Comparing (b) with XXIX. (d) it appears that the contraction of the cross section is to the increase in length in the ratio of |A. to A + /z, or of k to 1. Further, since T? O and do not involve x, we have from (b) rj = - kyz/p +/(*), = - kz*/2p + g(y), when / and g designate two unknown functions. From (4) we have -ty/p+f'(z) + g'(y) = 0, and hence f\z) = c, where c is an unknown constant. It follows that f(z) = cz + c', g(y) = kf/ 2p-cy + c" and T, O = - kyz/p + cz + c', = k (V 2 ~ Z<2 )/%P ~W + c". At the point 0, where y = z = Q, we have Vo = ^> & = 0> an d hence c' = and c" = 0. Since the prism does not turn about the ;r-axis during flexure, it follows that for y = Q, ^ = also, and consequently c = 0. We obtain therefore and further, from (a), (d) t = xzl* r i =- kyz/p, t=k(y*- These values for ^, rj, and ^ satisfy the equations XXX. (c), since by hypothesis X=Y=Z=0. The equations 1-6 show that the conditions of equilibrium are fulfilled. From (1) and (b) we get X, --=S = (3X/A + 2/* 2 )/(A. + p.) . zfp. If we introduce the general coefficient of elasticity E [XXIX. (d)], we have (e) S = Ez/ P . The resultant E of the forces S is (f ) R = Ejp . \z . dydz, and is equal to zero, if the a;-axis passes through the centre of gravity of the prism. If we assume this and then determine the moment M of the forces S with respect to a horizontal line passing through the centre of gravity, we will have M = \Szdydz = Ejp . \z z dydz = EJjp, where J is the moment of inertia of the cross section. In order to bend the prism so that an axis passing through the centre of gravity of the prism becomes a circle of radius p, a rotating force of moment M must act on each end surface ; the axes of the rotating forces are perpendicular to the plane of the circle, and are oppositely directed. SECT, xxxin.] MOTION OF AN ELASTIC BODY. 89 The cross section of the prism is noticeably altered by the flexure. Since the parts on the convex side of the prism are extended, and the parts on the concave side compressed, the former tend to contract in the directions of the y and .s-axes, the latter to expand. If, for example, the cross section is a rectangle ABCD, as in Fig. 45, ABCD takes the form A'FC'D'. The two plane surfaces whose projections are represented in the figure by AB and ' r "~ CD are transformed into surfaces of double curvature. We may consider A'B' and C'D' as arcs with the centre E, while A'D' and B'C' are straight lines which intersect at E. The lines A' I? and B'C' are not changed in length, AB is shortened and FlG - 45 - CD lengthened. If z = %BC, it follows from the definition of k [cf. XXIX.] that A'B' = AB(\ - kz/p), C'D' = CD(l + kz/p). UOE = p, then A'B' /C'D' = (p' - z)I(p +z) = (l- kz/ P )/(l + kz/p), from which it follows that p = kp. This relation has been applied to the determination of k for glass prisms. SECTION XXXIV. EQUATIONS OF MOTION OF AN ELASTIC BODY. The resultant with which the elastic forces act on an infinitely small volume-element dv of an elastic body in the direction of the 3-axis is [XXV.], (dX x /^x + 'dX y !dy + 'dX i pz)dv. If the body is acted on besides by attractions or repulsions, whose component in the direction of the z-axis is X, the element dv is also acted on by the component of force X.dv.p, where p is the density of the body. Hence the .r-component of the acting forces is (dXjVx + VXJVy + VXfiz + pX)dv. If this resultant is not equal to zero, motion occurs in the direction of the .T-axis, and the momentum imparted to the part of the body under consideration in unit time is pdvd-(x + g)/dt? pdvd' 2 g/dt 2 , where t denotes the time. Hence we have = VXJdx + VXJciy + "dXJdz + P X. 90 THE THEOEY OF ELASTICITY. [CHAP. ir. If the components of stress are expressed by , ry, and as in XXX. (b), we obtain the equation (a) p = (* + /*) 39/3* + )"V 2 + pX The equations for V) and are similar. As in XXX. (e) the equations (a) take the form (b) p= (A + 2/*) . 36/3x + 2/i(3A,/3 - 3/* 2 /3y) + /oA'. If the force whose components are AT, F, and Z has a potential, and if therefore X= -3*/3a:, F= -3/3y, 2T = -3/3*, by differentiation of equation (b) with respect to #, y, 2 respectively, and by addition, we have (c) P e = (A + 2/*)V 2 e- / >V 2 . In what follows we assume that no external forces act, so that the components X, Y, Z are zero. Therefore V 2 ^ drops out of equation (c). SECTION XXXV. PLANE WAVES IN AN INFINITELY EXTENDED BODY. Lame* treated this form of motion in the following way. Suppose a plane wave propagated in a direction which makes the angles a, J3, y with the axes; let the velocity of propagation be V, and let the direction of vibration make with the axes the angles a, b, c. If u represents the distance of a point from its position of equilibrium, U the amplitude, and T the period of vibration, the vibration at the origin may be expressed by u= Ucos(2trt/T). At any other point, whose coordinates are x, y, z, we have (a) = tfcosW. We have further (b) = u cos a, rj = u cos b, = ti cos c. If the angle between the direction of propagation and the direction of vibration is represented by <, we have cos = cos a cos a + cos bcos/3 + cos c cos y. For brevity we set - tfsin jar/ **, t - * Lame, Theorie de 1'elasticit^, p. 138. Paris, 1866. SECT, xxxv.] PLANE WAVES. 91 and obtain 9 = 2KS/T V. cos , 36/ar - - ^ 2 u/T z V- .cos a. cos & V 2 = - 47r 2 M/r 2 V 1 . cos a, = - 47r 2 M/r 2 . cos a. By the help of these relations and corresponding ones for t\ and we obtain from XXXIV. (a) r (A + p.) cos a cos < + (p. - pV 2 ) cos a = 0, (c) I (A + /*) cos y cos < + (/x - pV 2 ) cos c = 0. If these equations are multiplied by cos a, cos/2, cosy respectively and then added, we have (X + 2/X-/3/ 72 ) cos < = 0. We therefore have either (d) (e) pV' 2 = \ + 2fj. or cos< = 0. In the first case, equations (c) become cos a = cos a cos <, cos b = cos (3 cos <, cos c = cos 7 cos <. If the right and left sides of these equations are squared and added, we obtain (f) cos 2 (/>=l, so that either < = or < = TT. The vibrations therefore occur in the direction of propagation ; they are called longitudinal vibrations. In the second case < = JTT, that is, the vibra- tions are perpendicular to the direction of propagation ; they are called transverse vibrations. Longitudinal Vibrations. The velocity of propagation 12 of these vibrations is determined by (d), (g) tt = . From formulas (g) and (k) the velocity 12 of the longitudinal vibrations is always greater than the velocity w of the transverse vibrations. In liquids and gases the only vibrations which can occur are longitudinal, since for these bodies p. = 0. For gases we have A. = J PJXXIX. (g)], and hence the velocity of sound in air is (m) 12 = *JP/p. P must here be expressed in absolute units. According to Regnault the density of atmospheric air at Paris equals 0,0012932 under a pressure of 76 cm. of mercury, and At a temperature of 0C. Since the acceleration of gravity at Paris is 980,94, the pressure of the air on a square centimetre equals 76.13,596.980,94 in absolute units. Hence the density p of the air under a pressure P in absolute units is has the value given in (d), these equations are also satisfied if is replaced by 30/3x, or by another differential coefficient taken with respect to one or more coordinates. Vibrations Due to Torsion. Let the axis of a circular cylinder coincide with the 2-axis, and its separate parts oscillate in arcs 94 THE THEORY OF ELASTICITY. [CHAP. n. about the same axis. The components of displacement of a particle from its position of equilibrium may be expressed [XXXII. (a)] by {e) = -<^y, fi = x, =0, where < is a function of z. From XXVIII. (e) we have = 0; therefore condensation and rarefaction do not occur. The equations of motion are [XXXIV. (a) and XXXV. (k)], = w 2 V 2 , 77=0,^,, whence we again obtain (f) = w?'d 2 /'dz' 2 . This equation is satisfied by (g) < = asin {27T/T. (t-zjw)}. Hence w = v //x//j is the velocity with which a wave motion is propagated in the direction of the axis of the cylinder. From XXX. (b) the components of stress are Z t = - Apx, X z = + AM where A = Zira/Tta . cos {2ir/T. (t - z/u)}. The other components of stress are zero. If the cylinder is of finite length, stationary waves can exist in it, that is, waves such that certain definite points of the cylinder called nodal points are at rest, while on both sides of a nodal point the vibrations are in opposite phase. The amplitude of the vibration is greatest half way between two nodal points, at the ventral segments. Stationary waves are formed when waves which have passed over a certain point return to that point again in the opposite direction. To find the period T of these vibrations, we notice that equation (f) will be satisfied not only by (g), but also by <}> = b sin{2?r/T. (t + z />)}, and in general by < = B sin 27rt/T. cos (ZirzjTu) + Ccos (2irt/T) . sin (2vz/T=Ccos (2vtjT) . sin (2irz/Tw). If I represents the length of the cylinder, and if the points for which .z = l are also fixed, we will have < = when z = l, and therefore pTr, where p is a whole number. Hence If, on the other hand, one end of the rod is free, Y z = X 2 = when z = I. .Since X t = - w . 3/'dz, we have 9//x. If both ends of the rod are free, T=2l/ SECT, xxxvii.] VIBRATING STRINGS. 95 SECTION XXXVII. VIBRATING STRINGS. Although the problem of the motion of vibrating strings is only slightly connected with the theory of elasticity, a simple example of this form of motion will be considered here. We suppose a perfectly flexible string stretched between two fixed points A and B. If P is the stress in the string, / the length of the string before the application of the stress, and / its length while the stress is applied, p the cross section of the string, E the coefficient of elasticity, we have 1-1 = P1 /FE. Let the string be slightly moved from its position of equilibrium, that is, the straight line which joins A and B, and let the new form of the string be designated by ACDB. By this deformation the length of the string is increased by dl = dP . 1 'FE. It is here assumed that dP is infinitely small in comparison with P, so that we may set the stress in the string everywhere equal to P. For the sake of simplicity we suppose that the motion of the string is always in one plane, say the z#-plane. Let A be the origin of coordinates, and let B lie on the a>axis at the distance I from A. The distance of any point C of the string from A may be represented by 5, and that of the infinitely near point D by s + ds. The com- ponents of stress at C in the directions of the x- and y-axis respectively are Pdx/'ds and P'dy/'ds. For the point D the similar components are PCdx/'ds + 'd-x/^s 2 . ds) and P(byl^s + 3 2 y/3s 2 . ds). The infinitely short portion CD of the string is therefore acted on by the force P^x/cts- . ds in the direction of the z-axis, and by the force P&y/'ds 2 . ds in the direction of the y-axis. If the string is displaced only very slightly from its position of equilibrium, we can set s = x; the z-component then vanishes and the particles of the string oscillate perpendicularly to the z-axis. If m represents the mass of unit length of the string, the equation of motion is or, if we set ma? = P, (a) ij = a 2 d*y/3a: 2 . The integral of this differential equation is (b) y = A H cos (mrat/l) . sin (mrx/l), where n is a whole number. When 2 = and x = l we have y = 0, and when / = 0, y = A n sin(mrx/l) ; this is the equation of a sinusoid. If, in the general case, the form of the string when t = is given by the equation y=f(x), then f(x) = A l sin (TTX/I) + AZ sin (2irx/l) + A 3 sin (3vx/l) + ... . 96 THE THEORY OF ELASTICITY. [CHAP. H. The coefficients A v A 2 , A 3 ... are determined in the following way. Let the general term of the series be A n sinn, where = -xjl. If both sides of the last equation are multiplied by sin <, we will have /(/^>/7r)sinn^) = ^ 1 sin + .... If this equation is multiplied by dfa and integrated between the limits and IT, we will have ff(l/Tr) sin n . d = A n / s For if in and n are different numbers, we have ~w. / sin m sin nd(f> = \ I (cos (m - w)< - cos (m -f n)$)e?< = 0. But when they are equal, / sin% d = 2 // . f(x) sin (nc//) . rfz. If, for example, the string is so displaced from its position of equilibrium that a point in it at the distance p from the end A is moved through the distance h in the direction of the y-axis, we have f(x) = hxjp for < x i(a 2 + & 2 + c 2 ))cfo. If we introduce the principal stresses A, B, C [XXIX. (e)], we have (b) E p = ^{(A + + C) 2 /E-(AS + C+CA)lfi}dv. By this equation (b) the potential energy is determined, the com- ponents of stress and of elongation are known. We confine ourselves to the statement of the following relation, (c) E r = i Jfe + Yjy, + Z : z z + *Zjs, + 2A>, + 2YjJdr, from which the others can easily be deduced. Galileo was the first to study the properties of elastic bodies; he failed, however, to reach correct results. The physical basis for the theory of elasticity was given by Eobert Hooke, Avho in 1678 published a treatise, De potential restitutira, in which he showed by experiment that the changes of form of an elastic body are proportional to the forces applied to it. Among earlier investigations those of Mariotte and Coulomb deserve especial mention. More recently the theory of elasticity has been developed principally by the French mathe- maticians, Cauchy, Poisson, Lame, Barr6 de Saint-Venant, and others. 98 THE THEORY OF ELASTICITY. [CH. u. SECT, xxxvin. We owe to Cauchy the theory of the components of stress in the form here given. For more extended accounts of the theory of elasticity we may mention Lame, Theorie Mathe'matique de V Elasticity des Carps Bolides. Paris, 1866. Clebsch, Theorie der Elasticitdt fester Korper. Leipzig, 1862. Among the more important recent treatises on the theory of elasticity we mention : Boussinesq, Application des Potentiel a I'Etude de I'Equilibre et du mouvement des Solides Elastiques. Paris, 1885. Barr6 de Saint- Venant, Me'moire sur la Torsion des Prismes. M6m. d. sav. Mr. T. XIV. Paris, 1856 ; Mtmoire sur la Flexion des Prismes. Liouville I., 1856. William Thomson, Elements of a Mathem. Theory of Elasticity. Phil. Tr. London, 1856; Dynamical Problems on Elastic Spheroids. Phil. Tr. London, 1864. Further researches on the theory of elasticity have been carried out in recent years by W. Voigt. CHAPTER III. EQUILIBRIUM OF FLUIDS. SECTION XXXIX. CONDITIONS OF EQUILIBRIUM. THE principal difference between solids on the one hand and liquids and gases on the other consists in the fact that the latter do not, like the former, offer a great resistance to change of form. A force is always needed to change the form of a fluid mass, but the resistance offered by the fluid is determined by the rate at which the change of form proceeds, and will be infinitely small if it proceeds very slowly. We assume that the motion by which the condition of equilibrium is attained proceeds very slowly, and we may therefore assume, in hydrostatics, that a fluid offers no resistance to change of form, so long as this does not involve change of volume. Each infinitely small change of form of an infinitely small part of the body may [XXVIII.] be treated as if it were produced by the dilatations , b, c in three directions perpendicular to each other. The lengths u, v, w drawn in these three directions become u(\ + a), v(l +b), w(\ +c). If A, B, C are the corresponding normal forces per unit of surface which act on the surfaces vw, uw, iw respectively, the work done by the normal forces in this change of form is Avwua 4- Buwvb + Cuvwc or (Aa + Bb + Cc)u .v.w. The change of form considered will, in general, involve an increase of volume, given by uvw(\ + a}(\ +b)(l +c) - uvw. Since a, b, c are infinitely small, the increment of the volume equals (a + b + c)u. v. w, if we neglect infinitely small quantities of a higher order. If we start from the assumption that the work done by the forces equals zero if the volume is not changed, we have at the same time Aa + Bb + C'c = Q and a + b + c = 0. These equations can both be true only if A = B = C. 100 EQUILIBRIUM OF FLUIDS. [CHAP. in. The equations for the components of stress [XXVI. (i)] give X t =Y, = Z, and Z, = 0, X, = 0, Y x = 0. There are, therefore, no tangential forces in a fluid in equilibrium. If we start from the condition that the only forces which act on fluids in equilibrium are perpendicular to their surfaces, we reach the same result, namely, that the normal stresses are all equal. To show this we set Z y = Q, A' 2 = 0, 1^ = 0, and have, from XXVI. (a), P=X x cosa, Q=Y,cos/3, fi = Z;Cosy. P, Q and R are the components of stress for a surface whose normal makes the angles a, (3, y with the axes. The stress acting on this surface is exists, such that (d) -d3>px = pX, -d3>py = pY, ?&fiz = f>Z. Equations (c) are the essential conditions of equilibrium ; if they are satisfied, p may be determined from the equation dp = p(Xdx + Ydy + Zdz). If the forces have a potential i/s, so that we will have (e) dp= - pd^. In gases p is a function of p in liquids p may be considered constant. In the latter case we obtain (f) p = c - pif>, where c is constant. SECTION XL. EXAMPLES OF THE EQUILIBRIUM OF FLUIDS. The conditions of equilibrium of a liquid mass contained in a vessel, and acted on by gravity only, may be determined in the following way : Suppose the position of the particles of the liquid referred to a system of rectangular coordinates, whose 2-axis is directed perpendicularly upward ; we then have X = 0, F-0, Z=-g, and therefore $ = gz. Since the density p is considered constant, equilibrium can obtain under the action of gravity. From XXXIX. (f) we have p = c-gpz. Hence the pressure at the same level is every- where the same. We now determine the pressure in a liquid contained in a vessel, which rotates about a perpendicular axis A with constant angular velocity w. The fluid will turn, like a solid, about the axis A with the same angular velocity as that of the vessel. A particle at the distance r from the axis A is acted on both by gravity and by a centrifugal force whose acceleration is w 2 r. We 102 EQUILIBEIUM OF FLUIDS. [CHAP. HI. SECT. XL. refer it to a system of rectangular coordinates whose 2-axis is directed perpendicularly upward, and coincides with the axis of rotation. We then have X=tfx, Y=^y, Z=-g, and the potential ^ is ^= -% V+ lTt dt > W +-^ the components of velocity at P. The quantities dU, dF, dW repre- sent the increments which U, V, W respectively receive during the time dt, if our attention is confined to the motion of a particular particle. 103 104 MOTION OF FLUIDS. [CHAP. iv. The other symbols, u, v, iv, represent the components of velocity at a definite point in space, where one particle replaces another in the course of the motion. If x, y, z are the coordinates of the point considered, u, v, w are the components of velocity of a particle situated at that point at the time t. After the lapse of the time dt the same point is occupied by another particle, whose components of velocity are u + du/'dt.dt, v + 'dv/'dt.dt, w + 'dw/^t.dt. A particle situated, at the time t, at a point whose coordinates are x + dz, y + dy, z + dz, has a velocity whose projection on the ,r-axis is u + 'du/'dx . dx + "dufdy . dy + "dufdz . dz. The velocities u, v, w are everywhere functions of x, y, z and /. If u, v, w are the components of velocity, at the time /, at the point P, whose coordinates are x, y, z, the components at the time t + dt at another point P, whose coordinates are x + dx, y + dy, z + dz, will be u + 'du/'dt . dt + 'du/'dx . dx + "dufdy . dy + 'du/'dz . dz, etc. If the fluid particle is situated at the time t at P and at the time t + dt at P', then we have U=u and U+ d U/dt .dt = u+ oupt . dt + 'duj'dx . dx + ^fdy . dy + 'du/'dz . dz, or (a) d U/dt = Vu/?)t + Vu/'dx . dx/dt + 'du/'dy . dyfdt + 'dufdz . dz/dt. The particle considered traverses the distance PP' in the time dt, hence its velocity is PF/dt, the projections of which on the coordinate axis are evidently dxfdt = u, dy/dt = v, dz/dt = w. Thus we obtain d U/dt = 'du/'dt + u . du/'dx + v . 'du/'dy + w . 'du/'dz. The equations for dVjdt and dlVjdt are similar. To find the equations of motion of a fluid let us cut from it a parallelepiped dw, whose edges are dx, dy, dz, and on which a force acts whose components are X, Y, Z. In the time dt the parallelepiped receives an increase of momentum, whose components are pXdudt, pYdwdt, pZdwdt, when p denotes the density of the fluid. The pressure p acting at the point .r, y, z, as has been shown in a former chapter, imparts to dto the components of momentum - 'dp/'dx'. d . dt. By a similar argument applied to the two other pairs of faces it appears that the total difference between the quantities of fluid which leave and enter the parallelepiped is (d(pu)fix + ^(pv)fiy + -d(pw)fiz)d< . dt. The parallelepiped at first contained the quantity p . do> ; after the lapse of the time dt it contains the quantity (p + ~dpfit . d)d(pv)fiy + ^(pw^fiz = 0. If the density p of the fluid is constant, the equation of continuity becomes (e) 'dU'/'dx + "do fiy + Vwfiz = 0. Euler's equations are specially suited to investigations of the motion in fluid masses with fixed boundaries. If the surface of the fluid changes there will be points which will lie sometimes within and sometimes without the fluid ; the velocity at such a point cannot be determined by the method here given. Lagrange's method is the one then employed. To this we will return later. In equations (c) and (e) there are contained four unknown quan- tities u, v, 10 and p, for whose determination we have four equations given. To determine the constants of integration the conditions of the motion of the fluid must be given at a definite time. If the fluid is bounded by a fixed surface, the components of velocity in the direction of the normals to the bounding surface are zero. If u, v, w are the components of velocity of a particle at the boundary of the fluid, and if the normal to the bounding surface makes the angles a, ft, 7 with the axes, we have (f ) u cos a + v cos ft + w cos 7 = 0. 106 MOTION OF FLUIDS. [CHAP. iv. SECTION XLIL TRANSFORMATION OF EULER'S EQUATIONS. In k fluid in motion an elementary parallelepiped, whose edges are originally dx, dy, dz, not only changes its position in space but may also rotate and change its form at the same time. Its motion at any instant is determined by the components of velocity u, r, v: ; the rotations and changes of form may be determined in the following way : In the theory of elasticity the component of rotation h x of such an element is expressed by h x = ^(dC'/'dy - ty'/'dz), if ' and rf are the infinitely small changes of the coordinates z and y introduced by the motion. We may set ' = iv.dt, rj' = v.dt, and obtain If is the corresponding angular velocity, we will have h x = %. dt and hence (a) = (3tp/By-a0/az); 77 = $ The equations for t] and may be derived in the same way as the first; , /, are the components of angular velocity in a rotation about the three coordinate axes. If no rotation exists in the fluid, we have = y = {=Q or These equations are the conditions for the existence of a function 4> of x, y, z and t which has the property that it = - 3^/Da;, v = - 3 is called by v. Helmholtz the velocity potential, since the components of velocity U, v, w, are related to each other in the same way as the components of a force if it has a potential. The equation of continuity [XLI. (e)], on the assumption that a velocity potential exists and that the fluid is incompressible, becomes * = V 2 = Q. The velocity h of a particle is From equations a it follows that 'du['dy = 'dv/'dx - 2 'du/'dz = 'dtv/'dx + 2r ) . The first of equations XLI. (c) becomes dw/3* + 2(ui) - vQ + u . 'du/'dx + v . dr/da: 4- w . 'dwj'dx = X-l/p. 'dpjox. This equation may be written f 3tt fit + 2(v) - rfl = X - 1 Ip . \ We have similarly I -dv fit + 2 -w)=Y-\lp. SECT. XLII.] VORTEX MOTIONS. 107 where h is the velocity of a particle. "We may eliminate p from equations (b) by differentiating the second of those equations with respect to z, and the third with respect to y and subtracting. We thus obtain If we use the equation of continuity 'du/'dx + 'dv/'dy + 'dw/'dz = 0, and the relation following from (a) 3/3z + cfy/3?/ + 3/?z = 0, we obtain 3/3< + u . 3/3z + v . 9/3y + w . m'dz - . 'du/ox - rj . 'duf'dy - If & 7 /' represent the components of rotation at a point in the region containing the fluid at the time /, we may use &, H, Z to represent the components of rotation of a particle at the time t + dt, whose components at the time t were , -rj, . The connection between the components , 77, and &, H, Z may be established in the same way as that previously used to find the relation between the velocity at a point in space and the velocity of a particle of the fluid. We have = S, dS/dt = 3/3rf + Using this equation we obtain (c) dS/dt = . 'du/'dx + rj . -duj-dy + . 'du/'dz + tfdZ/'dy - If at any time no rotation exists in the fluid, and if therefore = y = =0 at any point in the fluid, a rotation may still be set up if Z and Y have no potential. If, on the other hand, Z and Y have a potential so that Z= -W/oz and F=-3/3y, we will have dSjdt = 0. If besides X= -o/'dx, we have dU/dt = Q and dZ/dt = Q. Hence no rotation can be set up in an ideal fluid if the fwces have a potential In this case, the particles which rotate already continue to rotate, but the particles which do not rotate from the beginning will never rotate. This theorem was first given by v. Helmholtz. SECTION XLIII. VORTEX MOTIONS AND CURRENTS IN A FLUID. In researches on the motion of fluids it is important to observe whether the particles rotate or not. If there is rotation it is called cortex motion. We then have (a) = $ From this it follows at once that (b) 3/3z + 3^/3y + 3f/3^ = 0. The equation of continuity is (c) 'du/'dx + 'dv/'dy + 'dw/'dz = 0. 108 MOTION OF FLUIDS. [CHAP. iv. If the forces have a potential it follows from equations XLIL (c) that C dS/dt = . 3n / cte + 17 . 'dw I'dy + . (d) { dZ fdt = . Sw/dx + 77 . dw?/3# + . 'dwj'dz. In these equations , 77, are the components of rotation at the point x, y, z; S, H, Z are the same components for a particle which at the .time t is situated at the point x, y, z, but which at the time t + dt is situated at the point x + dx, y + dy, z + dz. On the other hand, if the components , 77, are zero at every point in the fluid at a definite instant they are equal to zero at any time, from equations (d). In this case we call the motion a flow. It is characterized by the equations (e) 'dw/'dy = Zv/'dz, ?m/dz = 'dw/'dx, Vvj'dx = 'du/'dy. From XLII. u, v and w then have a velocity potential , which depends in general on x, y, z and t. The equation of continuity is (g) 3 2 /dr. Since = 0, 77 = and u, r, w are independent of z, the equations of motion (d) are satisfied. We assume f = for r r , where is a constant. In the first case, we have w = f + Cjr 2 , where C is a new constant. SECT. XLIII.] STEADY MOTION. 109 C must vanish, because otherwise the particles at the axis would have an infinitely great velocity. Hence o> = for the part of the fluid lying within a circular cylinder whose radius is ?- () , and whose axis coincides with the 2-axis. These fluid particles therefore rotate about the -axis, just as if they formed a solid body. If, on the other hand, r>r , and hence {=0, the angular velocity w' will be ot'=C'/r z . The linear velocity is rw' or C'/r, and therefore inversely proportional to the distance of the particle from the axis. On the condition that there is no discontinuity in the motion of the fluid, we have for r = r , = C'/r 2 . Hence for r > r , we have rw' = r 2 f /r. If r is infinitely small and infinitely great, we obtain a so-called vortex filament. The action of the vortex filament on the surrounding fluid depends on its cross-section and its angular velocity. If we set m = irr ^ , the velocity h of a fluid particle which does not belong to the vortex is h = rw = m/Trr. Vortex filaments may have other forms ; they were first investigated by v. Helmholtz,* and afterwards by William Thomson, and several others. We see from this example, that the separate parts of the fluid do not need to turn about themselves as their centres of gravity describe circles : although the fluid surrounding the vortex filament revolves about the 2-axis, the separate drops, into which the mass may be divided, do not rotate about themselves. SECTION XLIV. STEADY MOTION WITH VELOCITY-POTENTIAL. If the components of velocity are independent of the time, or if the condition of motion at any definite point in the fluid does not change, the motion is called steady. If a velocity-potential < exists, we have (a) u= 'd/'dz, v c)(/3y, w= -'dtfij'dz, where is a function of x, y, z only. The same holds for the potential M* 1 of the forces ; hence the function T in XLIII. (i) must be constant. If we set T= -C, we have (b) V+p/p + $h 2 = C. If the only forces which act are pressures within the fluid, we may set ^ = and conclude that the velocity of the particles increases as they pass from places of higher pressure to places of lower pressure, and inversely. For a motion for which there is a velocity-potential, the equation of continuity is (c) V 2 < = 0. As an example of such a motion, we will consider a sphere at rest in an infinitely extended fluid. The particles of the fluid which * Helmholtz, Crelle's Journal, Bel. 55, S. 25, 1858. 110 MOTION OF FLUIDS. [CHAP. iv. are at a great distance from the centre of the sphere move with equal velocities in the same direction. Let the sphere be placed so that its centre is at the origin of coordinates 0, and let the radius of the sphere be R. The particles whose distance r from is infinite are supposed to move in a direction parallel to the positive 2-axis with the velocity w . We set the velocity-potential (d) $=V- u-^z, in Avhich V = when r = oo. Then (e) u = -'dF/'dx, v= -'dF/'dy, 10 = -'dP r jdz + tc . Using equation (c) we have (f) V 2 F=0. If we set P=l/r, or equal to a differential coefficient of 1/r taken with respect to x, y, or z, the equation (f) will be satisfied. Since the arrangement around the .r-axis is symmetrical, we will consider if the assumption will satisfy the given conditions. The particles of the fluid move over the surface of the sphere, and hence the component of velocity in the direction of the radius is equal to zero, that is (3/<3r) r=J j = 0. If we set z/r = cosy, we have =-(? cos y/r 2 - ra' cos y and d/dr = 2C cos y/r 3 - w? cos y. From this it follows that (h) C=w Q R*. From equations (d), (g), and (h), it follows that Using equations (e) we obtain u = - %w Q R 3 zx/r>, v = - f w R?zyli* t w= - ^R^Sz^/r 5 - 1/r 3 ) + zr . If we set tt,- 2 + t> 2 = s 2 and x 2 + y 2 = q-, we will have s= -^w^qz/r 5 . If q and z are the coordinates of the path of a particle, the equation of the path will be (k) dq/dz = s/w. Remembering that r 2 = q 2 + z 2 , we may integrate equation (k) and obtain q~(\ -R?/r 3 ) = c. If c is constant, this is the equation of a stream line. If c = 0, we will have either r = R or q/E. Hence the pressure p on the part of the sphere which lies toward the positive side of the 0-axis, is as great as that on that part of the sphere which lies on the negative side of the 2-axis ; the moving mass of fluid will therefore impart no motion to the sphere. And SECT. XLIV.] LAGRANGE'S EQUATIONS. Ill further, a sphere which moves with constant velocity in any direc- tion in an infinite mass of fluid experiences no resistance during its motion. This result, which is at first sight so startling, is explained by the fact that the resistance offered by friction is not taken into account. SECTION XLV. LAGRANGE'S EQUATIONS OF MOTION. Suppose a particle P of a fluid to be originally situated at the point whose coordinates are a, b, c, and after the lapse of the time dt, to have reached the point x, y, z. The general coordinates x, y, z, are functions of t, a, b, c; if t alone varies in these functions, we obtain the path of a particular particle. If, on the other hand, we give to the coordinates a, b, c, all possible values, and keep t constant, we have the positions of all the particles of the fluid at the same time. If the pressure is designated by p, and the density of the fluid by p, and if we set U=x, V=y, JF=z, we obtain from XLI. (b) (a) x = X-l/ P .dp/dx, ij=Y -Ijp.dpldy, z = Z-l/p.dp/dz. In order to eliminate the differential coefficients with respect to x, y, z, we multiply these equations respectively by dx/da, dy/da, dz/da, by dx/db, dy/db, dz/db, and finally by dx/dc, dy/dc, dz/dc. By addition we then get the following equations : f (x - X) . dx/da + (y-Y). dy/da + (z-Z). dz/'da +l/p. dp/da = 0, (b) - (x-X). dx/db + (y-Y). dy/'db + (z-Z). dzjdb + l/p . dp/db = 0, ( (x - X] . dx/dc + 0/-Y). dy/dc + (z-Z). dz/'dc + l(p. dp/dc = 0, These equations are due to Lagrange. To these equations there must be added a relation which expresses the fact that the volume of the fluid does not change. The particles originally situated in a rectangular parallelepiped with the edges da, db, dc, are at the time t contained in a parallelepiped, the pro- jections of whose edges are dx/da . da, dy/da . da, dz/da . da ; dx/db . db, dy/db . db, dz/db . db ; dx/dc . dc, dy/dc . dc, dz/dc . dc. The volume of the parallelepiped at the time t will therefore be dx/da, dy/da, dz/da dx/db, dy/db, dz/db dx/dc, dy/dc, dz/dc .dadbdc. 112 MOTION OF FLUIDS. [CHAP. iv. Since the fluid is assumed incompressible, the equation of con- tinuity is 'dxj'da, 'dy/da, 'dz/'da (c) -dxj-db, -dy/cib, -dz/Zb =1. 'dx/'dc, 'dy/'dc, 'dzj'dc To apply Lagrange's equations, we will consider a fluid mass, which turns with a constant angular velocity , b = r sin , and further, x = r cos ( + (at) = a cos ut -b sin W, y = r sin (< + W) = b cos tat + a sin wt. From these relations it follows that dx/'da = cos (at, ~dx/db = - sin (at, 'dx/'dc = 0, ~dy/da = sin (at, ~dy/db = cos (at, 'dy/'dc = ; i 1 = o> 2 a;, y (a 2 y, z = 0. The equation of continuity (c) is satisfied and the equations of motion (b) are 'dp/'da = /ooj%, 'dp/'db = p(a 2 b, 'dp/'dc = gp. Hence we obtain by integration p= C + p(iw 2 (a- + J 2 )+^c). This solution agrees with that given in XL. SECTION XLVI. WAVE MOTIONS. Lagrange's equations may be used to advantage in investigations on wave motions in a fluid acted on by gravity. All the particles of the fluid may be assumed to move in plane curves parallel to the x^-plane ; let the a;-axis be horizontal, and the -axis be directed perpendicularly downward. Then, if we set y = b, we obtain 'dx/'db = 0, 9y/3a = 0, dy/3& = l, 3y/3c = 0, "dz[db = and y = 0. If we further set p = pP, the equations of motion given in XLV. (b) become ,. ( x. 'dx/da + (z-g). 'dz/'da + ?>P/da = 0, t x . dx/'dc + (z-g). dz/dc + 'dPj'dc = 0, and the equation of continuity XLV. (c) takes the form (b) 'dx/'da . 'dz/dc - 'dzj'da . "dx/dc = 1 . Suppose the particle B (Fig. 47), having, while in its position of equilibrium, the coordinates OA=a and AB = c, to move in a circle SECT. XLVI.] WAVE MOTIONS. 113 DFE, whose centre is at C. Let D be the position of the particle, represent the angle between CD and the perpendicular CE by 0, and set BC=s, CD = r. We then have (c) x Here m and n are constants and r and s are functions of c. We therefore obtain "dzfda = 1 + nr cos 0, 'dz/'da = - nr sin 0, 'dxj'dc = 'dr/'dc . sin 0, "dz/dc = 1 + 'dsj'dc + 9r/3c . cos 0. By these relations, equation (b) takes the form 3s/3e + nr . 'dr/'dc + {nr(l + 3s/3c) + 3r/3c} cos = 0. Since this equation holds for all values of t or 0, we have (d) 3s/9e + nr.3r/dc = and 3r/3c + r(l+a*/3c) = 0. We obtain further from equations (a) the relations (e) -mh:'drl'dc- the last of which is transformed by the help of equations (d) into (f ) - (w 2 - gnfrCdrfdc + ( 1 + ^s/'dc) cos } - g + 'dP/'dc = 0. If the pressure depends on c only, it follows from (e) and (f) that (g) (h) m 2 = gn, P = gc, if the constant is set equal to zero; the pres- sure therefore disappears for c = 0. This condition must hold at the free surface of the fluid. The paths of the particles are circles. If the time required by the particle to traverse its path is T, that is, if T is the period of oscilla- tion, we have m = 27r/7 7 and 6 = 2ir/T. 114 MOTION OF FLUIDS. [CHAP. iv. SECT. XLVI. If X is the wave length and h the velocity of the wave, we will have h = Tg/2Tr and h = \/T, from which it follows that (i) h = *Jg\j'2ir and n = 2ir/X. The motion of the particle is such that it describes a circle whose centre lies a little above the position of equilibrium of the particle. From the first of equations (d) we have s= -^nr 2 , where the con- stant disappears, since s and r vanish simultaneously. Hence also (k) s= -7rr 2 /A. From the second of equations (d) it follows that d log r + nd(c + s) = 0, and hence, by integration, log r + n(c + s) = k, when k is a constant. For a particle on the surface we have c = ; if the values of r and s for this particle are designated by E and S, we have log B + nS=k. We have further log (r/E) + n(c + s-S) = Q. The factor c + s-S=H is the perpendicular distance between the centre of the path of the particle considered and the centre of the path of a particle in the surface. We have therefore (1) r = Ee~ Zira i' x . If ds/dc is eliminated from equations (d) we obtain dr + nr(dc - nrdr) = 0, and by integration 1 ,'n . log r + c - |nr 2 = Tc'. For the particles on the surface we have 1/n. log E - ^nE 2 = k' . Hence (m) c = A/27T . log (E/r) - ir/A . (E 2 - r 2 ). The free surface can be thought of as formed by the rolling of a circular cylinder on the under side of a horizontal surface AB (Fig. 48), FIG. 48. which lies at the height OA = X/2ir over the centres of the paths which the particles in the surface describe. The free surface is then represented by a straight line whose distance from the axis of the cylinder is E, CHAPTER V. INTERNAL FRICTION. SECTION XLVII. INTERNAL FORCES. IN the discussion of the motion of fluids, the friction among the fluid particles has not been considered. Friction is excited in different degrees between the particles of the fluid when they move among themselves at different rates. In consequence of friction the viscosity of the fluid is more or less great. We will try to determine the friction caused by the motion of the fluid. We will suppose that the particles of a fluid mass are moving in a direction parallel to the ar-axis, and that those situated at the same distance from the ar^-plane have the same velocity. The velocity increases in proportion to the distance from the xy-pl&ne. One sheet of the fluid glides over another and thereby gives rise to a definite frictional resistance, which, according to Newton, may be assumed proportional to the rate of change of velocity with respect to the distance from the ^-plane, so that du/dy = e. The friction between two contiguous sheets is then propor- tional to the difference of their velocities, and inversely proportional to the distance between them. We therefore set the velocity u = fix + *dufiy). "We know by experiment that ft is independent of the pressure. The meaning of the other quantities in (b) is clear without explanation. By the help of the formulas given in (b), we can determine the tangential forces which must act in the fluid to overcome the frictional resistances. We will now determine the magnitudes of the normal forces which are necessary for G d' D J)' the extension of a viscous fluid in a given direction. Let the fluid move in a direction parallel to AB (Fig. 50), and let the velo- city of a particle situated at the distance i/ from this line be equal to u. As before, we may set u = ?/ o + *y- After the lapse of the time dt y A has traversed the distance u dt, and G the distance (u + fAC)dt. CC' represents the motion of the point C relative to A, and we have CC' = e . AC . dt. If we designate the angle GAG' by d, we have (c) d = f.dt. The rectangle EFGH, described in the rectangle ABCD, transforms into the parallelogram EF'G'H', and we determine the increments which the sides EH and EF receive by this transformation. Eepre- senting the angle HEB by ^, and noticing that HH' and FF are SECT. XLVII.] INTERNAL FORCES. H7 parallel to AB, we have EH' = EH '+ HH 1 'cos & EF' = EF- FF'sin +. We further have HH' = BH. d = EHsint.d, FF' = AF. d = EFcos $ . d, and hence (EH' - EH)/EH= sin ^ cos $d ; (P" - EF)/EF = - sin ^ cos ^ty. If the increment of length per unit-length of EH is designated by ds, we have (d) ds = sin ^ cos \W< ; ds is also the diminution of length per unit-length of EF. To bring about the deformation considered, a tangential force T must act on ABCD, which is, from (a), (e) T=p.e. This force acts on the surface corresponding to CD in the direction CD, and on that corresponding to AB in the direction BA ; on the other two surfaces the forces act in the directions CA and BD. To determine the normal force N acting on the surface EF, we 'set in XXIV. (a) a = ^, /3 = |TT - ^, y = TT, and X y =Y x = T. Since all the other components of stress are equal to zero, we obtain (f) N=Pcost + Qsinf = 2TsiniI>cost. From (c) and (d) we have ds = sin ^ cos ^ . e . dt, and from (e) and (f ) JV=2/xesin ^cos \j/. Hence, we have (g) N=2p..ds/dL The stress acting on the surface EH is -N. It has been shown that a unit of length, in the direction EF, is increased by - ds. If a normal stress were to act on the surface of ABCD, it would have no influence on the deformation; but a normal force S + N would act on EF, and a normal force S-N on EH. If the normal stresses X M Y^ Z t act on a rectangular parallelepiped whose edges are parallel to the coordinate axes, when the fluid is in motion, they cause deformations and a change of volume. If, as in the theory of elasticity, we set the volume dilatation Q = x x + y, + z a then x x - JO is the part of the increase in the direction of the ic-axis which is here considered. Similarly, we set 3S = X x + Y y -f Z a and X x - S is the part of the normal force in the direction of the a;-axis which causes the deformation. By the help of (g), we obtain If, finally, we set for - S a quantity p, which may be considered a pressure, on account of its analogy with the pressure in ideal fluids and gases, and remember that x x = 'dl'dx, 'dx t /'dt = 'du/'dx, etc., we will have (h) X x = -p + 2/j. . 'duf'dx - ^('du/'dx + dv/3y + "dwfdz). Analogous expressions hold for Y y and Z f 118 INTERNAL FRICTION. [CHAP. v. From equation (h) the dimensions of ju, are ML~ l T~ l . The co- efficient p. has been determined for many fluids and gases. It changes very much with the temperature. The following values hold for C. : Water, 0,01775; Alcohol, 0,01838; Air, 0,000182. SECTION XLVIII. EQUATIONS OF MOTION OF A Viscous FLUID. We will now present the equations of motion of a fluid exhibiting internal friction. From XXV. the components of stress act on the unit of volume in the direction of the x-axis with the force (X) = -dXJ-dx + aX,/3y + *dXJVz. If U is the velocity of a single particle of the fluid in the direction of the z-axis, then pU=(X) +pX, where, in the usual notation, A" denotes the component of force in the direction of the x-axis. From XL VII. (b) and (h), we have (a) pU=pX- dpfdx + p-V 2 u + /x . 3(3tt/3z + dv/dy + 'dw/'dz)/'dx. This equation, and those analogous to it, which hold for the velocities V and W in the directions of the y- and -axes, are due to Stokes.* They hold in connection with the equation of continuity (b) 3/0/3* + 3{pw)/3a; + 'd(pv)/Vy + 3(pw')/^ = 0. We assume that the fluid is incompressible, and have (c) 'du/'dx + 'dv/'dy + 'dw/'dz = 0. ( p(u+u. 'du/'dx + v . 'duj'dy + w . 'du/'dz) = pV 2 u + pX - "dpfdx, (d) J p(v + u . ?>v fdx + v . 'dv fdy + w.d>v fdz) = /*V 2 ? +pY- 3p/3y, { p(w + u . 'dw/'dx + v . 'dw/'dy + w . 'dw/'dz) = p.V 2 w + pZ - 'dp/'dz. The equations are simplified if the motion is steady, that is, if ft = 0, v = 0, w = 0. If the velocity is very small, the terms u . 'du/'dx, v . 'du/'dy, etc., may be neglected ; we then have (e) nV 2 u + P X-'dpl'dz = 0, f j.V 2 v + pY-'dp/'dy = 0, p&w + pZ - 'dp/'dz = 0. If the forces have a potential , (f) V 2 p + pV 2 ^ = 0. If we introduce the components of rotation = (dw/dy - 'dv/dz), y = $(dufdz - dw/dx), and if the forces have a potential, we have from (e) (g) V 2 = 0, V 2 / = 0, V 2 C=0. Further we have (h) * Stokes, Cambridge Phil. Tr., Vol. vin., p. 297, 1845. SECT. XLVIII.] MOTION OF A VISCOUS FLUID. 119 With reference to the boundary conditions, it is assumed that the particles of the fluid which are in contact with solid boundaries have no relative motion with respect to them ; at the boundary of the fluid we have therefore w = 0, v = 0, w = 0, if u, v, and w represent the components of velocity at the bounding surface. If solids are present in the moving fluid, we may generally assume that each particle in the surface of the solid has the same velocity as the particle of the fluid which is in contact with it. SECTION XLIX. FLOW THROUGH A TUBE OF CIRCULAR CROSS SECTION. We consider a viscous fluid moving slowly through a narrow tube, which is set horizontal, so that gravity does not influence the motion. Let the axis of the tube be taken as the z-axis, and suppose that the particles of the fluid move parallel to it. We then have Equations XLVIII. (e), (c) and (f) then become (a) 9p/a = 0, 3p/3y = 0, ^w^'dp/'dz; (b), (c) dw/'dz = ; \7 2 p = 0. From (c) it follows that (d) d 2 p/dz z = Q and p=fa+p () , where / and p Q are constant. It follows further from (a) that ju,y%=/. Since w depends on the distance r of the particle from the axis of the tube, we have, since r 2 = x 1 + y 2 , V 2 w = d 2 w/dr 2 + l/r . dw/dr. Hence we obtain d^/dr^+l/r.dw/dr^f/fjL. By integration w = c log r +fr 2 /4:fj. + W Q . Since w has a finite value for r = 0, the constant c must, equal 0. Therefore (e) w = w l) +fr 2 /4/JL, where W Q is the velocity in the axis of the tube. If the pressure is equal to p when z = and to p l when z = l, we have from (d) f=(pi-p )/l- If we substitute this value of/ in equation (e), we have w = w -r z . (p -pi)/4fd- For all particles of the fluid which are in contact with the wall of the tube, we have w = 0. Representing by R the radius of the tube, we will therefore have 0=W -^.(p -j^)/4/tZ. We obtain finally 120 INTERNAL FRICTION. [CHAP. v. SECT. XLIX. The volume m of the fluid which flows in one second through a cross-section of the tube is given by (f) m = I 2irrdr . w = 7r(p Q -p^W/Spl, that is, the volume of the fluid is directly proportional to the fourth power of the radius of the tube, inversely proportional to its length, and inversely proportional to the constant /*. Poiseuille was the first who investigated the flow of a fluid through narrow tubes ; he was led to results which agree with the above formulas.* * Among recent works on hydrodynamics are to be mentioned : Lamb, Treatise on the Motion of Fluids. Cambridge, 1879. Auerbach, Die Theoretische Hydro- dynamik. Braunschweig, 1881. CHAPTER VI. CAPILLARITY. SECTION L. SURFACE ENERGY. THE form of a fluid mass on which no external forces act is determined by the forces with which its particles act on one another. If the mass is very great, it will take the spherical form, in consequence of the gravitational attraction of its parts ; if, on the other hand, the mass is small, the force of gravitation between the particles will have no perceptible influence. If the force of gravitation can be neglected, the force of cohesion, which acts in every fluid mass, tends to bring it into the same spherical form. From researches which have been made on the mode of action of this force, it appears that it acts only between particles which are at very small distances from one another. The law of its dependence on the distance between the particles is not yet known. We may nevertheless develop the laws of capillarity by the use of a method which does not require a knowledge of that law. If the form of a fluid mass is originally a sphere, work must be done to change it into any other form. If the fluid offers no frictional resistance, this work can be due only to the fluid particles situated in or near the surface ; since the only particles which can act on a particle at a greater distance from the surface are those which immediately surround it; these either remain in their positions or are replaced by others which act in the same way as those replaced. The work done is therefore expended in adding new particles to those already present in the surface, or, what is the same thing, in enlarging the surface. To increase the surface S of the fluid by the infinitely small quantity dS, the work CdS is necessary ; C is constant and may be called the capillary constant. Two bodies in general meet in a surface, and C depends on the character of these two bodies. In the case of a falling raindrop the 121 122 CAPILLARITY. [CHAP. vi. two bodies in contact are water and air. At the surface of a drop of oil which floats in a mixture of water and alcohol, as in the well-known experiment of Plateau, two liquids are in contact. Even when a fluid is in contact with a solid, or when two solids are in contact, the common surface possesses a definite surface energy. Let the capillary constant of two bodies a and b be C ab , and let S be the surface in which the two bodies meet. The potential energy E p of the surface S is (a) E p = C ab . S. Since C ab = E P /S, and since the dimensions of E p and S are L 2 T~ 2 M and L- respectively, the dimensions of the capillary constant are T~' 2 M. The surface of the fluid is under a definite tension somewhat analogous to that of an elastic membrane. If a rectangle DEFG is described in a plane surface of a fluid, and if three sides of it retain their positions unchanged while the fourth side FG, along with the particles of the fluid present in it, is moved through the distance FH in the direction EF, the surface is enlarged by the area FG . FH, and the surface energy is increased by C . FG . FH, where C is the capillary constant. To produce the motion considered, a force K must act on FG ; the work done is therefore equal to K . FG . FH. It follows that (b) K=C, or the tension per unit-length of the fluid surface is numerically equal to the capillary constant. This tension existing in the surface exerts a pressure in the fluid. Let P (Fig. 51) be a point in the surface which is supposed convex in the neighbourhood of P. Suppose two plane sections erected at P, which contain the normal to the surface at that point. One of these planes cuts the surface in the curve PA, the other in the curve PR PA and PB shall intersect at right angles and their radii of curva- ture shall be the principal radii of curvature of the surface at the point P. A third plane containing the normal to P cuts the surface in the curve PF, whose radius of curvature R is determined from Euler's theorem by the equation (c) l/E = cos 2 /fi l + sin 2 (f)/R 2 , where (f> is the angle between PA and PF. About the point P as a centre we suppose described a sphere of infinitely small radius which cuts the surface in the curve AFBDE. The element FG of this curve is acted on by the tension C . FG, proceeding from the adjacent parts of the surface. FG may be set equal to rd and the tension to Crd. The surface tension therefore draws the surface-element ABDE toward the interior of the fluid with a force Cr 2 [*'d/lt. Jo Therefore if P represents the pressure on unit of surface we have PTTT- = Cr 2 l^d^jR and P = C/ir . f^'d^/B. If for R we introduce the value given in (c), we obtain (d) P It is probable that in addition to the pressure here found, which arises from the curvature of the surface, there also exists a constant pressure M which acts in the fluid when its surface is plane. The total pressure due to capillary forces is therefore M+C(l/R l + l/R 2 ), where M and C depend on the character of the two bodies which are in contact in the surface. Since the phenomena of capillarity do not permit of the measurement of this quantity M, it need not be further considered. At 20 C. the value of C for the surface of contact between water and air is 81, between mercury and air 540, and between mercury and water 418. SECTION LI. CONDITIONS OF EQUILIBRIUM. Equilibrium exists in a fluid mass if its potential energy remains unchanged when the position and form of the mass are changed by an infinitely small amount. Since the energy depends on the extent of surface we must obtain an expression for the increment SS of the surface. Suppose a fluid A surrounded by another fluid B, the two fluids being such that they do not mix. If no external forces act on them, A will assume the spherical form. Suppose the surface S to be concave toward A and to move toward B so that it undergoes an infinitely small change of form. Let s be the contour of the surface S, and S' represent that surface after the change of form has occurred. The contour of S' may be repre- sented by s'. Erect at all points of s normals to S which cut the surface S' in a new curve o-, which may be supposed to lie within s'. If we designate the infinitely small distance between a- and s' by 81, the part of S' which lies between o- and s' will be given by (b) j/ . ds. 124 CAPILLARITY. [CHAP. vi. We now erect at a point P in S the normal PP which cuts S" at F ; set PP' = 8t>, and draw through P on the surface S two curves PE and PF, one of which corresponds to the maximum curvature of the surface at P, the other to the minimum. These principal curves and two others infinitely near them will bound a rectangle PEQF, whose sides PE = a and PF=b are infinitely small. If P^ and 72 2 are the principal radii of curvature, there will always be two angles a and (3 such that a = R l a, b = R. 2 (3, and therefore dS = a.b = fi l R 2 a/3. The normals to S erected at E, Q and F, intersect S' at E', Q\ F'. We set FE' = a', P'F = b' and obtain a' = (R 1 + 8v)a, b' = (R z + 8v)/3 If Si is the part of S' bounded by represents the angle between ds and the xy-pla.ne, and if all points of the surface S 3 are elevated by the same infinitely small amount Sz, where 8z is constant, we will have 8S 3 = Q, &!= -8S 2 = \cos8zds. The equation (a) then becomes (b) gp\ \zdxdy = (C M - C7 fc ) J cos 2 ) = 0. If the curvature of the surface is' expressed by the differential coefficients of z with respect to x and y, we will obtain from (c) a differential equation for the determination of the form of the surface. If the contact angle is also given, the surface is completely determined. If the cross-section of the tube is circular and very narrow, we may assume that approximately B^ = R. 2 = r/ cos a, where r is the radius of the tube and a the contact angle. The height z to which the fluid rises is then z = - 2 cos a/gpr . C^. We obtain the same result from equation (b) if we set ^zdxdy = Trr 2 z, Jc08Py vanishes if R and n are infinitely small. It has been shown [XIV.] that a 2 (i/r)/az 2 + a 2 (i/r)/a# 2 + a 2 (i/r)/a^ 2 = o. FIG 53 ^ e ^ ave tnere f re > from (b), for a point outside of L, (d) a^pyaa; 2 + a^/a^ 2 + a^/a^ 2 = v' 2 ^ = o. On the other hand, if P lies within the body we have [XIV. (h)] (e) V 21 ? + 4irp = 0. If we designate the normals to the surface drawn inward and outward by v s and v a , we have for the surface density o- [XIV. (1)], (f) W,/3v, + a^ /av a + 47TO- = 0. The horizontal lines over the differential coefficients indicate that their values are to be taken at the surface. Hence if v represents the electrical density on a surface, the sum of the forces acting in the direction of the normals drawn outward from both faces equals lira-. These properties of the potential hold for every system of bodies charged in any way with electricity. If the distribution is given, the potential can be determined either from (b) or from (e) and (f). If the potential is given, the densities p and o- are determined from (e) and (f), while the components of the electrical force are given from (c). The force F, acting in any direction ds, is F= - SECTION LV. THE DISTRIBUTION OF ELECTRICITY ON A GOOD CONDUCTOR. If a charge e is communicated to a good conductor, it distributes itself over the conductor. \Ve will determine its volume density p SECT. LV.] DISTRIBUTION ON A CONDUCTOR. 131 and the surface density = 0. Hence the electricity is distributed only on the surface of the conductor. From (b) ^ is constant in the interior of the conductor and equal to ", where is the value of the potential on the surface. We may determine rt from the equations y 2 ^ f a = and ' = * for all points of the surface. The surface density is given by 3*V9i' + 'd m [dv il + 47TO- = 0. Since , is constant we obtain (d) 4iro- = -^J?>v a . If we represent the force acting outward at the surface of the conductor by F, we have (e) F - 'd j a /dv a = 47ro-, that is, the force acting at a point on the surface of the conductor in the direction of the normal is equal to the surface density a- at that point multiplied by 47r. If ds is an element of a curve drawn on the surface of the con- ductor, we have 3^/3$ = 0, since "*" is equal to ^f everywhere on the surface, as has just been shown. Hence F has no components in the surface. The surface of the conductor is a surface of constant potential, and the direction of the force F is everywhere perpendicular to it. To determine the potential of the conductor its charge e must be calculated ; we have e = J Jo- . dS, and therefore , and hence c x = 0. The electrical force at the distance r from the centre is F= -d^ r jdr = c. 2 /r 2 . From LV. (e) we have further The potential and the capacity C are therefore a =0/r, C=QI^ = R. The dimensions of capacity are therefore those of a length. 2. The Ellipsoid. Eepresent the semi-axes of the ellipsoid by a, b, c, and its charge by Q. It is most natural to assume that the surfaces of constant potential are confocal ellipsoids. The equation of a system of such surfaces is (a) E = 3? I (a? + A) + f/(b 2 + A) + * 2 /(c 2 + A) = 1 . On our assumption the potential must be a function of A, so that we will write =/(A). To find this function / we proceed from and the analogous expressions for y and z. These give 2 /a*; 2 = 2/(tt 2 + A) - ^ . 9 2 A/3.7: 2 - 2ar/(a 2 + A) 2 . or, using equation (c), (e) 3 2 /ac 2 = 2/(a 2 + A) - A . 3 2 A/3x 2 + '2B. From equations (c) and (a) it follows that (f) A 2 ((d\px)* + (3A/3y) 2 + (9A/o?) 2 ) = 4A, and from (e) and (f) that (g) A . V 2 A = 2/(a 2 + A) + 2/(ft 2 + A) + 2/(c 2 + A). By the help of equations (f) and (g) it follows from (b) that . The potential at any point (x, y, z) is therefore given by + A)(6 2 + A)(c 2 + A), where A is known from the equation z 2 /(a 2 + A) + ?/ 2 /(&2 + A) + s 2 /(c 2 + A) = 1 . If the charge on the ellipsoid is Q, the potential at a point at a great distance from the ellipsoid is Qj^JX ; by comparison with (k) we then obtain (1) = Q/2 . d\lJ(a* + A) (ft 2 + A)(c 2 + A). The electrical force F and its components X, Y, Z are determined from the equations X = - dVdX . 3Aar Y= - dVdX . 9A3 Z = - F = - From (f) and (1) we have F= - If we represent the perpendicular let fall from the origin on the plane tangent to the ellipsoid at the point x, y, z of its surface by N t we have JA-\IN> and hence F=Q. JV/>/(^+A)(6* + A)(c 2 + A). The surface density o- on the ellipsoid itself is determined by the equation 47ro- = F, and A = for this ellipsoid, so that (m) a- = N. Q/lirabc. Hence the electrical density at a point on the ellipsoid is proportional io the perpendicular let fall from the centre on the plane tangent to the ellipsoid at that point. We will now consider several special cases. In the case of an ellipsoid of rotation a = b, and therefore from (1), if ^ is the potential of the ellipsoid, we have % = Q/2 . JfU/(a 2 + AjVc^+A: Hence for a > c, (n) = Qf-Ja* - c 2 . (\TT - arctg c/V 2 - c 2 ) ; for = c, (o)^P = Q/a; and for a/! -x 2 /a 2 -y' 2 /b' 2 . If the plate is circular, that is, if a = b, and if we set x 2 + y' 2 = r 2 , we have Ja? - r*. At a point whose distance u from the edge is very small, we have o- = Q/4:Tra. l/*/2au. In this case, therefore, the density is inversely proportional to the square root of the distance of the point from the SECTION LVII. ELECTRICAL DISTRIBUTION. If several charged conductors are present in a region, the distribu- tion of electricity on the conductors is determined not only by their form and magnitude, but also by their mutual action. The deter- mination of the conditions of electrical equilibrium is, as a rule, very difficult. The most important work on this subject has been done by Poisson and William Thomson. We will here make use of the method of electrical images given by Thomson. (a) Distribution on a Plane Surface. Suppose the quantity of electricity e present at the point (Fig. 55) ; let AS be the plane surface of a very large conductor L, which is in conducting contact with the earth. The potential ^ of L is therefore zero, since we assume the potential of the earth equal to zero (cf. VII.). We are to determine the surface density o- of the distribution on the surface. Let the potential at an arbitrary point in space due to the conductor L be "*",., so that ^ is the work done by the electrical forces of FlG - 55 - the conductor if unit quantity of electricity, which is supposed to be merely a test charge and to have no effect on the electrical distribution 136 ELECTROSTATICS. [CHAP. vu. on L, is transferred from the point P to infinity. If OP = r, the potential ^F at P will be ^ = e/r + V,. We now suppose a quantity of electricity - e situated at the point 0' (Fig. 55), which is the image of the point with respect to the plane AB. This imaginary quantity at 0' would act on all points lying on the same side of the plane AB as 0, in the same way as the quantity of electricity which is distributed on AB ; for the potential which arises from the quantities at and 0' satisfies Laplace's equation at all points which lie on the same side of the FIG. 56. plane AB as 0, except at the point itself. Further, the potential vanishes at all points of the plane AB, since all points of that plane, which passes perpendicularly through the middle point of the line 00', are equally distant from the points and 0', and hence for all points of the plane AB we have e/r - e/r' = 0, where r and r' represent the distances of a point in the plane from and 0'. Now if a function satisfies Laplace's equation and assumes assigned values over a given surface, and if the function itself and its differential coefficients are continuous, it is single-valued and determinate. This theorem is known as Dirichlet's Principle. SECT. LVII.] DISTRIBUTION. ELECTRICAL IMAGES. 137 It should be noticed that "9~ e = e/r', and that the potential at the point P is V = e/r -e/r'. A unit quantity of positive electricity lying at a point P (Fig. 56) in the plane AB is acted on by two forces, K=e/r 2 and K = e/r'' 2 , whose directions coincide with OP and PO' respectively. Hence the direction of the resultant force F is parallel to 00' and equal to F= - 2eOI/OP 3 , if we consider the force positive when it is directed toward the region in which lies. Now since 4:Tra- = F we obtain 11 e 1 +^ 21 e 2 +^) 31 e 3 + ...)Se r Com- paring the formulas (d) and (e), we obtain Therefore, (f) p. 2l =p lv p sl =^ 13 , and in general p mn =p nm . The electrical energy W, of the system may therefore be expressed as a homogeneous quadratic function of the charges by (g) W * = &n e i 2 + \Pv e * + fc e 3 2 + +Pi2 e i e 2 +Pi where the coefficients p mn are called coefficients of potential. If equations (a) are solved for e v e. 2 , e s , we obtain (h) If the charge 8e l is communicated to the conductor A l and the charge 8e 2 to the conductor A^ etc., so that the potential ^ is increased by 8 v while the other potentials retain their original values, we have The increment of the energy is therefore (i) 8/ SECT. LXII.] A SYSTEM OF CONDUCTORS. 149 From equation (h) the energy W is given by (k) If ^ is increased by 8W V we have Comparing equations (i) and (1), it follows that (m) q mn = q nm . The energy JF^,, expressed in terms of the potentials, is therefore given by " +028*^8+ The coefficients J9 12 , in general p nn >p mn and p mm >p nm . Further, the value of p nm lies between p nn and zero, and since p nn is positive, p nm is also positive. The potentials of the two conductors are equal only when the charged conductor encloses the uncharged conductor. If one conductor does not enclose the other, we will always have Pnn>Pmn an( i Pmm>Pmn- SECTION LXI1I. MECHANICAL FORCES. Let us suppose a set of insulated conductors ; their charges will remain unchanged in quantity when the conductors are displaced. Their potentials depend on the charges in the manner given in LXII. The forces acting on the charged surfaces tend to set the conductors in motion. We assume that all the conductors except A l retain their relative positions; that A l can move in the direction of the z-axis ; and we then determine the force which tends to move A l in this direction. Let the displacement of A^ be 8.1: The energy W e of the system will be diminished in consequence of this displacement by X8x. At the end of the motion the energy is W e + W n and hence we have W.-X. 8x= W. + 8JT., and (a) X= -8J7 f /8.v. Now, from LXII. (g) we have (b) X = |e 1 2 8/> 11 /&c + %e 2 ~8p 22 /8x + . . . + 1 e 2 8p 12 /Sj: + . . . , because the charges do not change during the motion, and are inde- pendent of the displacement 8x. This method may be always applied if the motion of the conductor is. one for which the mechanical work done by the system may be repre- sented in the form X8x. We now determine the force with which one of the conductors will move in the direction of the .r-axis if the potentials remain con- stant. Let A v A 2 , A 3 , ... be the FIG. 70. given conductors (Fig. 70). They are supposed to be connected by very thin wires with the very large conductors B v B 2 , B 3 , ..., whose potentials are ^ f 1 , ^o, ^ 3 , ... respec- tively, and which are so remote from the system of conductors A SECT. Lxiii.j MECHANICAL FORCES. 151 that they have no influence upon it by induction. If the conductor A l is displaced by Sx, the charges e v e 2 , e s , ... increase by 8e lt 8e 2 , 8e 3 , ..., and we have from LXII. (h) The electrical energy of the system thus increases by SJr= ^tej + 2 & 2 + 3 Se 3 + . . . , , + . . . . But from LXII. (n) the energy is equal to W^, + Sf^ in the new configuration of the system, where The work done is JTSa;. The sum of the energy 7F^, originally present and the energy 8fF supplied is equal to the sum of the energy in the new position and the work done. We therefore have tf* + 877=17^, + SIT* + X . 8x, X.8x = 8fT - 8!^. If we substitute the expressions found for 8JF and 8JF^,, we have v + ^2^3^23 + . . Hence, we obtain (d) X.8x = 8JF^, or X=8fTyjSx, and further 8JF= '2X . 8x. The electrical energy supplied to the system is therefore twice as great as the mechanical work done. Now, if during the displacement of the conductors their potentials do not change, energy must flow from B lt By B 3 , etc., to the conductors A v A%, A 3 , etc. One-half of the energy SJF supplied is expended in doing the mechanical work, the other half in increasing the electrical energy. SECTION LXIY. THE CONDENSER AND ELECTROMETER. 1. Parallel Plates. If two bodies at different potentials are placed near each other, a relatively great quantity of electricity can be collected on the surfaces which face each other. If A and B are two such bodies, 152 ELECTROSTATICS. [CHAP. vn. whose potentials are ^ and Mfg respectively, and if the opposing surfaces of the bodies are planes, the electrical force in the inter- vening space is everywhere constant, except near the edges of the plane surfaces. If a represents the distance between the planes, and if ^f l is greater than ^f y so that this force is directed from A to B, we have from VII. (c), (a) 1 = 2 + J F a and F=( 1 - 2 )la. The surface-density on A l is determined from 471-0- = ^ or l -V 2 ) . S/faa. The charge e 2 on B is equal to - e r The electrical energy W^ of the system is or (c) JT^ = I/Sir. (^-^^.S/a, where the energy is expressed in terms of the potentials. From (b) and (c) it follows that (d) JF e = 2irae l 2 /S. If the z-axis is perpendicular to the plane surfaces of the conductors, and if it is directed from A to B, we have, representing the x-co- ordinate of the plane face of A by x v and that of the plane face of B by x 2 , a = x 2 -x l , and W* = 1/87T . ppj - / . S/(x 2 - xj ; W. = 2we^(x 2 - xJ/S. The mechanical force which acts on A is [LXIIL] X l = - SWJSa^ = 2^/S ; X l = 2^ = $Fe v This corresponds to LIX. (e). We further have [LXIIL (d)] X 1 = Sfr+ISXi = 1/87T . (^ - 2 ) 2 . S/(X 2 - Xrf- = 1 1 Sir . F* . S. This agrees with LIX. (f). From the expressions which have been given for W^,, we also have W^,= l/8-n-. F 2 . S(x<, - xj, which agrees with LXI. (i). The capacity C is C=e 1 /(^ r l -^ r 2 )- ) if ^ 2 = . we have C=S/4ira. 2. Concentric Spherical Surfaces. If a sphere A v whose radius is R, is enclosed by the concentric spherical shell, whose internal and external radii are 7? 2 and R z , and if A l is given the charge e v and A 2 the charge e 2 , the inner surface of the charge SECT. LXIV.] CONDENSER AND ELECTROMETER. 153 The potentials within the sphere A l and within the spherical shell A s are therefore " l = eJP^ - e l /R 2 + (e l + e 2 )/E 3 ; 2 = (e 2 + e l )jR 3 ; and hence e x = R^(R Z - RJ . ^ - JR 2 . R l f(R a - RJ . / and These equations agree with those given in LXII. (h). The potential ^ in the space between the two spheres is where / is the distance of the point considered from the common centre of the spheres. The potential outside the spherical shell is ^P = (e 1 + e 2 )/r. The capacity C of the inner sphere is determined by e l = C^ v when 2 is set equal to zero ; we have therefore C = R^RJa t if we represent by a the distance between the surface of the inner sphere and the inner surface of the spherical shell. 3. Coaxial Cylinders. Suppose two coaxial cylindrical surfaces, A l and A^ confronting each other. Let their potentials be ^ and ^ 2 , and their radii R l and R respectively. Let a point in the space between A l and A 2 be at the distance r from the common axis of the cylinders. The potential ^f must satisfy the equation V 2x F = for this space. Since the equi-potential surfaces in the space considered are cylindrical surfaces coaxial with A lt the equation V 2 ^" = may be given the form [cf. XV.] d*P/dr 2 + 2lr.dV/dr = (), and we obtain by integration * = c log r + c r For r = R^ we have = lf and for r = R^ = , therefore *. = *! (log R 2 - log r)/(log E 2 - log RJ + V, . (log r - log ^)/(log R, - log Rj). For a point outside the outer cylinder, the potential is where r is the distance of the point considered from the axis of the cylinder. The constant c cannot be determined from the potentials alone. The electrical force F in the intervening space is F = - P 2 <'I r 3 . If A 3 is displaced by the distance 8x in the direction from A z to A v and therefore in the direction of its length, the area b8x is displaced from right to left, if b represents the width of the I 1 Jt Az \ plate Ay If the distance of A s from A l and A 2 is a, the force acting between A l and A 3 is ^ = ("*F 3 "ty^/a, and that acting between A. 2 and ^4 3 is ^ = (3 - ^ 2 )/a, on the assumption that the points considered do not lie near the edges of A s , or in the space between A l and A y From LXI. (i) the electrical energy is 7F=l/87r. \F-d-r. During the motion considered, the energy on the left side of the pair of plates is increased by that on the right side is diminished by l/8ir. (M / 3 -'*J / 2 ) 2 /a 2 . 2ab8x. The gain of energy is therefore This expression does not fully represent the gain of energy, since no account is taken of the relations at the edges. We therefore set SECT, LX.IV.] CONDENSES AND ELECTEOMETEE. 155 The force X which tends to move A 3 from right to left is then [LXIIL (d)] X=k(V t -V l ).(V i -Mp 1 + VJ). We may apply this result to the quadrant electrometer. If, in this instrument, the movable aluminium plate turns through the angle 6, we may set approximately 6 = aC*, - Yj) . (^ 3 - C*-j + * 2 )) where ^ and ^ are the potentials of the quadrants, 3 is the potential of the aluminium plate, and a is a constant whose value depends on the form and dimensions of the apparatus. SECTION LXV. THE DIELECTRIC. We have assumed until now that the bodies considered were either good conductors or perfect insulators, on which the charge was immovable. Experiment shows, however, that there are no perfect insulators. Electricity on insulators is often lost by conduction, which for the most part is due to the film of fluid deposited on them by the air. But even if this film is removed by careful drying, con- duction still persists. If a charge of electricity is communicated to one part of an insulator, it is distributed after a considerable time in the insulator in the same way as it would be in a good conductor. Besides this, another action also exists which is instan- taneous. When a movable insulator is brought into the neighbour- hood of a charged conductor, the insulator sets itself in the same way as a good conductor, from which it follows that an instantaneous distribution of electricity takes place in it. According to Faraday, insulators con- sist of very small conductors which are separated by an insulating medium. The capacity of a condenser is increased by replacing the air which serves as the insulator between its surfaces by other insulators, such as glass, shellac, calc spar, etc. Let A and B (Fig. 72) be two conducting plates which (^ j are separated by the insulator CD. Let A be brought to the potential ^f v and B to the potential ^ 2 ; let the surface-density on A be a-, that on B is then taken within BC. Hence, when the quantity Q is introduced through the closed surface D, the same quantity flows out through the same surface. We are therefore justified in assuming that the quantity enclosed by the surface D is always zero. This holds for the closed surface E, which may be drawn in the insulator surrounding BC, and thus we obtain the general theorem that the total quantity of electricity contained within a closed surface is equal to zero. If the quantity of electricity >, which flows through a unit of area perpendicular to the direction of electrical force, is proportional to that force at every point, we will have (d) ) = Kjkir . F. If /, g, and h are the quantities which pass through three units of area in an isotropic body taken perpendicular to the three coordinate axes, that is, if they are the rectangular components of the displacement ), and if X, Y, Z are the components of the electromotive force F, we have (e) f=K/4:Tr. X, g=K/4ir.Y, h = K/4:Tr.Z. These expressions are consistent with the relations (f) Z=-3*/3ar, Y= -3*"/3y, Z= - SECTION LXVL CONDITIONS OF EQUILIBRIUM. Suppose the closed surface AS' to enclose a portion of the electrical system, passing partly through the dielectric, partly through the conductors. We will make use of our previous conclusion, that the total quantity of electricity within S is zero. If we represent by e the total quantity enclosed by S, and by )' the quantity which flows out through a unit of area in consequence of the displacement in the dielectric, we have efo'.dS, The normal to the surface S directed outward makes angles with the axes whose cosines are I, TO, 7i. We have ^)' = K/^ir. Fcose, if e is the angle between the normal to dS and the direction of the electromotive force F. Since F cos e = XI + Ym + Zn, we obtain (b) (c) e = 1/47T . \K(Xl + Ym + Zn)dS = \(fl + gm + hn}dS. If we apply equation (c) to an infinitely small parallelepiped, whose edges are dx, dy, and dz, we find that the quantity which enters, it is fdydz + gdxdz + hdxdy, and that which leaves it '/+ dydz + g + dydxdz + h + dz 158 ELECTROSTATICS. [CHAP. vn. if p is the volume-density, the charge contained in the parallelepiped is pdxdydz; now the total quantity of electricity within it equals zero, and therefore fdydz + gdxdz + hdxdy + pdxdydz = (f + ? f dx}dydz + (g + ^dy}dxdz + (h + fdz}dxdy. \ ox J \ oy J \ oz J From this it follows that the volume-density p of the free electricity within the body [cf. LXVIIL] is given by (d) p = 'd/J'd -and from LXV. (e) we obtain In terms of the potential, this becomes /dr represents the increment of r which results from the pressure on the surface, we have from XXXI. (h) d/dr = ^ar+b/r 2 , where a = 3/(3 A + 2/0 . (prf -pff)/(r* - r*) ; b = 1 /4/ . ( Pl -pjr* . r 2 */(r s - rf). From this it follows that If we set K^^/Sir . fjf j/[(*j **i)(^2 3 ~ r i 3 )] = -^ r we w ^ have The volume contained by the hollow sphere will be increased by the action of the electrical force. If we represent by 6 the increment of the unit of volume, we have 47T/3.(r + d JF=-1/87T. By integration by parts, we obtain W= 1 1 Sir . J J f tf((3/aB) 2 + Cd pyy- + where the integration is extended over the entire region, and it is assumed that the force and the potential vanish at infinity. If F represents electrical force, we have (g) JF=lj8ir. Electrical Double Sheets. We have seen in LVIII. that two conductors which are very near each other, and are kept at the two different potentials ^if l and W^ are oppositely charged. The surface-densities -. The potential due to the element d% . d^ . d arising from the com- ponent of magnetization A is, from (a), Addr]d/r 2 . (x - )>. B and C SECT. LXIX.] THE MAGNETIC POTENTIAL. 167 give rise to the potentials Bdgdrjdflr 2 . (y - rf)jr, Cdgdrjdflr 2 . (z - )/r. The sum of these three potentials, integrated over the whole magnet, gives the total potential V, (b) V= \\\\A(x - ) + B(y - r,) + G(z - fl^W- Since r'dr/'dx = x-g, rdrj'dy = y% rdrfdz = z-, we have (c) V= - \\\(A .3(l/r)/3a; + .3(l If we set ^ = Jff^/r.deMt *,= \\lBfr. we have (d) F"= -(3 t = 2ir(It 2 -r 2 /3), where R is the radius of the sphere. If we represent the magnetic potential for points outside the sphere by ^a> an( l for points inside it by V iy we have (c), (d) V a = 47T/3 . ^/r 3 . (Ax + By + Cz) V i = 47r/3 . (Ax + By + Cz). Let / be the intensity of magnetization, and let its direction make angles with the axes whose cosines are A, /*, v. Let 6 be the angle between the direction of J and the line r. We then have Xx/r + fjiy/r + vz/'r = cos and (e) V a = 47r/3 . 1PJ cos 0/r 2 ; F; = 47r/3./rcose. Hence, the potential outside the magnet is the same as that which is set up by an infinitely small magnet, whose magnetic moment is 3K = 4ir/3.W [LXIX (a)]. If the a-axis lies in the direction of magnetization, the potential in the interior of the magnet is V i = 4?r/3 . Jx. The magnetic force ^) inside the sphere is therefore constant, and is expressed by (f) $=- -Air/3. J. Outside the sphere we divide the force into two components, one of SECT. LXX.] POTENTIAL OF A MAGNETIZED SPHERE. 169 which, P, acts in the direction of the line r, the other, Q, perpendicularly to that line. We then have P= -Vrjdr, Q= -l/r .'dTJ'dQ, or P = 87T/3 . Wcos 0/7- 3 = 2%R/r 3 . cos ; Q = 47T/3 . E 3 Jsin 0/r 3 = Wfr 3 . sin 0. From LXVIII. the surface-density is determined by cr = /cos9. The resultant forced is F^^ft/r 3 .^/! + 3cos 2 6. We have further tgQ = 2Q/P. If is the angle between the direction of the force F and the direc- tion of r, we have tg/dz. We now determine the moment of the forces which tend to turn the magnet about one of the coordinate axis, say the z-axis, on the assumption that the magnetic forces are constant. Let the coordinates of the point (Fig. 77) be x, y, z. The force which acts on the surface BO' in the direction of the z-axis has a moment with respect to the x-axis equal to Bdxdz . y. (y + dy). The force acting on OB' has a moment with respect to the same axis equal to - Bdxdz .y.y. Neglecting small terms of higher order, the resultant moment is Bydxdz , dy. The forces acting on the surfaces O'C and OC' give rise to the moment - Cftdxdy . dz. The moment L, which tends to turn the magnet about the z-axis, is therefore (c) L = \\\(By-C($)dxdydz. The moments of rotation M and N, with respect to the two other axes, are determined from analogous expressions. If the magnet is subjected to the action of the earth's magnetism only, the magnetic force may be considered as constant both in magnitude and direction. The components a, (3, y are then inde- pendent of x, y, z, and therefore X=F=Z=Q. The centre of gravity of a magnet does not move under the action of the earth's magnetism. The magnet is, however, acted on by a moment of rotation, which may be determined in the following way : Let the magnetic moment of the magnet be 9)?, and suppose its direction with respect to the coordinate axes to be determined by the angles whose cosines are I, m, n. We then have Wl = \\\A.dT, mm = \\\B.dr, Wn = \\lC.dr, and hence L = Wl(ym - /3n) ; M=$tt(an - yl) ; N= Wl(/3l - am). From these equations, it follows that La + Mtf + Ny = and LI + Mm + Nn = 0, that is, the resultant moment is perpendicular to the magnetic force and also to the magnetic axis of the magnet. If the direction of the force is parallel to the or-axis, and if the magnetic axis lies in the xy- plane and forms with the ,T-axis the angle 6, we will have (d) Z = 0, M=Q, N= -2tta.sine. If a magnet can turn about a vertical axis, the moment which tends to increase the angle between the magnetic axis and the magnetic meridian is -~9RlT.dll 6, where H denotes the horizontal component SECT. LXXI.] FORCES ON A MAGNET. 171 of the earth's magnetism. If w is the angular velocity of the magnet and / its moment of inertia, we have, from XXII. (c), d(Ja>) = - WH .s'mQ.dt, or since w = dQ/dt = 0, (e) je= -WlH.sinQ. If the angle 6 is very small, the period of oscillation of the magnet is given by XXII. (e), (f) T SECTION LXXII. POTENTIAL ENERGY OF A MAGNET. By the potential energy of a magnet is meant the work which is needed to transfer the magnet from a position in which no magnetic forces act on it to the position in which the magnetic potential is V. We will first consider an infinitely small parallelepiped (Fig. 78), whose components of magnetization are A, B, C. In order to bring the magnetic surface OA' to the position in which the potential is V, the work - A . dydz . V must be done. The opposite surface O'A is brought to the position in which the potential is F'+'dV/'dx.dx, and the work done on it is A . dydz. (P'+'dT/'dx.dx). The work done on these two surfaces therefore amounts to A . 'dVj'dx.. dxdydz. If we obtain in like manner the work done on the two other pairs of surfaces, we find that the whole work W done in transporting the magnet is (a) W= \\\(A.^ Vfdx +B.3 Vj'dy + C . 9 Vfdz)dxdydz, or since a, /?, y, the components of the magnetic force, are FlG 78 we have (b) JF= -\\\(Aa + B(3 + Cy)dxdydz. We will apply this equation to the case of a magnet subjected to the action of the earth's magnetism only. Let its magnetic moment be 3ft, and let the direction of its magnetic axis make angles with the coordinate axes whose cosines are /, m, n. We then have . dxdydz = m, \\\B .dxdydz = nm, \\\C .dxdydz = nW, 172 MAGNETISM. [CHAP. vui. Representing the magnetic force by , and supposing its direction to make angles with the coordinate axes whose cosines are A, p, i>, we obtain W= - Wl^lX + mp + nv). Letting 6 represent the angle between the magnetic axis of the magnet and the magnetic force, we have (c) W= - 3J?^) . cos0. If the direction of the force is parallel to the z-axis, as in LXXf., and if the magnetic axis lies in the ay-plane, the work done in turning the magnet through the angle dQ is dW= + 9tf . sin 6 . dQ. This agrees with LXXI. (d). We will now consider a very small magnet situated near a very strong magnet. If the small magnet has sufficient freedom of motion, it will turn so that its magnetic axis is parallel to the direction of the magnetic force. In this case we have 6 = and its potential energy W is W= - 3ft$. Since the motion of the small magnet involves the loss of potential energy, it moves in such a way that W diminishes. This occurs by the last equation, when ^) increases ; the magnet therefore moves in the direction in which the magnetic force increases. A particle of a paramagnetic substance therefore tends to move towards the place where the magnetic force is greatest. On the other hand, diamagnetic bodies move toward the place where the magnetic force is a minimum. In order to find the magnetic energy residing in a system of magnets, we proceed in the following way : The potential at every point within the system varies proportionally with the values of the components of magnetization. We assume that the components of magnetization change only in such a way that, in successive instants, they always increase by the same fraction of their final values. On these con- ditions the potential increases in the same proportion. If the com- ponents of magnetization arc originally zero, the potential is also originally equal to zero. Let the final values of the components of magnetization be A, B, C. At a particular instant during the increase of the components of magnetization, let these be represented by nA, nB, nC, where n is a proper fraction. At the same time the potential at any point is equal to nV. If the components of magnetization increase by A.dn, B.dn, G.dn respectively, the potential at the point considered increases by Fdn. If A, B, C increase by A . dn, B.dn, G.dn respectively, the work needed to accomplish this is, \>y (a), \\\(A . dn . ridF/dx + B.dn. ndVj'dy +C.dn. n = n.dn\\\(A . SECT. LXXII.] POTENTIAL ENERGY OF A MAGNET. 173 Now, if n increases from to 1, we have / ndn = ^, and the work done is (d) W= \\\\(A . -dVftx + B . -dFj-dy + C . VFj?)z)dxdydz, or, by introducing the components of the magnetic force, (e) W= - \\\\(A* + Bfl + Cy)dxdydz. The energy of a magnetic system may be expressed in another way. The same method by which we before obtained an expression for the energy, shows that (f) W=*\\ = 0, and hence W = - 1 /STT . J f J V( 3 2 Vfdy? + 9 2 Vfif + 3 2 Vfiz^dxdydz. By integration by parts over the whole infinite region, we obtain or (g) W=-\l&ir. ^\(a z + /3' 2 + y 2 )dxdydz. Similar expressions hold for dielectric polarization [cf. LXI.]. SECTION LXXIII. MAGNETIC DISTRIBUTION. A piece of soft iron brought into a magnetic field becomes magnetized by induction. "We assume that the intensity of magnetization at any point is a function of the total magnetic force acting at that point. We assume that the intensity of magnetization is proportional to the magnetic force, or that (a) A=ka, B = kf3, C=ky, where k is a constant. The magnetizing force proceeds partly from the per- manent magnets present in the field, and partly from the quantities of magnetism induced in the soft iron. The potential due to the former may be designated by V, that due to the latter by U, so that A = - k . 3( F+ U)fix, B = - k . 3( F+ U)fdy, C = - k . 3( F+ U)fiz. Now, in the space not occupied by permanent magnets, we have V 2 F"= 0, and therefore 'dA/'dx + 'dB/'dy + 'dC/'dz^ -kV 2 U, or since, from LXVIII. (e), p = -(dAj'dx+oBj'dy + 'dCI'dz}, we have finally V 2 U-p/k = 0. Since the potential W is due to the components of magnetization A, B, C, the equation that holds within the soft iron is V 2 U + 4-rp = 0. From the last two equations we obtain (b) (1 + irk)p = Q, and hence p = ; that is, there is no free magnetism present within the soft iron. The 174 MAGNETISM. [CHAP. vm. magnetism present is therefore situated on the surface of the iron. We will now determine the surface-density cr of this distribution. For this purpose we use the equation where v a and v t are the normals drawn from any point on the surface of the iron inward and outward respectively. U t and U a are the values of the potential due to the induced magnetism inside and outside the iron mass. Now we have 3 Vfi Vl = - 3 F/3v rt , and hence 4 + 3 7,/3v, + 3 UJdv a = 0. The magnetizing force just outside the surface of the soft iron is -3(P"+ 7,)/3v 4 in the direction of v t . The free magnetism on the corresponding surface-element is therefore cr .dS = k.'d(V+ U i )/'dv i .dS. Hence we have (c) 4vk . 3 F/'dvt + ( 1 + 47r)3 U^v, + 3 Ujov a = The relation (c) in connection with the equations (d) V 2 U t = Q, V 2 f/ = serves to determine the potentials U t and U a . As an example of the theory here presented, we will consider the magnetization of a sphere subjected to the action of a constant magnetizing force ^) which acts in the direction of the -axis. Let the intensity of magnetization of the sphere in the direction of the a-axis be A ; the force due to the magnetization and acting in the direction of the z-axis is [LXX. (f)] equal to -4TT/3.A. From equation (a) we have, therefore, A = k(5g> - 4ir/3 . A), and hence ^=&

>0, if the direction of motion coincides with the direction of the force. If the magnetic potential is V, we have, however, ^)= -dF/ds, and therefore, for a closed path, since the potential is a single-valued function of the position of the point. Any tube of magnetic force must begin and end on the surface of a magnet. If the tube ends with the cross-section PQ (Fig. 80), so that a magnetic force is present in the tube TUQP, while it is zero outside the tube at R and S, we may apply Jt,---,S equation (a) to the region TUQSRP. Since a magnetic force acts at the surface TU, but not in the region PQSB, the surface integral taken over TUQSRP cannot be zero. Magnetism must therefore be present within the closed surface, which contradicts our assumption. Therefore, any tube of magnetic force ends at the surface FlG of a magnet. In order to represent the magnitude and direction of magnetic forces, Faraday used lines of magnetic force ; he assumed that the lines 176 MAGNETISM. [CHAP. vm. of force are continued in the body of the magnet. His mode of repre- sentation has become of very great importance. If a magnet is broken, and the surfaces exposed by the fracture are placed so as to face each other and separated by only a small distance, a strong magnetic T FIG. 81. force acts in the region PQEU (Fig. 81). This force is due partly to the free magnetism in the interior of the magnet and on its original surface, and partly to the free magnetism on the newly-formed sur- faces. The force due to the former cause is directed from the north pole n to the south pole s, that due to the latter from s to n. The latter force is in practice the stronger, so that we may say with a certain propriety that the magnetic tubes of force are produced through the interior of the magnet along the path D'F'FD (Fig. 81). FIG. 82. If S (Fig. 82) is a closed surface lying outside all the magnets in the field, and therefore containing no magnetism, we have from XIV., It is customary to express this result in the following way : The integral J^) n . dS which is extended over a part of the surface may be SECT. LXXIV.] LINES OF MAGNETIC FORCE. 177 divided into the parts $> nl . dS lt ^) n2 . dS 2 , etc. Let them be so taken that they are all equal, and let their common value be taken as unity. Since the product ) n .dS is constant for the same tube or line of force, the integral J.) n . dS gives the number of lines of force which traverse the surface. If this integral is zero, as many lines of force entei' the sur- face as leave it. This holds for a surface which contains one or more magnets, for the sum of the magnetism in every magnet is zero. On the other hand, the theorem does not hold if the surface cuts through a magnet. Nevertheless, if the magnet is divided into two parts, MNQP (Fig. 81) and RSTU, and if they are situated infinitely near each other, the theorem holds for either of them if the surface considered contains one part, but excludes the other. Now, if this mode of division of the magnet produces no disturbance in its magnetization, the theorem can be expressed in the following way : We represent the components of the magnetic force by a, (3, y. In the part of the surface lying outside the cleft, no other magnetic force is acting; on the other hand, the free magnetism +0- on the element dS of PQ (Fig. 81), and o- on the corresponding element of RU, produce a force which can be determined in the following way : From XIII. a surface on which the surface-density is o- exerts an attractive force 27ro- on a unit of mass lying very near it. In the case of magnetism, the force lira- is a repulsion. If there are two parallel surfaces, on one of which the density of the magnetic distribution is o-, while on the other it is - o-, the magnetic force acting between the surfaces is 47r 1, in diamagnetic bodies /*< 1. SECTION LXXVII. MAGNETIC SHELLS. Suppose a thin steel plate to be magnetized so that one face is covered with north magnetism and the other with south magnetism. At any point A in the face N (Fig. 85) draw a normal to the plate which cuts the surface S at B. Let the plate be so magnetized FIG. 85. that -o- represents the magnetic surface-density at B, and +0- that at A. We set AB = e, and call o-e = 4> the strength of the shell* at the point under consideration. If the plate is infinitely thin and the surface-density infinitely great, has a finite value. Such a plate is called a magnetic "shell" The potential of such a shell may be expressed in the following way: Let LM (Fig. 86) be the shell, dS a surface-element on its positive face, BC the . normal to this surface-element, and P the point * In the original, the moment of the surface. TB. SECT. LXXVII.] MAGNETIC SHELLS. 181 at which the potential is to be determined. We represent the angle between BC and BP by e. The potential at the point P, due to that part of the shell whose end-surface is dS, is [LXIX.] dF=(r.dS.e.cosf/r z . Hence, if the strength of the shell is constant, (a) V= < . J{ cos e . dS/r 2 , where the integral is to be extended over the whole surface. If the solid angle subtended by dS at the point P is called dw, and if we set BP = r, we have dS . cos e = r^dw. Therefore dp r =or. e. e?w = . If PQP' (Fig. 87) is a curve which does not cut the shell, and whose ends lie infinitely near each other on opposite sides of it, the work done by the magnetic forces in moving a unit ---,. magnet pole over the path PQP' is equal to 4iri. This theorem holds even if other magnets are present in the field. They act on the pole with forces which have a single-valued potential, and the work done by them during the motion of the unit pole in the curve PQP' is equal to zero ; for this curve may be con- sidered as a closed curve, since P and P' are infinitely near each other. After obtaining an expression for the potential of a magnetic shell, we determine the force with which the shell acts on a magnet pole of unit strength. The normal to the shell makes angles with the axes whose cosines are I, m, n; let */> k e the coordinates of a point in the shell, and x, y, z the coordinates of the point outside the shell for which the potential is to be determined. We then have cos e = I. (x - )/r + m.(y- ^)/r + n . (z - f)/r, where r' 2 = (x - ) 2 + (y- rjf + (z- ) 2 . From equation (a) the potential is r= & . J J [(,- _ g) .1 + (y _ rj)m + (z - f )]/r . dS. 182 MAGNETISM. [CHAP. vm. Since 3r~ 1 /c3 = (as-)/r s i we obtain V= *f J(/ . Br- 1 /^ + wi . Br" 1 /^ + n . 'dr~ l j'dt)dS. Represent the components of the magnetic force in the direction of the .r-axis by a, we then have (d) a= -'dP'/'dx, and because 3r~ 1 /3= -Br" 1 /^. If the shell does not pass through the point .T, y, 2, 7- will never become zero, and we have 3V-i/32 + 3V- W + BV-'/Bf 2 = 0, (e) a = + *. J J[TO.3 2 i-V3i/3 + . 3 2 r-Y33 - l(Wr~ l l'dr From the theorem of VI. (f), we have f \(X. d/ds + Y . drj/ds + Z . dC/ds)ds + n WesetJf=0, Y= +*.3r~ 1 /9t 2= -^.'dr~ 1 /'dr i , by which the right sides of equations (e) and (f) become identical, and then obtain (g) a = * . {Br-VBf - *?/<** - Sr- 1 /^ . dC/ds)ds. Analogous expressions hold for (3, /. By carrying out the differen- tiation, we obtain (h) a = $ . J[(z - fl/r 8 . diy/rfe - (y - 1?)/?- 3 . d{/ds]ds. The force is therefore determined by the contour and the strength of the magnetic shell. This result follows from the fact that the potential is determined by the solid angle and the strength of the shell. In order to find the geometrical meaning of equation (h), we use the following method : Let , 77, be the coordinates of the point (Fig. 88) ; Oy and Oz represent the directions of the y- and s-axes respectively. Let the element ds be parallel to the 2-axis, and represented by OA=d; we then have drj = Q. Let the point P, for which the potential is to be determined, lie in the y^-plane, and let OP = r. We set and have y -rj = r . sin 6. The magnetic force due to ds = OA is, (i) a = - * . ds . sin 6/r 3 ; it is perpendicular to the y.^-plane. Its direction may be determined in the following way : If the right hand is held so that the fingers point in the direction of ds, and the palm is turned toward the pole P, the thumb gives the direction of the force. Finally, we determine the work which must be done to bring a magnetic shell from an infinite distance to a place where the magnetic potential is equal to V. Let the shell be divided into elements dS. In order to bring the surface-element which carries the quantity SECT. LXXVIL] MAGNETIC SHELLS. 183 o- . dS of south magnetism to its final position, work equal to - o- . dS . V must be done. In order to bring the corresponding surface-element carrying the same quantity of north magnetism to its place, work equal to (F'+dF'/dv. e)ar . dS must be done, if v represents the normal to the surface-element dS. Hence the total work done is A = f \dVldv .&r.dS=3>. \\dVldv . dS. Now, since dVjdv = - (la + mfi + ny), we obtain for the work done A= -3>.l\(loi + mp + ny)dS. If JV represents the number of lines of force contained by the contour of the shell, we have A = - & . N. CHAPTER IX. ELECTRO-MAGNETISM. SECTION LXXVIII. BIOT AND SAVART'S LAW. OERSTED discovered that the electrical current exerts an action on magnets ; the law of the magnetic force which is due to an electrical current was discovered by Biot and Savart. Let AB (Fig. 89) be a conductor traversed by a current which is S*P measured by the quantity of electricity flowing /* in unit time through any cross-section. Let D 0,-' the quantity of magnetism p. be situated at the point P, and let the conductor AB be divided into infinitely small parts ds. If CD = ds is an infinitely small part of the conductor CP = r, and 6 the angle between r and the direction of the current in CD, the direction of the force will be perpendicular to the plane determined by r and ds. If the right hand points in the direction of the current and the palm is turned toward the magnet pole, the direction of the force exerted by the current-element on the pole is given by the direction of the thumb. The magnitude of the force K is (a) K=fi . i.ds/r 2 . sin0. The magnetic force which is due to any system of electrical currents, whose direction, strength, and position in space are known, may be calculated from (a). If the current forms a closed circuit, and if the intensity of the current is the same at all points in the conductor, we may determine the force due to the current and also the potential which the current produces. The force due to any current-element is equal to that exerted by a line-element of the same length, which forms part of the contour of a magnetic shell, whose strength is equal to the current- 184 CH. ix. SECT. Lxxviii.JBIOT AND SAV ART'S LAW. 185 strength. This follows from a comparison of equation (a) with LXXVII. (i). From LXXVII. (b), the potential V of a closed circuit of strength i, at the point P, is (b) V=iw, where w is the solid angle subtended by the circuit at P. If a, /?, y are the components of the magnetic force acting at P, we have, from LXXVII. (h), ( a = i . f ((* - {)/rS . d-nlds -(y- n)li* . dtfds)ds, (c) J8 = i . \((x - )/r3 . dflds -(z- f)/ 7 = * If ABC (Fig. 90) is a conductor through which a current i flows in the direction indicated by the arrow, and if a unit magnet pole moves around the current in the direction found by using the right hand in the manner before described, the work done by the mag- netic forces during the movement over the /j path DFED is, from (b), equal to 4iri. If the path of the pole encircles several currents i, i', i" etc., the magnetic forces due to these currents do upon it, during its motion, the work A, given by (d) A = 4ir(i + i' + i" + ...), in which the currents which flow in one FlG direction are to be reckoned positive, and those which flow in the opposite direction, negative. Hence, the potential which an electrical current produces at the point F is not determined only by the position of that point. If we bring a unit pole (Fig. 90) to F over the path GF from an infinite distance, the work which is done will be equal to V, the potential at the point F. If the pole then passes around the current over the path FEDF, the work k-n-i will be done, and the potential at F becomes V+iri. If the pole passes n times around the current in the same way, the potential at F becomes V+ 47rni. Hence, the potential at the point F has an infinite number of values. The differential coefficients of the potential with respect to x, y, z are nevertheless completely determined. If a pole of strength ^ passes once around the current, the work done on it is iTrip; if a magnet passes once around the current and returns to its original position, the work done is 47ri2/*, when 2/* represents the sum of the quantities of magnetism in the magnet. But since for any magnet 2/* = 0, the work done is in this case equal to zero. 186 ELECTRO-MAGNETISM. [CHAP. ix. Since an electrical current may be replaced by a magnetic shell, we can obtain the magnetic moment of an infinitely small closed current. If i is the current-strength, the strength of the equivalent magnetic shell is a-e=--i. If dS is the surface of the shell, we have i.dS = = faNi and ^) = INifR ; in the latter case, ^p = 0. SECTION LXXX. THE FUNDAMENTAL EQUATIONS OF ELECTRO-MAGNETISM. Up to this point we have considered the path of the electrical current as a geometrical line. In reality the current always occupies space, and is determined by its components along the coordinate axes. For example, if dy . dz is a surface-element perpendicular to the z-axis, and if the quantity of electricity u.dy.dz. dt passes through it in SECT. LXXX.] EQUATIONS OF ELECTRO-MAGNETISM. 189 the positive direction in the time dt, u is the component of current in the direction of the re-axis. The components of current in the directions of the two other axes are represented by v and w. If Oy and Oz are drawn through the point (Fig. 94), whose coordinates are x, y, z, parallel to the cor- responding coordinate axes, and if the rectangle OBDC is constructed with the sides dy and dz, the current u.dy . dz flows through the element OBDC. If the components of the magnetic force are represented by a, p, y, and if a unit pole o moves about the rectangle in the direction OBDCO, the work done by the magnetic forces will be (3.dy + (y + 'dy/'dy . dy) .dz-(p + 3/3/3z ,dz).dy-y.dz This is [LXXVIII. (d)] equal to 4?r . u . dy . dz. Hence, we obtain the equations (a) 4iru = (dy[dy-'dpfdz), 4m = (da[dz-'dy[dx) i 4irw = (d(J/'dx-'da/'dy). These equations express the current in terms of the magnetic force. In a region where there is no current we have u = 0, v = 0, w 0, and therefore o y /3y = 3/3/32, da/32 = 3y/3z, 'dfij'dx = 3a/3y, or a. dx + p.dy + y.dz= -dV. Therefore, in a region where there is no current the magnetic forces have a potential. In this case the forces arise from magnets. From equations (a) the magnetic force is not determined only by the components of current. If u, v, w are given and a, p, y so deter- mined that equations (a) are satisfied, these equations will also be satisfied if we replace a, p, y by where V is an arbitrary function. The potential due to the magnets present in the region is V. We will now consider a few simple examples : (a) Suppose the direction of the magnetic force to be parallel to the 0-axis, and its magnitude to be a function of the distance r from. this axis (Fig. 95). We then obtain from equations (a) 47TM = +dyjdr.y/r, 4irv = -dy/dr.x/r, w = 0. The current is parallel to the zy-plane and perpendicular to r. The current-strength J is J= u cos (uJ) + v cos (vJ), J -u. y/r + v . x/r = l/4?r . dy/dr. 190 ELECTRO-MAGNETISM. [CHAP. ix. If y is constant in the interior of a cylinder whose radius is OA=r l (Fig. 95), and equal to zero outside a cylinder of radius r. 2 , the current in the unit length of the cylinder is j . dr = - . dr FIG. 95. which agrees with LXXIX. (b). (b) If the current-strength is given, we can find the magnetic force by integrating equations (a). Let u and v be zero, and w be a function of the distance r from the ^-axis. We then have from (a) These equations will be satisfied if we assume that y = and that a and (3 are functions of x and y only. Suppose that the magnetic force is resolved into two components, one of which, R, acts in the direction of the prolongation of r, and the other, S, is perpendicular to r. We then obtain a = R . x/r - S . y/r, fl --= R . yjr + S . .r/r, and therefore 47rw = dS/dr + S/r =l/r. d(Sr)/dr. If the conductor is a tube bounded by two coaxial cylinders whose radii are R l and R 2 , and if w is constant in the conductor we have, if C v C 2 , C 3 are constants, S l r=C v 2Trwr 2 + C. 2 = S. 2 r, S 3 r=C 3 . The first of these equations holds for the interior, the second for the con- ductor, the third for the space outside the conductor. From the nature of the problem, S l must have a finite value in the axis ; we therefore have C\ = 0. Since the magnetic force changes continuously, we have , = when r = R ly and therefore C. 2 = - 2-n-wR^, S. 2 = Airier - Since irw(R. 2 2 - R^) is equal to the current-strength i in the conductor, we have S 3 = 2i/r. Therefore, an infinitely long straight linear current exerts a magnetic force at a given point, which is inversely pi-oportional to the distance of that point from the current. SECTION LXXXI. SYSTEMS OF CURRENTS IN GENERAL. The components of current and the components of magnetic force are connected by the equations [LXXX. (a)] (a) 4?rM = 'dyj'dy - 'dfi'fdz, ITTV = 'da/'dz - 'dy/'dx, lirw = 'dft/'dx - 'da/ay. SECT. LXXXI.] SYSTEMS OF CUREENTS. 191 From these equations it follows that (b) This equation corresponds with the equation of continuity in mechanics, and asserts that the total quantity of electricity contained in a closed region is constant. It thus appears that the current, whose components are u, v, w, moves like an incompressible fluid. There is never any accumulation of electricity, but only a displacement of it. This apparently con- tradicts experience ; in order to be consistent with our method of treatment we assume with Faraday that an electrical polarization or an electrical displacement occurs. We represent the components of this displacement by /, g, h. If one of the components, say /, increases by the increment df in the time dt, df/dt=f represents the quantity of electricity which passes in unit time through a unit of area per- pendicular to the z-axis, in consequence of the change of polarization. If p, q, r represent the components of the electrical current which is due to the flow of electricity through the body, we have (c) u=p + df/dt, v = q + dg/dt, w = r + dhjdt. These quantities, u, v, iv, are the components of the actual current, which is made up of the current conducted by the body and the current arising from the change of polarization or the electrical displacement. If the components of the current are finite, the components of magnetic force vary continuously when no magnets are present in the region. The components of force perpendicular to the surfaces of the magnets, if any are present, is in general discontinuous. We assume that currents of infinite strength do not occur in practice ; however we sometimes consider the flow in a surface, in which case we must assume that the components of current in the surface are infinite. In this case the components of force parallel to the surface vary dis- continuously on passage from one side of the surface to the other. If a : and a 2 represent these components of force, and J the quantity of electricity which flows through a unit of length perpendicular to the components, we have from LXXVIII. (d) 47r/=a 2 -a 1 . We may obtain the same result from (a) as follows : We consider two surfaces whose equations are z = c l and z = c y We obtain from the first two equations (a) 477 . Fu . dz = 33 .dz- + fi - . /"* . dz = a -cosa. Hence the increment of the potential energy of the conductor is, from (e), dW '= -iK. ds.dp, if i is the current-strength. In order to cause the motion here described, a force X must act on ds in the direction A A', which is determined by X.dp = -iK. ds.dp, X= -iK.ds. Hence the force -IK.ds acts on the current-element in the aforesaid direction, and if the current- element is free to move, the direction of its motion is perpendicular to the direction of the magnetic force as well as to its own direction. The direction of the motion is determined by laying the right hand on the current [cf. LXXVIII.]. It follows further that the force which acts on an element ds of the current i is perpendicular to the plane determined by the current and the direction of the magnetic force Q. If we represent the angle between the direction of the force and the direction of the current by <, this force will equal $>i . ds . sin <. SECTION LXXXIII. THE MEASUREMENT or CURRENT-STRENGTH OR THE QUANTITY or ELECTRICITY. (a) Constant Currents. To measure constant currents we generally use a galvanometer con- sisting of parallel circular conductors carrying the current whose strength is to be determined. A magnet whose dimensions are small in comparison with the radius of the coils, is suspended in the centre of the apparatus, which is so placed that the coils are parallel to the magnetic meridian. The current sets up a magnetic force whose value is Gi perpendicular to the direction of the earth's magnetic force, whose horizontal component is called H. G depends on the con- struction of the galvanometer. If G is constant in the region in which the magnet moves, the angle < by which the magnet is turned from its position of rest by the current is determined by (a) tg = GilH, i = H/G.tg, that is, the current-strength is, in this case, proportional to the tangent of the angle of deflection. SECT. LXXXIII.] MEASUREMENT OF CURRENT-STRENGTH. 195 (b) Variable Currents. It is very difficult to determine the strength of currents of short duration at any instant. We may, however, easily measure the total quantity of electricity Q which flows through the conductor. From' LXXI. (d) the moment which tends to turn the magnet about a perpendicular axis is - SCfta sin 0, if 9JJ is the magnetic moment of the magnet, a the magnetic force, and the angle between the direc- tions of 9Ji and a. Setting a = Gi, where G is the galvanometer constant, and assuming Q = ^TT, the directive J wee exerted by the current on the magnet is equal to 0J6ri. The total moment caused by the current is therefore \^llGi.dt = ^lG.Q, if we write Q = \i.dt. Q is the quantity of electricity which passes through the conductor during the discharge. If .7 is the moment of inertia of the magnet, and will be its moment of momentum. We thus obtain the equation (b) *$lGQ = Jo>. If the period of oscillation of the magnet is called T, we have, by LXXI. (f), (c), (d) T = Tr.Jj/WH and therefore Q = Hr^/Gir 2 . The kinetic energy which the magnet receives from the impulse given to it by the current is |/w 2 , in consequence of which it turns through the angle 6. Its potential energy thereby increases from -WlH to -WlHcosQ; the work done on it is 9Ji#(l -cos0). We therefore have iJo) 2 = 29JlH"sin 2 (0/2), or, if 6 is very small, (e) G-TW/TT and Q = Hr/TrG . 0, that is, if there is no damping action on the magnet, and if its angular displacement is small, the quantity of electricity flowing through a section of the conductor is proportional to the angular displace- ment of the magnet. (c) Damping Action. The oscillations of the magnet generally diminish rather rapidly in consequence of what is called damping or damping action. Damping arises from resistance of the air and the action of currents induced by the motion of the magnet in neighbouring conductors. If there is no damping, we have from LXXI. (e) and (f), when the oscillations are small, d' 2 Q/dt 2 = - 7r 2 /r 2 . 0, T is therefore the period of oscillation of the undamped magnet. We may assume that the damping action is proportional to the angular velocity dQ/dt. Taking the damping into account, we have, to determine the deflection 0, the differential equation (f) + 2?w0 + 7r 2 /r 2 .0 = 0. The factor m depends on the size and character of the oscillating magnet, on the density of the air, 196 ELECTRO-MAGNETISM. [CHAP. ix. and on the size, character, and position of the masses of metal in which currents are induced. If we set TT/T = n and 6 = e at , we have a 2 + 2ma + n- = and a = - m \/n 2 - m?*J - 1 in which it is assumed that n > m. Setting (g) n 2 -m 2 n- 2 / T i 2 > we have 6 = (A sin (rtfa) + B COS^/T^) . e~ mt . If = at the time / = 0, we obtain Q A . e~ mt . sin (irf/Tj). dQ/dt = <*> at the time / = 0, and therefore 9r 1 */ > .tf^"*..nn(vl/T 1 ), To find the magnitude of the deflection we set dQ/dt = Q, and obtain (h) tg^/Tj) = 7r/wr r If T O is the smallest root of this equation, the successive roots are T O + TJ, T Q + 2r v .... The oscillations are therefore isochronous. If we repre- sent the deflections by U 6 2 , 3 , ..., we have .e- mr <>. sin (^TO/TJ), . e - m ( T o+ T i> . sin (TT^/TJ ), ~ < T o+2Ti) . sin (*-T O /TJ ). If the position of equilibrium is designated by A , the first point of reversal by A v the second by A. 2 , etc., the ratio between the oscillations AA* and A.A is (ej-e^^Og-e,) and (e 1 -e 2 y/(e 3 -e 2 )=e"- r '. We set w-r^A, and obtain (i) A = log nat [(e x - 2 )/ '(0 3 - 6 2 )]. A is the logarithmic decrement, which can be very exactly determined from a series of oscillations. From (g) the period of oscillation T I} is (k) TJ = T . Vl + A' 2 --. Therefore the period of oscillation is increased by the damping action. If we set T = T O in equation (h), we have t g( 7rr o/ T i) = v l mr i and 7rT o/ T i = arctg(ir/X), mr = A/V . arctg(7r/ A), sm(irr /T 1 ) = l/Vl+A2/ir2. Hence we have further : = T^/TT . g-V*-.arctg(ir/A) , i/^/j + xay^-' a nd (1) fc) = TrOj/T! . Vl + A*/JT* . W" arctg (x/X), We obtain from (d) and (k) Q = Hr^jGir- . w/(l + A 2 /jr 2 ), and using equation (1), (m) # = e i .fir 1 /GV. x / r - 1TO * 0r / x >. l/Vr+A^. In order to determine the quantity of electricity sent through a conductor by an electrical current, whose duration is small in com- parison with the period of oscillation of the magnet, we must determine the logarithmic decrement and the period of oscillation of the magnet. Q is then determined from these quantities, if we know in addition the intensity of the earth's magnetism and the constant of the galvanometer. SECT. LXXXIII.] MEASUREMENT OF CURRENT-STRENGTH. 197 Setting arctg 7r/X = ^7r-x we have tg.r = A/7r. If A is very small, we have x = X.jir and arctg (jr/A) = \TT A/TT. If the damping action is insignificant, we can neglect higher powers in the series in which the exponential may be developed, and obtain 2 and Q = SECTION LXXXIV. OHM'S LAW AND JOULE'S LAW. We have up to this point assumed the existence of the electrical current and have not discussed the question of the way in which it is started and maintained. This mode of treatment in many respects lacks clearness. We will therefore state such facts as are well established by observation. The so-called galvanic elements can establish and maintain an almost constant current. In order to maintain a constant current in a conductor, an electromotive force must act in the direction of the current. If u is the quantity of electricity which flows in unit time through a unit of surface of the #>/-plane in the direction of the .r-axis, we can set w=C.X, if C is the conductivity and X the component of the electromotive force in the direction of the o-axis. C depends on the nature of the conductor, and may be supposed to have the same value in the conductor in all directions. If the com- ponents of current and of force in the other two directions are v, w, and Y, Z respectively, we have (a) u = CX, v=CY, w=CZ. Hence 'fafdx + 'dordy + 'du;l'dz=C('dX[dx + 'dY[dy + 'dZ[dz). If the steady state of the electrical current has been reached, the left side of the equa- tion equals zero, and hence the right side is also equal to zero. If the electromotive forces have a potential V, we have (b) V 2 F~=0. This equation states that no free electricity is present within the conductor- as soon as the current becomes steady. The electromotive forces must therefore arise from the free electricity on the surface of the conductor. Suppose ABC (Fig. 98) to be an electrical conductor. We will consider a portion of it which is bounded by the infinitely small cross-sections A A ' = S and BB' = S, separated from each other by the distance I, which is ______ ^r- 1 __ also infinitely small. If AB is parallel to ^ ]b -7- the ./--axis, the component of current u equals CX, and therefore the quantity of electricity FlG- 98 ' i = uS=CX.S flows through the cross-section S. If V and V" are the potentials at A and B respectively, we have i = C . S . ( V- 198 ELECTRO-MAGNETISM.. [CHAP. ix. SECT. LXXXIV. and further (c) t = (V- P)/(//(7S) = (F- V")/K The resistance I! is directly proportional to the length of the conductor and inversely propor- tional to its cross-section, and to the conductivity of the substance constituting the conductor. The difference of potential between A and B is V - V. Equation (c) contains Ohm's law, according to which the current- strength is directly proportional to the difference of potential and inversely proportional to the resistance. The quantity of electricity i . dt flows through the cross-section A A' in the time dt and passes from A to B under the influence of the electromotive force X. The work done is therefore The work done in this part of the conductor by the electromotive forces in unit time is (d) A = i(V- V) = i' 2 R This work is trans- formed into heat in the conductor. Therefore the quantity of heat developed in a conductor is proportional to the square of the current-strength and the resistance of the conductor. This theorem was proved experi- mentally by Joule and deduced theoretically by Clausius. CHAPTER X. INDUCTION. SECTION LXXXV. INDUCTION. FARADAY was the first to demonstrate that a current is set up in a conductor if a magnet or a conductor carrying a current is moved in its neighbourhood. F. E. Neumann discovered the laws of these induced currents. Faraday himself afterwards described a method of determining the strength and direction of the induced current which possesses great advantages, because it makes it possible to visualise the process. Suppose ABO to be a closed conductor (Fig. 99), and DE, D'E', etc., the lines of force enclosed by it. Let us designate an element of a surface bounded by the conductor by *"* dS, the components of the magnetic force (cf. LXXI.) by a, /?, y, and the angles which the normal to dS makes with the axes by I, m, n. An electromotive force arises in the conductor if the interal (a) changes its value. If magnetizable bodies are enclosed by the circuit which have a greater permeability for lines of force than air, the components of force must be replaced by the components of magnetic induction. An electromotive force then arises in the conductor if the integral (cf. LXXIV.) N= \(al + bm + cn)dS changes its value. An induced current arises if the number of lines of f wee enclosed by the con- ductor is changed. If the change in the number of lines of force is an increase, the induced current tends to diminish the number of the enclosed lines of force, for the direction of the induced current is such that its own lines of force are opposite to the lines of force formerly existing. If the direction indicated in the figure by the 200 INDUCTION. [CHAP. x. arrow is taken as positive, the induced electromotive force acts in a negative direction. According to Lenz's law, the current induced by the motion of a circuit tends, by its electrodynamic action, to oppose the motion, by which it is induced. In order to determine the magnitude of the induced electromotive force, we suppose that the current-strength at a given instant is equal to i. If we move the conductor in the magnetic field, we must, by LXXXIL, do the work -i.dN, and, at the same time, the quantity of energy Ei 2 . dt is transformed into heat. "We have at once - i . dN= EP . dt, and therefore, because Ei equals the electro- motive force e, (b) e = - dN/dt, that is, the induced electromotive force is equal to the decrease in unit time of the number of lines of foi'ce enclosed by the circuit. The induced electromotive force depends on the value of the magnetic induction, whose components are a, b, c, not on the magnetic force, whose components are a, /3, y. If there is no magnet near, and if the coefficient of magnetization of the region is & = (cf. LXXVL), the induction and the magnetic force have the same value, and a, b, c may be replaced by a, /?, y. For example, if the circuit ABC at the time t carries a current of strength i, the number N of lines of force passing through the circuit in the positive direction is N=Li. L is the number of lines of force if the current-strength is unity. It is called the coefficient of self- induction. If the current diminishes there arises an electromotive force (c) e = d(Li)/dt. According to Ohm's law we have e = 7iV. if we represent the resistance by E, and therefore (d) Ei = - d(Li)/dt = - L . flijdt, provided that the coefficient of self-induction L is constant. This coefficient depends on the permeability of the region and also on the form of the conductor. If i is the current-strength at the time / = 0, we have (e) i = i . e~ R!L -'. The current-strength therefore diminishes the more rapidly the greater the resistance and the smaller the coefficient of self-induction. We obtain from (c) J"eidt = -Zp.di = Ui 2 . From LXXXIV. (d), the left side of this equation is an expression for the work done, which appears as heat in the conductor. Hence we obtain for the electro-kinetic energy T of a conductor whose self-induction SECT. LXXXV.] INDUCTION. 201 is L and which carries a current of strength i, (f) T=^Li 2 . The electro-kinetic energy of the circuit is therefore equal to half the product of the coefficient of self-induction L and the square of the current-strength i. If ABC and A'B'C' (Fig. 100) are two conductors carrying currents whose respec- tive strengths are ^ and i 2 , the current ^ sets up a number L l i l of lines of force which pass through the conductor ABC. The current i<, also sets up lines of force, and the number of them which pass through ABC may be represented by M 2l i 2 . The total number of lines of force enclosed by ABC is therefore (g) N^L^ + Mrf* The number of lines of force enclosed by A'B'C' is (h) N 2 = From LXXXII. and the discussion at the beginning of this section, we have, in the usual notation, = * COS er ' dS^ n l c 2 )dS 1 = i 2 \ cos e/r . The integral with respect to dS 2 equals i^f^, that with respect to dS 1 equals i 2 M 2l . From LXXXII. we have M 12 = M 2l = | cos e/r . ds^dsy. M 12 = M 2l is the coefficient of mutual induction of the two circuits. If R 1 and R are the resistances of the conductors ABC and A'B'C' respectively, we have Rfr = 6l = - dNJdt = - L l . dijdt - M 2l . di 2 /dt, R t ,i 2 = 2 = - dN 2 /dt = - L. 2 . dijdt - M 12 . dijdt. Hence we have, for the electro-kinetic energy T of a system of two con- ductors which carry the currents z t and i 2 , T = J^i, + e 2 i 2 )dt = SLj* + M l2 ij 2 + \L^. For the electro-kinetic energy T of any system of conductors, we find in the same way, (i) T=i(L l i l 2 + L 2 i 2 ^ + L s i 3 2 + ... +2M, 2 i l i 2 + 2M l3 i l i 3 + 2M 23 i 2 i,+ ...). If N v N 2 , N 3 ... denote the number of lines of force enclosed respectively by the conductors 1, 2, 3... so that, for example, N 1 = L l i 1 + M 2l i 2 + M 3l i 3 + ..., the expression for the electro-kinetic energy T becomes (k) r=(JV 1 *' 1 + A~ 2 ' 2 + JV 3 3 + ...) = pM, that is, the 202 INDUCTION. [CHAP. x. electro-kinetic energy of a system of currents is equal to the sum of the pro- , ducts of the number of lines of force enclosed by each conductor' and the strength of the current present in the conductor. Electrical currents arise not only if the neighbouring currents change in strength, but also if they change their position, so that 7l/ 12 , M 13 . . . vary, and also if the conductor itself changes its form. In all cases the induced current is determined by the change in the number of lines of force enclosed by the conductor. SECTION LXXXVI. COEFFICIENTS OF INDUCTION. In the investigation of variable electrical currents flowing in wire coils, the coefficients of induction between different turns in any one coil and between separate coils are of great importance. The calculation of these co- efficients is in most cases very difficult ; we will consider only one simple case. We suppose two circular conductors whose radii are r : and r. 2 (Fig. 101). They have a common axis, and are separated by the distance b. Suppose r 2 >r 1 . We have to calculate the integral FIG. 101. . cos e. We first evaluate the integral m, m = ^dsjr . cos e. We have r- = b 2 + r 2 2 - 2r 1 ?' 2 . cos c + 1\ 2 . If p is the shortest and q the longest distance between points of the two conductors, we have and m = 2 I r a . cos e . dtj*ffP + (q* - p' z ) sm 2 . If a is a small angle, so chosen that qa is very great in comparison with p, we can set m = 2 . deJP + .J/i + 2 f. *, . df . (1 - 2 sin 2 |e)/? sin it. -a f/2 sin e - f sin ^ . efeT . [Iog(2r 1 a/p) - log(a/4) - 2], . [log(8r 1 //>) - 2] = 2[log(8r 1 / J9 ) - 2]. SECT. LXXXVI.] COEFFICIENTS OF INDUCTION. 203 With this value for m, we obtain M l . 2 = 47rr 2 (\og(8r l /p) - 2). On the assumptions which have been made we may set r l = r. 2 = R, so that (a) M n = ^E(\og(SE/p)-2). The coefficient of induction between two coils, the number of whose turns is n^ and ?i.> respectively, and for which the mean value of logp is expressed by logP, is given by (b) ^ 12 = 4r Jl%7 r J ??(lo g (8^/P)-2). On the same assumptions the coefficient of self-induction L of a single coil, if n is the number of turns, is given by (c) L = 4irn 2 E(\og(8B/P) - 2). We will not go further into the calculation of coefficients of induction ; in most cases they are determined experimentally by one of the following methods : Methods of Determining the Coefficients of Induction. (a) If the coefficient of mutual induction M of two coils Z x and L. 2 is known, we may determine in the following way the coefficient of mutual induction M' for two other coils Z/ and L. 2 '. Let Zj and L. 2 (Fig. 102) be the coils whose coefficient is known, and L and L. 2 those which are to be investigated. A current is FIG. 102. passed through the coils Z/ and L. 2 from the voltaic cell E. The coils Z 2 and L. 2 are joined by conductors, and conductors are joined from the points a and b to the galvanometer G. If the current / which passes through L^ and Z/ is suddenly broken, electromotive 204 INDUCTION. [CHAP. x. forces e and e' arise in L. 2 and L. 2 '. If / 2 and J. 2 ' are the strengths of the currents induced in L. 2 and L. 2 \ we have e = - d(L^J 2 + MJ)/dt, e = - d(L. 2 'J. 2 + M'J)/dt, where L. 2 and Z- 2 ' represent the coefficients of self-induction. Applying KirchhofTs laws to the circuit L. 2 G and L 2 G, it follows that - d(L. 2 / 2 + M J)/dt = R 2 J 2 + G(J 2 - J 2 '), - d(L 2 J 2 + M'J)/dt = R 2 'J. 2 r - G(J 2 - J 2 '), if the resistance of the galvanometer is designated by G, and the resistances of the coils L 2 and L 2 by E. 2 and R 2 respectively. We multiply these equations by dt and integrate from t = to t = T, where T is a very small time-interval. If the current / is broken at the instant t = 0, a current is induced in the circuit L. y 'GL 2 which, in the time T, sets in motion in the circuit L. 2 the quantity of electricity C 2 , in L 2 the quantity C 2 , and therefore in the galvanometer the quantity C 2 -C 2 . At the time t = Q, J=J and J. 2 = J. 2 ' = C 2 = C 2 ' = 0, and at the time t = r, J=0 and the induced current has also vanished, so that J 2 = / 2 ' = 0. Hence we have MJ= E 2 C, + G(C 2 - G,'), M'J= R 2 'C 2 - G(C 2 - C 2 ). C 2 - C 2 = J(M/E 2 - M'/R 2 )/(1 + G/R 2 + G/R 2 ). C 2 - C 2 is the induced quantity of electricity which flows through the galvanometer, that is, the total current. The galvanometer shows no deflection if the resistances satisfy the equation M'/M=R 2 '/R 2 . (b) The comparison between two coefficients of self-induction can be carried out in the following way : Let A BCD (Fig. 103) be a FIG. 103. Wheatstone's bridge- with a galvanometer inserted in the arm BD. L^ and L 2 are two coils inserted in the arms AB and BC, whose coefficients of self-induction are to be compared. The current entering at A and passing out at C distributes itself in the conductors, and causes a deflection of the galvanometer needle. Let the resistances SECT. LXXXVI.] COEFFICIENTS OF INDUCTION. 205 &L, R y Sj_ and S 2 in AB, EG, AD and DC respectively, be so adjusted that no current passes through the galvanometer ; we then have When the circuit is broken at E, an electromotive force arises by induction in : and L 2 , in consequence of which a current of strength g flows through the galvanometer. Let i lt ' y v y. 2 respec- tively be the current-strengths in the conductors AB, BC, AD, DC. They are connected by the relations i l = y v 2 = 7o- We then have - d(L&)ldt = (R l + Sfo + Gg, - d(L. 2 i 2 )/dt = (E 2 + S 2 )i 2 - Gg. At the instant = 0, when the circuit is broken at E, the same current i was in AB and BC ; whence LJQ = (R l + SJ f\ .dt+GTg.dt; L,i = (R z + S 2 ) l\ .dl-oTg. dt, Jo Jo Jo >>o where r denotes a very short time. No deflection is caused in the galvanometer by the current g if I g.dt = Q. Since ^ = i. 2 + g we have i : = i 2 , if no current flows through the galvanometer, and hence in that case LJL 2 = (E l + changes, during the rotation of the coil, from + i~ to ^TT, and since i is zero both at the beginning and at the end of the motion, we have (b) 2SH= RQ, where Q denotes the total quantity of electricity which flows through the conductor. If Q is measured by the method described in LXXXIIL, we have The absence of H from this expression shows that it is not necessary to know the intensity of the earth's magnetism in order to determine the resistance. Sir William Thomson's (Lord Kelvin's) Method. If the coil above described turns with a constant angular velocity w, we have by (a) - SHw cos < - L . di/dt = Ri. The integral of this equa- tion is i = i . e~ KiL - A . cos(^-a). If the rotation is continued for a considerable time, the exponential term vanishes and need be no longer considered. To determine A and a, we have (c) A=SHt/(Rcosa + Lwsma); tga^Lu/R, and therefore A = SHu/Rjl + LW/R* = SHu/R . cos a. It thus appears that the self-induction appar- ently increases the resistance. If ON (Fig. 104) is the magnetic meridian and if the line OM is perpendicular to it, the coil acts on a magnetic needle at its centre with the force Gi, whose direction is that of the line OP perpendicular to the plane of the coil. The components of this FIG. 104. force are OM= a = Gi . cos < and SECT. LXXXVII.] MEASUREMENT OF RESISTANCE. 207 Let a a and b l denote the mean values of these forces. We then have a, = 1 /2ir . / Gi . cos . d. d. Now * Jo I cos ( a) . cos < . d(f> = TT . cos a, / cos ( a) . sin . d = TT . sin a, and hence fl x = - \GA . cos a, 1^=- \GA.sm a. The magnet at the centre of the coil turns from the meridian in the same sense as that in which the coil rotates. If its angular displacement is represented by B, we have tg 6 = - !/(# + bj) = GA cos a/(2H- GA sin a), or, introducing the value of A, tg 6 = GS< cos 2 a/(27i - GSw sin a cos a). This equation, in connection with (c), serves to determine the resistance R. If a is very small we have B=GSot/'2tgQ. L. Lorenz's Method. Suppose that a metallic disk ABC (Fig. 105), whose radius is a, turns with constant velocity about an axis passing perpendicularly through its centre. Around the rim of the disk, and concentric with FIG. 105. it, let there be placed a coil EF, through which flows an electrical current of strength i, arising from the voltaic battery H. This current sets up a magnetic force, whose component perpendicular to the plane of the disk may be set equal to mi, where m is a function of the distance from the centre of the disk 0. If the disk turns from B to A, and if the current flows in the same direction, the electromotive force induced in the disk is directed from the centre to the periphery. A spring is placed at the point B, and is connected 208 INDUCTION. [CHAP. x. by the conductor BGDEO with a rod which touches the disk at its centre. DE is the conductor whose resistance E is to be determined. If the current from the battery also flows through the conductor DE, we may so adjust the angular velocity of the disk that no current flows through the conductor which connects E with the disk. When this condition is attained, the galvanometer needle shows no deflection. If the electromotive force induced in the disk is represented by e, we then have e = Ei, where i denotes the current flowing through the resistance. To determine e, we consider the disk replaced by a ring BAC and a straight conductor OA. The circuit OAGDEO is divided into the two circuits OAB and BGDEOB. The number of lines of force passing through the latter circuit is not changed during the motion. On the other hand, the number passing through the circuit OAB increases in unit time by / mino . dr = iw I mr . dr. Jo JQ This change in the number of lines of force gives the induced electromotive force. We therefore have ,-a -a Ei = i

    = 'lirn and n . 2irr . dr. The integral gives the coefficient of mutual induction M between the coil EF and the disk ABC, or the number of lines of force which pass through the disk if a unit current is flowing in the coil. Hence we have E = nM. Therefore to measure the resistance we determine the number of revolutions per second which must be given to the plate in order that no current shall flow through G. SECTION LXXXVIII. FUNDAMENTAL EQUATIONS OF INDUCTION. We have hitherto determined the electromotive force induced in the conductor s by the change in the number N of lines of force which are enclosed by s. N represents the number of lines of force which pass through an arbitrary surface containing the conductor S ; it is therefore determined by the conductor alone. Hence three quantities, F, G, H, can be so determined that the line integral l(F . dx/ds + G . dy/ds + H. dz/ds)ds SECT. LXX xvm.] EQUATIONS OF INDUCTION. 209 is equal to the surface integral N=\\(al + bm + cn}dS. It is necessary for this, by VI. (f), that (a) a = a#/3y-3fiya?, b = 'dF/'dz--dH/'dx, c We may also obtain these equations of condition by the assumption that, for example, dS is equal to the surface- element di/dz, which is represented by OB DC (Fig. 106). The line integral then becomes , dz)dy - Hdz = (dH/dt/-'dG!'dz)dydz. FIG. 106. Since the surface integral in this case is a . dydz, we obtain the first of equations (a). If these equations are supposed solved for F, G, H, we have (b) N= ^(F. dx/ds + G . dy/ds + H. dz/ds)ds. If the part of the region considered is at rest, the induced electro- motive force e is determined by (c) e = - dNjdt = - \(dF/dt . dx/ds + dG/dt . dy/ds + dH/dt . dz/ds)ds. If we set (d) P= -dF/dt, Q = -dG/dt, R= -dH/dt, we may consider P, Q, E, as the components of the electromotive force (, and if the integration is extended along the whole conductor, we obtain as an expression for the total induced electromotive force (e) e = $(P.dx+Q.dy + R.dz). These components may also be determined directly by variation of the value of the magnetic induction, whose components are a, b, c. We thus obtain from (a) and (d) - da/dt = ?>R[dy - VQfdz, (f) - db/dt Suppose that the electrical current, as has been remarked in LXXXI., is made up of two parts, namely, the current of conduction, which is proportional to the electromotive force, whose components are p, q, r, and the current due to the changes in the electrical polarization, whose components are /, g, h. Then for the components of the total electrical current, we obtain u =p + df/dt, v = q + dg/dt, w = r + dh/dt. o 210 INDUCTION. [CHAP. x. If C represents the conductivity, we have (g) p = CP, q=CQ, r=CR and (h) ' f=kP/4v, g = kQ''ir, h = kR/4Tr, where k is the dielectric constant measured in electromagnetic units. Hence we have (i) u=CP + k/4ir.dP/dt, v=CQ + k/ir.dQ/dt, w=CR + kjlir .dR/dt. From LXXVI. the components of the magnetic induction 33 and the components of the magnetic force ^p are connected by the equations (k) a = p.a, & = ju/3, e = /ry, where p. is the magnetic permeability of the substances. We have already found the following equations connecting the magnetic force and the components of current (cf. LXXX.) : (1) 47Ttt = 3y/3y-30/3s, 4 = 'dafiz - "dy/'dx, 4-n-w = 3/3/3a; - "daffy. SECTION LXXXIX. ELECTRO-KINETIC ENERGY. By LXXXV. (k), the electro-kinetic energy of any system of con- ductors is expressed by T=^Ni. By using LXXXVIII. (b) we obtain T=^\(Fi.dx/ds + Gi .dy/ds + Hi. dz/ds)ds. If u, v, w, are the components of current, a current i, in a conductor whose cross-section is A, may be expressed by i = A. -Ju 2 + v* + w 2 , and we have also i.dx/ds = uA, etc. If we set dxdydz = A.ds, we obtain (b) T= $ J J JCFw + Gv + Hw}dxdydz. If u, v, w are here expressed by the components of the magnetic force [LXXXVIII. (1)], we have T= 1/87T . J Jf |7(3y/3y - 3/3/3) + GCdafdz - Vy/dx) + HCdfi/'dx - 'da/'dy)] . dxdydz. If the separate terms are integrated by parts, and the integration is extended over the whole infinite region, it follows, since at the boundary of this region a, (3, y are infinitely small of the third order, that f \\H . 3a/3?/ . dxdydz = - f J fa . ^Hffy . dxdydz and Jf f G . Vafdz . dxdydz = - J J J a . VGfdz . dxdydz. Analogous expressions hold for the other integrals. By reference to LXXXVIII. we therefore obtain (c) T= I/Sir . f f f (oo + /3b + yc)dxdydz. If no magnets or no bodies which can acquire an appreciable magnetization are present in the region, we have a = a, b = f3, c = y, and (d) T= I/Sir. Hl SECT, xc.] ABSOLUTE UNITS. 211 SECTION XC. ABSOLUTE UNITS. In Physics we generally take the centimetre, (/mm and second as units of length, mass and time respectively. These are the units which are used in the theory of electricity. We will now proceed to express in terms of them the most important electrical and magnetic quantities. They are designated by the symbols L, M, T respectively (cf. Introduction). (a) The Electrostatic System of Units. In electrostatics the force y, with which two quantities of electricity ^ and e. 2 act on each other, is expressed by (cf. LIII.) ^ = 1 2 / r2 where r is the distance between the quantities. If 1 = e 2 = , we have e = r J$, and hence the dimensions of a quantity of electricity e are [e] = [LL*M*T-*] = [tfM*T-*]. The electrical force F, which acts on a unit quantity of electricity, has the dimensions of the quantity e/r 2 , and therefore TJie electrostatic potential (cf. LIV.) has the dimensions of the quantity e/r, and therefore [] = [L^M^T^/L] = [L*M*T~ 1 ]. The capacity C [cf. LV. (g)] has the dimensions of the quantity /", and therefore [] = []. The capacity, therefore, has the dimensions of a length. The surface-density cr (cf. LIV.) is the quantity of electricity present on the unit of surface, and therefore [F m = 47r)/ V*F. = K! V 2 . Hence in the electromagnetic system the equations connecting the components of dielectric polarization with the electrical force are We thus obtain ground for the assumption that V\;>' ; " i* "" G i A fc b, G 1 2 FIG. 109. In the present case the integral which is to be evaluated becomes / | ds.ds'/r, because (Fig. 109) cose = l. In order to find first the -'o ^o value of the integral P= I ds/r we draw FG perpendicular to CD and AB, and write FG = a. If AG = b v BG = b 9 FA = r 1 , FB = r. 2 , we have P = \ ds/r + I ds/r, where s is the distance of any point on AB from G. Now since \dsfr = Ids/Jat + s* = log nat(s/a + Jl+s z /a z ) we obtain P = \ogna,t[(r l + b l )(r 2 + b 2 )/a 2 '] = log nsA[(AF + AG)(BF+ BG)/a?]. 218 ELECTRICAL OSCILLATIONS. [CHAP. xi. If a is very small in comparison with I, we may write AF=AG=*' and FB = GB = l-s', and have / > =lognat[4s'(^ -s')/ 2 ]. We then calculate the value of the integral \P.ds and obtain P.ds' = 2l. [log nat(2//a) - 1]. We will now calculate the coefficient of self-induction of a wire with circular cross-section. Let AB (Fig. 110) be the cross-section of a cylindrical conductor, of length / and radius R. Let the current-density u be constant throughout the cross-section. The current- strength i is then given by i = r-iru. We will first consider the inductive action due to a filament D whose cross-section is ds; this filament is supposed to act on a line which is parallel to the axis of the cylinder, FIG. 110. an( j p asses through the point c. If DC=a, the inductive action is obtained by the variation of the integral u.dS. 2/(log nat(2//a) -l) = u.dS. (M+Nlog nat a), where, for the sake of conciseness, we use M and N as symbols for the quantities Jl/=2(lognat2i-l) and N=-'2l. We must distinguish between two cases : OD = r may be either greater or less than OC=r v First let r>r r The elements dS may be taken so as to form the surface of a ring whose area is 2irr . dr. From the demonstration of XIII. log a equals the logarithm of half the sum of the greatest and least values which a can take. These values are respectively r + r l and r - r r The mean value required is therefore logr. Hence we have the integral *'2irr . dr . (M+ N\og nat r) = mt{M(Sr - rtf + N[R 2 log nat R - r* log nat r x - $(R 2 - r-f)] }. For that part of the cylinder whose distance from the axis is less than 1\ the mean value of the greatest and least values of a will equal r r Hence the integral for this part is u I Jfcw . dr . (M+ N\og nat r : ) = Tru(Mi\ 2 + Ni\ z . log nat r x ). The sum of both integrals is [ -' SECT, xcn.] CALCULATION OF THE PERIOD. 219 In order to obtain the mean value of this quantity for all the filaments composing the cylinder, we need only find the mean value of r x 2 , since all the other quantities are constant. But since it follows that the mean value sought is iruR 2 (M + .ZV(log.E - J)}. If we introduce into this equation the values M= 2J(log 21-1) and N= - 21 and set iruR 2 = i, we obtain for the quantity, the variation of which gives the self-induction, 2/i(log(2?/^) - f ). We therefore obtain for the quantity L, L= '2l(\og(2l/E) - f). In Hertz's investigation, 1=150, .# = 0,25 and therefore Z=1902, where all lengths are expressed in centimetres. In order to calculate the period of oscillation, we will next determine the capacity of a sphere with a radius of 15 cm., such as Hertz used. If Q is its charge and >F its potential, the capacity C in electrostatic units is C=Q/y. Eepresenting the charge and the potential in electromagnetic units by Q' and " respectively, and using V=3. 10 10 , the velocity of light in vacuo, we have Q=7(^ and = 7^". The capacity c in electromagnetic units is therefore c=Q' l W = C/V 2 . The period of oscillation is then given by T=2irjLC/V. Using the symbol introduced at the end of XCL, we have c = ^c v where Cj is the capacity in electromagnetic units of each of the two large spheres. Hence we must set =15/2, and obtain r=2,5/10 8 seconds. The corresponding wave length in air is 2,5.10- 8 .3.10 10 SECTION XCIII. THE FUNDAMENTAL EQUATIONS FOR ELECTRICAL INSULATORS OR DIELECTRICS. Maxwell shows that it follows, as a consequence of his theoretical views of the nature of electricity, that a change in the electrical polarization of the dielectric can set up electrical oscillations. The results which he obtained are so important that we will consider some of them here. For this purpose we will follow Hertz in sub- stituting in the fundamental equations of LXXXVIII. electrostatic units for the electrical quantities, while the magnetic quantities shall be measured in electromagnetic units. The quantity of electricity which is displaced by the electrical force at a point in the dielectric 220 ELECTRICAL OSCILLATIONS. [CHAP. xi. through a surface-element which stands perpendicular to the direction of the force, is, according to LXV., equal to K!TT multiplied by the magnitude of the force F. Representing the components of the electrical displacement by /, g, h, and the components of the electrical force by X, Y, Z, we have f=KXJ^ g = KY/4ir, li = KZ^. If the component X increases by dx in the time dt, the quantity of electricity df flows through unit area in the direction of the z-axis ; the com- ponent u of the current-strength in the dielectric is eqiial to dfjdt, and we have (a) u = K/ir.'dXI-dt, v=K/4ir.c)YI'dt, w = K/4ir . -dZfdt. The equations LXXX. (a) express the fact that the work done by the magnetic forces in consequence of the movement of a unit pole about the current is equal to the current-strength multiplied by 4?r. If the current-strength is measured in electrostatic units, we have (b) 4irM/F=3y/3y-a)8/as, 4ir/F=3a/3-3y/aB, 4irw/F=3/3/az-'9a/3y, since the electromagnetic unit of quantity of electricity equals V electrostatic units. The electromotive force induced equals - dNjdt, if N represents the number of lines of force enclosed by the circuit. Since the electro- motive force in electromagnetic units equals the electromotive force in electrostatic units multiplied by F, we have from LXXXVIII. (f) and (k) f - >*/ F. da/3/ = ?)Z /-dy - -dY/-dz ; - * (c) - ,*/ F . 3(3^ = -dXfdz - 'dZ/'dx ( - /*/ V. From (a) and (b) we obtain KIV. 'dX/'dt Jf/F. -dYpt = 'da/'dz - If we now set J='dX/'dx + ?>Y/'dy + ~dZfdz, we obtain from (c) and (d) fj.K/F' 2 .d 2 X/'dl 2 = V 2 X-'dJ/ox. If we are dealing with a region in which K is constant, K/4ir . /= ?//3.c + 'bgj'dy + "dhfdz. If there is no electrical distribution in the region, we have, from LXVI. (d), J=0; and hence (e) pKIF*.&XIW=V*X', pK/r*.&Y/df 2 = V*Y; nK/r 2 .VZ/-dP = VZ. These equations, in connection with equations (c) and (d), give (f) f iK/F^.^a/'df = V 2 a- (JiK/F 2 .-d 2 pfdt* = r-p; nKj V*- . 3 2 y/^ 2 = V 2 y. at the same time 3o/daf-f dj9/3y+dy/d*0 from LXXVI., if u is constant. SECT, xciv.] PLANE WAVES IN THE DIELECTRIC. 221 From LXVII. (g), the electrical energy W is expressed by (g) W= l/8ir . J Jf K(X* + Y* + Z*)dxdydz. The electrokinetic energy T, according to LXXXIX. (c), is (h) T=l/8ir. J{jM(a 2 + p 2 + y 2 )dxdydz. SECTION XCIV. PLANE WAVES IN THE DIELECTRIC. We will now investigate the movement of plane waves in a dielectric. Let the plane waves be parallel to the ys-p\&ne. The components of the electrical force are then functions of .r only, and from the equations XCIII. (e), we have At the same time also 'dXj'dx + VYfdy + 'dZ/'dz = 0. Since Y and Z are independent of y and z t we have 'dX/'dx = 0, and since, in this case, the only forces which occur are periodic, X=0. The direction of the electrical force is therefore parallel to the plane of the wave. By a rotation of the coordinate axes we can make the y-axis coincide with the resultant of the components Y and Z. We therefore need to discuss only the equation /JT/F 2 . 3 2 7/9^ = 3 2 F/3a; 2 . The integral of this equation is (a) Y=bsin[2Tr/T . (t - a;/o>)], where T is the period of oscillation and w the velocity of propagation. The differential equation is satisfied if w= F'/Jp-K. For vacuum /*= 1, K= 1 ; hence V is the velocity of propagation of plane electrical oscillations in vacuo. For ordinary transparent bodies, /*=!. The velocity of propagation in such bodies is therefore V/\/K. Maxwell assumed that electrical oscillations are identical with light waves. It has been shown by experiment that o> = V/N, where N represents the index of refraction of the dielectric. The electromagnetic theory of light gives w = Vj^K; hence we have K=N 2 , that is, the specific inductive capacity of a medium is equal to the square of its index of refraction. The fact that this theorem holds for a large number of bodies is a strong confirmation of Maxwell's hypothesis. From this hypothesis almost all of the pro- perties of light can be deduced. According to XCIII. (c) we have, under the above conditions, a = 0, 13 = 0, and (b) /*y = 7b/u> . sin [27T/J . (t - ar/w)] = Nb sin [lirjl . (t - x/u)]. The direction of the magnetic force is therefore parallel to the plane of the wave and perpendicular to the direction of the electrical force. 222 ELECTRICAL OSCILLATIONS. [CHAP. xi. We will supplement this discussion by the following examination of the relation between the electrical and the magnetic forces. Let an electrical force act in the y^-plane, parallel to the axis Oy (Fig. Ill), and suppose it to increase uniformly from & ,7 zero to Y Q in one second. In consequence of this an electrical current v is set up in the same direction, and because 'dY[dt= Y , we have from XCIII. (a), This electrical current will set up magnetic forces, which are parallel to the ^-axis. r IG. 111. ___ .__ We Avill assume that the magnetic force increases uniformly from zero to y . In consequence of this an electromotive force will be induced in the surrounding region. We will assume that the electrical and magnetic actions advance in one second over the distance O.r = w (Fig. 111). The magnetic force decreases uniformly from ;c = to x = u; the same statement holds for the electrical force OD=Y . The electrical current, on the other hand, has the same strength everywhere between and x. This is explained by remarking that the electrical force at a point F, whose distance from x equals l/n. Ox, has acted only during Ijn seconds, and has, during this time interval, increased from to l/n. Y n ; its increase in one second is therefore equal to F . Let a unit pole move in the rectangular path OzBxO (Fig. 111). The magnetic force y acts only in the path Oz and acts in the direction of motion ; hence the work done by the magnetic forces is equal to y . Oz. The quantity of current, measured in electromagnetic units, which the unit pole has encircled, is v/F~. Oz. Ox. From LXXX. we have therefore y . OZ = TTV. Oz . Ox/F~, or because OX = (D, we obtain (d) Fy = 47rz;w. The electromotive force, measured in electromagnetic units, which is induced by the motion about a closed path, is e = - dNjdt, if A 7 " represents the number of lines of force enclosed by the path. We have therefore JV= ^e . dt. The mean value of the electromotive force in the direction Oy is | . F . V, in electromagnetic units. The value of the^ before-mentioned integral, extended over the rectangular path 0>/Cx, is |F F. Oy. The mean value of the magnetic force per- pendicular to the surface OyCz is Jy ; hence the mean value of the magnetic induction is \ . fj.y . We therefore have the equation SECT, xciv.] PLANE WAVES IN THE DIELECTRIC. 223 and hence (e) FT = /ry w, NY Q = p.y () , if the index of refraction N is substituted for F'/w. In this connection it must be noticed that the wave is propagated in the direction of the .r-axis, that on our assump- tion the electrical force acts in the direction of the y-axis, and that then the magnetic force acts in the direction of the -axis. Hence if the right hand is held so as to point in the direction in which the wave is propagated, with the palm turned toward the direction of the electrical force, the thumb will point in the direction of the magnetic force. If we represent the magnetic force by M and the electrical force by F, NF= pM. From (c) and (d) it follows that Fy = -KT w. Substituting in (e) the value of y , we obtain V 1 = /JTu> 2 . Hence the velocity of propagation is w = VjJp.K. From (a) and (b) it follows that the relations (e) between the electrical and magnetic forces hold also in the case of plane waves. In vacuo both forces have the same numerical value. SECTION XCV. THE HERTZIAN OSCILLATIONS. H. Hertz succeeded in producing very rapid oscillations in a straight conductor, which also caused oscillations in the surrounding dielectric. We can form some idea of the nature of these oscillations in the following manner, due to Hertz : Let the middle point of a conductor coincide with the origin of coordinates, and let the oscillations take place along the 2-axis. The magnetic lines of force are then circles, whose centres lie on the -axis. The electrical lines of force have a more complicated form. We start with the differential equation XCIII. (f) for the magnetic forces. For the sake of conciseness we set VjKp- = w. We first investigate an integral of the differential equation (a) l/o> 2 .c> 2 w/3/ 2 = V%, on the hypothesis that u is a function of t and of r = >Jx 2 + y 2 -j- z*. We have then, from XV. (1), V% = 1/r . 3 2 ()/3r 2 , and therefore l/o> 2 .3 2 (rM)/3/ 2 = 9 2 (?-tt)/3r 2 . If we set k=2Tr/T and J = 2jr/7w, where I 7 is a constant, then (b) u = a/r . sin (kt - lr) is a particular integral of the differential equation. The function u, as well as its differential coefficients taken with respect to x, y and z, therefore satisfies the differential equations XCIII. (f) for the components of magnetic force a, /?, y. In the case under consideration y = 0, and hence we have 'da/'dx + 'dft/'dy = 0. As the simplest solution of the differential 224 ELECTRICAL OSCILLATIONS. [CHAP. xi. equation we obtain (c) a = - fPu/'dfdy, /3 = ^ufdfdx, where the differ- entiation with respect to t is introduced for use in the subsequent calculation. From (c) we obtain a = - 'd-uj'dfdr .yjr; (3 = 'd-u/'dfdr . x/r. The resultant magnetic force is therefore M= a*M/a/3r . Jx 2 + f/r = Wupfdr . sin 6, where is the angle between the radius vector from the origin and the s-axis. The force M is perpendicular to the plane which contains the point considered and the .e-axis. If we set kt-lr = , we have M = ka(l. sin r/27r, we have that is, the force is determined by Biot and Savart's law, the oscillations in the conductor acting like a current element. For greater distances the magnetic force is (d) M= 4*-%/rW . sin [2-rr/T . (t - r/o,)] . sin 6. Hence the magnetic waves proceed forward in space with the velocity of light. We will now calculate the electrical forces. From (c) and XCIII. (d) we obtain Since u depends only on r and f, the carrying out of the differentiations , xz/r 2 ; (e) KZI F= 3 2 t*/3r 2 . (r 2 - z 2 )/7- 2 + cto/'dr . (r 2 + z 2 The electrical force E in the direction r is (Xx +Yy + Zz)jr, and hence from (e), KRj V= 2/r . 'duf'dr . cos 6 = - 2a(l cos )]dt = F^r/fywrw. -0 We may set F=o> and /* = !, and obtain for the quantity of energy which passes in one second through a unit area perpendicular to the plane of the wave, the quantity F6 2 /87r. The quantity of heat which a square centimetre receives in one minute from the light of the sun is equal to about three gram-calories. This quantity of heat corresponds to the energy 3 . 4,2 . 10"/60 de- veloped in one second. If we now set F=3. 10 10 , we have 6 = 0,04. Since the unit of electrical force in the electrostatic system equals 300 volts, we obtain for the maximum electrical force of sunlight 12 volts per centimetre. The maximum magnetic force is 0,04, and amounts therefore to a fifth of the horizontal intensity of the earth's magnetism in middle latitudes. The experimental basis for the mathematical treatment of electro- statics was given by Coulomb. Poisson handled a number of problems in electrostatics and gave the general method for their solution. Sir William Thomson (Lord Kelvin) also treated the same problems in part by a new and very ingenious method ; his papers are specially recommended to the student (Reprint of Papers, 2nd Ed., 1884). Faraday (1837) developed new views of electrical polarization or dis- placement. On the foundation of these concepts, Maxwell constructed his development of the theory of electricity (Treatise on Electricity and Magnetism, 1873). Helmholtz treated electrostatics in a different way and solved new problems. His papers may be found in Wiedemann's Annalen. The theory of magnetism advanced parallel with the theory of electrostatics. The same authors, and sometimes even the same works, deal with both subjects. 228 ELECTEICAL OSCILLATIONS. [CHAP. xi. SECT. xcvi. Ampere in his Thforie MatMmatique des Phe"nomenes Electrodynamiques, Paris, 1825, discussed the theory of electrical currents. This work forms the foundation for all the recent development of that theory. The new concepts of the magnetic and inductive actions of electrical currents, developed by Faraday, were given a mathematical form by Maxwell in his work : Treatise on Electricity and Magnetism, 1873. We have followed Maxwell's methods in dealing with these topics. On the other hand Gauss, W. Weber, F. E. Neumann, Kirchhoff, and Lorenz, proceed from Ampere's theory. We are indebted to William Thomson and G. Kirchhoff (Poggendorff's Annalen, 121) for the theory of electrical oscillations in conductors. Maxwell and Lorenz showed that electrical oscillations may also exist in the dielectric. By the investigations of H. Hertz, the theory of electrical oscillations has been given such extension and significance, that no one can predict the consequences to which it may lead. CHAPTER XII. REFRACTION OF LIGHT IN ISOTROPIC AND TRANSPARENT BODIES. SECTION XCVII. INTRODUCTION. As the number of facts discovered by the study of light increases, and as additional relations are found between light and other natural phenomena, it becomes increasingly difficult to construct a theory of light. According to the emission theory, which in its main features may be attributed to Newton, and which was handled mathematically by him, energy is transferred by minute bodies, called light corpuscles, which pass from the luminous to the illuminated body. It was supposed that these light corpuscles carried with them not only their kinetic energy but also another kind of energy, to which the luminous effects were due. In the last century the emission theory was sufficient to explain the phenomena then known. But its develop- ment could not keep pace with the advances of experimental knowledge; this became evident early in this century in connection with the great discoveries in optics which were made by Young, Fresnel and Malus. In opposition to this theory, Fresnel developed his first form of the wave theory, which originated with Huygens ; in this form of the theory, the light waves were supposed to be longitudinal. According to the wave theory, the space between the luminous and illuminated bodies is filled with a material medium. By the action of the particles of this medium on each other, the energy emanating from the luminous body is propagated from particle to particle through this medium to the illuminated body. Hence energy resides in the medium during the transfer of light from one body to the other. The wave theory has many advantages over the emission theory. Notably the phenomena of interference are explained by it in a perfectly natural way. It succeeds also in explaining some of the phenomena of double refraction. But the explanation of the 229 230 REFRACTION OF LIGHT. [CHAP. xn. polarization of light by this theory offered difficulties which could be overcome only by the assumption that the direction of the light vibra- tions are perpendicular to the direction of the rays. Since Fresnel retained the idea that the medium in which the light vibrations are propagated, the ether, is a fluid, he encountered an obstinate resistance to his new form of the wave theory ; Poisson rightly maintained that transverse vibrations can never be propagated in a fluid. Although the wave theory, in its original form, was some- what open to criticism, and in many respects was insufficient, in that, among other matters, it could not explain the dispersion of light, yet it was a decided advance on the emission theory. Since the phenomena of optics cannot be explained on the assumption that light is due to vibrations in an elastic medium, not even when this medium is supposed to be a solid, we must endeavour to explain them in another way. Among recent efforts in this direction, the electromagnetic theory of light, developed by Maxwell, has special advantages. In Maxwell's view, light is also a wave motion, but it consists of periodic electrical currents or dis- placements, which take the place of the etherial vibrations of Fresnel's theory. Maxwell determined on this assumption the velocity of light in vacuo and in transparent bodies, and reached conclusions which agree very well with the facts. Polarization and double refraction can also be readily explained by Maxwell's theory, and it has even been applied successfully to the study of dispersion. Since Fresnel's formulas are of great importance for our subsequent study, we will develop them at the outset. We may here recall briefly the principal laws of light which hold for isotropic and perfectly transparent bodies. The knowledge of these laws is neces- sary for the deduction of Fresnel's formulas, but is not sufficient. I. Light is propagated in any one medium with a velocity which depends on the wave length of the light, but not on its intensity. The velocity of light has different values in different media. II. If a ray of light falls on a plane surface, separating two different media, both refraction and reflection occur at this surface. All three rays that is, the incident, the refracted, and the reflected lie in the same plane, which is perpendicular to the refracting surface. If a represents the angle of incidence, ft the angle of refraction, and y the angle of reflection, we have 7 = 0. and sin a/ sin (3 = N. The index of refraction N is constant for homogeneous light. SECT, xcvn.] INTRODUCTION. 231 III. If co represents the velocity of light in the medium con- taining the reflected ray, and w' its velocity in that containing the refracted ray, we have JV= to/to', and therefore sin a/sin /2 = w/w'. IV. Light can be considered a wave motion in a medium, called the ether. It is a matter of indifference whether we here consider the bodies themselves, or an unknown substance, or perhaps changes in the electrical or magnetic condition of the bodies. We wish only to indicate that the luminous motion may be expressed by one or more terms of the form a cos (2irt/T +<(>), where a is the ampli- tude, T the period of vibration, the phase, and i the variable time. The intensity of light is then expressed by a 2 . V. The motion of the ether is perpendicular to the direction of the ray of light; that is, the vibrations are transverse. Either the motion takes place always in the same direction, in which case the ray is rectilinearly polarised, or two or more simultaneous rectilinear motions may give the ether particles a motion in a curve, which is in general an ellipse. Rays of light of this sort are said to be elliptically polarized. If the path of the ether particle is a circle, the light is circularly polarised. Fresnel's conception of natural light was that its vibrations were also perpendicular to the direction of the ray and rectilinear, but that they changed their directions many times, and on no regular plan, in a very short interval of time. SECTION XCVIII. FRESNEL'S FORMULAS. Suppose the plane surface OP (Fig. 113) to be the surface of separation of two transparent isotropic media. We represent the velocity of light in the medium above the surface of separation OP by w, and that in the medium below the surface by w'. If N represents the index of refraction of the ray of light in its passage from the first to the second medium, we have w = JVw'. We select the point in the bounding plane as the origin of a system of rectangular co ordinates, and draw the z-axis perpendicularly upward and the y-axis in the plane of incidence, that is, the plane passed through the normal to the surface at the point of incidence and the incident ray SO. The 2-axis is therefore perpendicular to the plane of incidence. Further, let SO be the incident, OT the reflected, and OB the refracted ray. We designate the angle of incidence by a, the angle of refraction by ft. The amplitude of the vibrations 232 KEFRACTION OF LIGHT. [CHAP. xn. of the incident ray may be called w a , and that of the vibrations of the refracted and reflected rays u 2 and u 3 respectively. The planes of vibration of these rays make angles with the plane of incidence, which are represented by v < 2 , 3 respectively. The components of motion along the coordinate axes are 1? tj v { v for the incident ray ; 2 , rj y 2 , for the refracted ray ; 3 , ?; 3 , 3 , for the reflected ray. It is further advantageous to introduce symbols for the components of motion which lie in the plane of incidence and are perpendicular to the direction of the rays. These components of motion for the three rays are designated by s v s y s s , respectively. We then obtain the following equations : (a) = -s 3 sma, 77 3 = s 3 cosa, VV + Cii tg'fcj = W s v In order to express s^ s 3 , and 2 , 3 , in terms of s l and ^, we must make certain assumptions with regard to the behaviour of the light at its passage from one medium to another. I. Fresnel assumed first, that no light is lost by reflection and refraction, or, the sum of the intensities of the reflected and refracted light is equal to that of the incident. This law is only a statement of the law of the conservation of energy in a particular case, it being merely the assertion that the kinetic energy of the incident ray is equal to that of the reflected and refracted rays. Let OPSS' (Fig. 113) be a cylinder, the area of whose base OP is A, and whose slant height SO is equal to w, the velocity of light. If we represent the density of the vibrating SECT, xcvm.] FEESNEL'S FOEMULAS. 233 medium by p, the kinetic energy L v of the light contained in the cylinder considered is L^ = J . p . o> cos a . A . u^. After the lapse of a second this kinetic energy is divided between the reflected and refracted rays. The kinetic energy L 3 in the reflected ray is L 3 = \ . p . w cos a . A . u 3 2 , and the kinetic energy L 2 in the refracted ray is L 2 = \p . o> cos ft . A . u 2 2 , when p represents the density of the vibrating medium below the bounding surface. Therefore, on the assumption made by Fresnel, we have L l = L 2 + L 3 or /3w(w 1 2 w 3 2 ) cos a = pV . u 2 2 . cos (3. Taking into account the relations 3 . Brewster found that tg< 3 = tg< : . cos(o, - /i?)/cos (a + (3). This follows from Fresnel's formulas, since tg< 3 = 3 / s s = fi cos ( a ~ P)/ s \ cos ( a + /*) = fc g^i cos ( a " /*)/cos ( a + /*) This agreement argues for the correctness of Fresnel's formulas. Fresnel assumed that the elasticity of the vibrating medium is the same on both sides of the refracting surface. We have seen that this assumption is to some extent arbitrary. On the other hand F. E. Neumann assumed that p = p. We obtain on this latter assumption from (g) and (h) f s 9 = 2 sin a . cos a . Sj/sin(a + /3), (1) J 2 = 2 sin a. cos a. / sin (a + /3) cos (a -J3), [ 53 = sin(a-/3). 5l /sin(a + ^, & = tg (a-/3). ^/tg (a + /3). These equations agree with the results of experiment, on the assumption that the vibrations occur in the plane of polarization, as well as those of Fresnel. We will now consider the components of motion along the normal during the passage of light from one medium to the other. This SECT, xcviu.] FEESNEL'S FORMULAS. 235 component above the bounding surface is x + 3 , that below it is g y It follows from (a) and (g) that j 4- 3 = 2/>'. sin a . cos a . sin ft . s^(p . sin a . cos (3 + p . cos a . sin /3), 2 = 2p . sin a . cos a . sin )6 . s-^Kp . sin a . cos (3 + p. cos a. sin (3). From these equations it follows that i + g s = i 2 - P/P- If we assume with Neumann that p = p, we have x + 3 = 2 i ^ a ^ i g > ^ e components of vibration perpendicular to the bounding surface are equal above and below it. On the other hand we obtain from Fresnel's assumption (m) x + 3 = N 2 . ,. Fresnel's equations agree fully with experiment only when the index of refraction is about 1,5; it is in this case only that s 3 = at the angle of polarization. In other cases S 3 is a minimum, but does not vanish. Several attempts have been made to explain this fact. Thus, for example, Lorenz assumed that the passage of the light from one medium to another occurs through an extremely thin intervening layer of varying density, and that therefore the change of density is not discontinuous. SECTION XCIX. THE ELECTROMAGNETIC THEORY OF LIGHT. In XC1V. it was proved that electrical vibrations are propagated in vacuo and in a great number of insulators with the velocity of light. This fact suggests the assumption that light consists of electrical vibrations. It was also shown that, when the wave is plane, the electrical and magnetic forces lie in the wave front. In the most simple case the electrical force F is perpendicular to the magnetic force M. If the permeability //, of the medium is equal to 1, which is approximately the case for most dielectrics, we have (XCIV.), (a) M=NF, where N is the index of refraction. We will now develop the usual expressions for the reflected and transmitted light, considering first the boundary conditions. Since no free magnetism is present, and since the electrical current strength is everywhere finite, the magnetic force varies continuously during the passage from one medium to another. "\Ve take the refracting surface as the y^-plane, and the z-axis a$ the normal to it. Hence if a, /3, y are the components of the magnetic force on one side of the refracting surface and a', /?', y those on the other side, we have (b) a = a',P = (3',y = y, 236 REFRACTION OF LIGHT. [CHAP. xn. The electrical force arises partly from induction and partly from the free electricity on the refracting surface. Let a- denote the density on this surface, X, Y, Z the components of the electrical force immediately above, and X', V, Z those immediately below this surface. We then have, from LXVL, 4ir 2 . WXfdt* = VIST, 1/w 2 . 3 2 F/9 2 = V-Y, \ l/w2 . wzrdP = V 2 Z, -dXfdx + ^Yj-dy + VZ/Vz = 0. The boundary conditions are obtained by remarking that the com- ponents of the electrical and magnetic forces parallel to the bounding surface are equal on both sides of it. Therefore, if the z-axis is perpendicular to the refracting surface, we have )]. We obtain for the refracted ray, if w' is the velocity of light in the second medium, = ^ 2 .sin/3.cosF 2 , F= F 2 . cos /3 . cos T 2 , Z=Z.cosT (d) l rr , = [27r/r.(^-(-a;cos^ + ysm/3)/o>')]. For the reflected ray we have r =^ 3 .cosr 3 , 1 ^ j = [2-ir/T . (t - (x cosa + y sin a)/o>)]. To simplify the calculation we replace the trigonometrical form (f) cos(^- ( - z cos a + y sin a)/o>) by the expression (g) ^(M-scosa+ysineO/a^ where i = \/- 1 and k=%Tr/T' In the final result we use only the real part of (g), namely (f). Both expressions satisfy the same differential equation, and therefore in calculations one of them may be replaced by the other. If the refraction occurs at a plane surface, we may replace the expressions (c), (d) and (e) by the following : X=F l .$ina. (b) *,< 240 EEFEACTION OF LIGHT. [CHAP. xn. , X=F a .s\n(3.e ki ( t -<--* coa l 3 +y s (i) {X= F 3 . sin a . **{*-(* Y=F 3 .cosa.e ki ( t -<- XC(iS gki(t-(xcosa+ysin 3 * a+y sin These equations express the components of the electrical force for the incident, refracted, and reflected rays. In order to satisfy the conditions (b) and (c) it is necessary that sin a/a> = sin /3/w'. Since the velocities of propagation w and w' are constant, we can set N= sin a/sin (3, where N is the index of refraction. From equations (b) we have (1) (F l + F 3 )cosa = F. 2 .cos(3; Z 1 + Z s = Z y From (c) we obtain (m) (F l -F 3 )siu/3 = F 2 .sina- (Z l - Z 3 )cosa. sin/? = ^ 2 . sin a. cos (3. From (1) and (m) we obtain Fresnel's equations [XCVIII. (k)] for the reflected and refracted waves. The problem is solved when (3 is not imaginary. /3 becomes imaginary when sin /?> 1, and therefore when sin a > N. In this case we must use the complete expressions (i) and (k). If the electrical force is perpendicular to the plane of incidence, the reflected wave is determined by the real part of the expression (n) - Z l . Sin(a - /3)/sin(a + /?) . e i-i(<-(xcosa+ysina)/ W ) > We get this expression by the use of the last of equations XCVIII. (k). But since cos(3 = Jl - siri 2 a/N'\ and therefore (o) Ni . cos /3 = Jsirfa-N 12 we have - sin (a - (3) /sin (a+(3) = (cos a + i*Jsin?a - N 2 )/(cos a - K/sin-a - ,/V 2 ). If we now set (p) cos a = C . cos Jy, J sin 2 a - N' 2 = C . sin |y, we obtain (q) tg | y = N/ sin' 2 a - N 2 /cos a, C?=l-N 2 ; (r) -sin(a-/3)/sin(a + ) = ^. Hence the real part of the expression (n) is (s) Z^ . cos [k(t -(xcosa + y sin a)/w) + y]. In this case the reflection is total, since the component Z l appears in the expression for the incident as well as in that for the reflected SECT, c.] EQUATIONS OF ELECTROMAGNETIC THEORY. 241 wave. But while, in the case of ordinary reflection, no difference of phase arises between the two waves, we have in this case a difference of phase 7, which may be determined from (q). If the electrical forces far the incident wave are parallel to the plane of incidence, we determine the real part of the expression (t) - F l . tg(a - /3)/tg(a + /2) . ett(-Cscosa+3/sina)/w). Using equation (o), we have tg(a - /2)/tg(a + /3) = (TV 2 . cosa If we set (u) JV~ 2 .cosa so that (v) tg 8 = Vsin' 2 a - N 2 /N 2 cos a, we obtain (x) tg(a- /3)/tg(a + p) = e iS . Hence the real part of the expression (t) is (y) -F 1 cos[k(t- (zcosa + ysina)/w) + S]. The reflection is therefore total. To determine the difference of phase 8 between the reflected and incident waves, we may use equation (v). We obtain from (q) and (v) tg |(8 - y ) = V sin 2 a - N 2 / sin a tg a. If a is the limiting angle of total reflection or critical angle, we have N = sin . Hence the real part of (a) is (/3) 2 cos a/(7 . e tov/sini!a -^/w . z i . cos [k(t - y sin o/u) + ly]. Since C 2 = 1 - JV 2 , we obtain 4 cos 2 a/(l - JV 2 ) . Z* . (**#/***>- W\ t for the intensity of the transmitted light, where X denotes the wave length. The expression shows that, in this case also, a motion exists which corresponds to the refracted ray in the case of ordinary reflection; it is, however, appreciable only within a very small distance from the refracting surface. Similar results are obtained in the investigation of the refracted ray if the electrical force of the incident light is parallel to the plane of Q 242 REFRACTION OF LIGHT. [CHAP. xn. Remark: In order to obtain the real part of an expression of the form (n) we may use the following method. The expression (n) is thrown into the form (A + Si) . 0* = (A + Bi)(cosV + i sin ). The real part of this is R--=A . cos^- B. sin^. Now if we set A = C.cosy, B = C.siuy, we have (y) li= C. cos fF + y), where and y are determined by f |tgy= The expression (n) then takes the form Z^ . (cos a + iv/sin 2 a - JV 2 )/(cos a - ijsin' 2 a - N' 2 ). In the case considered, therefore, A+Bi = Z l . (cos a + i\/sin 2 a _ JV 2 )/(cos a - z'v/sin-a - N' 2 ). If + i and i are interchanged, AVC obtain A - Bi = ^(cosa - iv/sin-a N' 2 )/(cos a + iv/sin 2 a- By multiplication of the two expressions we obtain C~ From (8) it further follows that tg y = 2 cos o\/sin 2 a N 2 /(co& 2 a - sin 2 a + N' 2 ) . This equation may be also obtained from (q). SECTION CI. REFRACTION IN A PLATE. We will consider the case of a plane wave of light falling on a plane parallel glass plate, whose thickness is a and whose index of refraction is N. We can determine the intensities of the reflected and transmitted light in the following way. We choose one surface A of the plate as the y^-plane, and draw the positive a-axis outward from this surface. Let a represent the angle of incidence, /5 the angle of refraction, w and a/ the velocities of the light inside and outside the plate. A part of the refracted ray is reflected toward the surface A at a point E of the surface B. This part is again divided at the surface A, a part of it passing through that surface in the direction FG, while the other part is again reflected toward B. The light is thus reflected within the plate repeatedly. Since the plate is bounded on both sides by the same medium, the angle of exit is equal to the angle of incidence a. Now plane waves which move in the same direction may be compounded into a single plane wave. SECT. CI.] REFRACTION IN A PLATE. 243 Besides the incident wave we have to consider four others, namely, the wave reflected from A, the wave passing through B, and two waves in the plate itself. I. We will first consider the case in which the electrical force of the, incident wave is perpendicular to the plane of incidence. The component of the electrical force outside A is expressed as in the former para- graph by FIG. 117. Similar expressions hold for the component Z' of the electrical force in the plate, which are obtained by replacing a by /?, u> by to', and introducing the new constants Z^ and Z. Thus we obtain Z' = Z 2 . gW(-(-*M|8+y8iij8)/w') + Z^ . gKft-fccos/S+if sinjSyW). We have for the component Z" of the transmitted ray Z" = Z- 3 . e l ' l ( l - ( - x cos -+y sin a >/ w ). The boundary condition Z=Z' when x = Q, gives (a) Z l + Z 3 = Z 2 + Z 4 . Similarly Z' = Z" when x= -a, or Z 2 .e~ kia cos jS/w + Z . e kia cos /3/ w/ = Z 6 .e~ kia cos a / w . Now if we set ka . cos /5/w' = ?/, ka . cos a/w = v, we can write the last condition in the form (b) + Z . e ui = Z 5 . e~ vi . We have, further, when x = 0, ?>Z'/'dx = 'dZ/'dx, and similarly, when x = -a, 'dZ'/'dx = 'dZ" ' fdx. These equations of condition give (c) (Z l - Z 3 ) cos a . sin /? = (Z 2 - Z 4 ) sin a . cos /3 and (d) (Z 2 . e~ ui - Z . e ui ) sin a . cos ft = Z 5 . e'". cos a . sin /?. It follows from (b) and (d) that ZJZ. 2 = e- 2i ". sin (a - /3)/sin(a + /5), 244 EEFEACTION OF LIGHT. [CHAP. xn. or if sin(a-/3)/sin(a + /3) = , ZJZ^ = t . e~* ui . From (a) and (c) we have also Z 3 /Z l = (- eZ 2 + Z 4 )/(Z 2 - fZj, and therefore (e) Z 3 /Z, = - (* - O/(V * ui - e O- If N is greater than 1, u is always real, and we may set ZJZi = - 2w . sin /[(! - 2 )cos M + (1 + e 2 )f . sin ]. Designating the intensity of the reflected light by C 2 , we obtain, by the method indicated at the end of C., C* = Z l *. 4e 2 . sin 2 /[(l - e 2 ) 2 + 4e 2 . sin 2 tt]. But because k = 2-!r/T = -2ino/\ and u = 2-nr . N. cos /3 . a/A, it follows that / f) (C Z = Z*. 4e 2 . sin 2 (27rJV. C os /3 . a/A)/[(l - 2 ) 2 \ + 4 2 . sin 2 (27rA r . cos . a/A)]. Hence no light is reflected if 2-n-N . cos /3 . a/A =pr, where p is a whole number. This result is of special importance in the study of Newton's rings. On the other hand, if N<1 and at the same time sma>N, [3 will be imaginary. In this case we can no longer use equation (f). We then have [C. (o)] Ni . cos (3 = >/sin 2 a - N' 2 , and hence ui = 2ira/X. . *Jsiri 2 a-N 2 . If we set m = ui and e= -e+ { 7 [C. (r)], equation (e) takes the form Z 3 /Z, - (e-- e-)/(e>"-v - -*+*); Designating by C 2 the intensity of the reflected light, we obtain in the same wajr as before, (g) C 2 = Z*. 1/[1 + 4 sin 2 y/(e m - e-) 2 ], where tg |y = >/sin 2 a - 7V 2 /cos a and m = 27ra/A. . Vsin 2 a - JV 2 . The relations which we have here considered occur in the case of two transparent bodies which are separated by a layer of air. If the thickness of the layer of air is very much greater than the wave length of the light, total reflection will occur. This is in accord with equation (g), which in this case gives C 2 = Z l 2 . On the other hand, if a is small in comparison with the wave length, all the light passes through the layer of air. In consequence of this a black spot is seen if the hypotenuse of a right-angled glass prism is placed on the surface of a convex lens of long focus. If the angle of incidence a in the glass prism is less than the critical angle, a dark spot appears surrounded by coloured rings ; but if the angle of incidence is greater than the critical angle, the rings disappear while the spot remains. SECT, ci.] KEFRACTION IN A PLATE. 245 The spot is larger for red than for blue light. This result is contained also in the expression for the intensity of the reflected light. The transmitted light is complementary to the reflected light. II. If the direction of the electrical force of the incident light is parallel to the plane of incidence, the disturbance outside the surface A is determined by X=F l . Sin a . e i-i(-(-*>sa+y 8 ina)/ W ) - F s . Sin a . gK(t-(*coea+y sinetf/w), Y= F l . COS a . e ki (t - ( - * cos a+y sin o)/w) 4 7^ 3 . cos a . e ki ( l ~ (x cos a +y sln a )M. The disturbance inside the plate is given by X' = F Z . Sin ft . (-(-* cos p+y sin 0)/w) - F 4 . Sin ft . 6('-(* cos/3+y sin / 3)/W) ) F = 1*2 . COS . **( -<-* cs/3+y sin/3)/u>') + /^ . COS /3 . (*-(* cosp+y sin )/&>') . and outside the surface .5 by X" = F K . Sin a . gti{-(-arcoso+y sinoyw^ Y" = F- . COS a . ^('-(-^cosa+J/sinaVw). We must now determine the constants ^ J^, jF 4 , F b . When x = the boundary conditions give F= F, or (h) (^ + .F 3 ) cos a = (F 2 + F 4 ) cos /8. Similarly, when z= -a, ,F 2 . cos ft . e~ kia co80/w + ^ 4 . cos ft . e kia s pl"' = F 5 .cosa. e~ kia cos a / w . Using the same notation as before, we have (i) F 2 . cos ft . e~ ui + F . cos ft . e ui = f 5 . cos a . e-"'. We have, further, when # = 0, or (k) (^ - F 3 ) sin ft = (F 2 - Fj sin a. The same condition holds when x = - a, or (1) f^ . sin a . e-" 1 - .P 4 . sin a . e'" = ^ . sin ft . r. We obtain from equations (i) and (1) FJF 2 = e~- ui . tg(o-j8)/tg(a + )8). But if we set tg(a-/3)/tg(a + /3) = e', we will have FJF 2 = ^.e~ M . It follows from (h) and (k) that ^/^ = ( - e' + e' . e- a ")/(l - '2. e - 2 '), or (m) ^3^= -(e !(i -e- ui )/( 1 /'"* -e'.e~ Mi )- We thus obtain the intensity Z> 2 of the reflected light in the same way as we obtained the expression (f) from (e), .. 1 ' ( I - e' 2 ) 2 + 4e a . sin a (2^V cos /3a/ A)' 246 REFRACTION OF LIGHT. [CHAP. xn. If sin a > N and if (3 is therefore imaginary, we obtain the in- tensity of the reflected light in the following way : We have = tg(a- /?)/tg(a + f3) = e 8i , if, as in C. (u), we set JV 2 . cos a = Z) . cos JS, >/sin 2 a -N' 2 = D. sin |3. If we further set ui = m, it follows that The intensity D 2 of the reflected light is then (o) Z> 2 = FS . l/( 1 + 4 sin 2 8/( w - e~ m f-), in which expression tg |<5 = x/sin 2 u - N' 2 /N 2 cos a, m = '2-n-a/X . The expressions (n) and (o) for the intensity of the reflected light when the direction of the electrical force is parallel to the plane of incidence, lead to essentially the same results as equations (f) and (g), which hold when the direction of the electrical force is perpen- dicular to the plane of incidence. We only remark that, from equation (n), D- vanishes if e' = or (a + /2) = |TT. In this case the angle of incidence is equal to the angle of polarization. SECTION CII. DOUBLE REFRACTION. Up to this point we have assumed that the value of the dielectric constant K is independent of the direction of the electrical force. Boltzmann, however, has shown that the dielectric constant of crystals has different values in different directions, and depends on the direction of the electrical force. Let K v K 2 , JT 3 represent the value of the dielectric constant in three perpendicular directions which are those of the coordinates a;, y, z. Then in place of equations XCIII. (a), we use (a) u = K l /4v.'dXI'dt, v = K 2 /4ir .'dY/dt, w = K 3 /4ir . 'dZ/'di. Equations XCIII. (d) and (c) become KJ V. 'dX/ot = 3y/3y - 3/2/3,2, A' 2 / V. 3 Yfdt = 3a/3^ - K 3 / V. 'dZj'dt = 3/3* - 3a/3y, and if we set the magnetic permeability /* = 1, we have - I/ V. 3a/3* - *dZfdy - 'dY/'dz, -\\V. 3/?/3* = 3JST/32 - -\\V. 3y/3* = 3F/3.C - SECT, cn.] DOUBLE REFRACTION. 247 Further, if we set we obtain f I/a 2 . - (b) J 1/42. 327/9*2 We will consider a plane wave moving through a body to which these equations apply. Its direction of propagation is determined by the angle whose cosines are I, m, n; the direction of the electrical force / is determined by the angle whose cosines are A, /*, v. We then have (c) X=\f, Y=rf, Z=vf, f=F.cos[2ir/T.(t-(lx + my + nz)/u>)']. F is constant, and the velocity of propagation w depends only on the direction in which the wave is propagated. It follows from (c) that V 2 A"= - 47T 2 A//T 2 7r/ro) . F . cos 8 . sin [2ir/r . (t - (Ix + my + )/)]. We obtain from the first of equations (b), (d') A - I. cos 8 w 2 . A/a 2 . This equation and the two similar to it take the forms (e) (a 2 - 2 - w 2 ) + c% 2 /(c 2 - w 2 ) = 1. For brevity we will write for this equation 2a 2 2 /(a 2 - w 2 ) = 1 . But because a 2 = a 2 - w 2 + w 2 , we also have - w 2 ) = 2/ 2 + 2a> 2 / 2 /(a 2 - a> 2 ) = 1. 248 REFRACTION OF LIGHT. [CHAP. xn. Since 2Z 2 = 1, it follows that (f ) I 2 /(a 2 (o 2 ) + m 2 /(i 2 - o> 2 ) + % 2 /(c 2 - or) = 0. This equation is of the fourth degree in w. Since two of its roots are numerically equal to the other two but of opposite sign, the electrical wave has two velocities of propagation, u^ and 0)3. We may give equation (f) the form (g) <*-(l 2 (b 2 + c 2 ) + m 2 (a 2 + c 2 ) + n 2 (a 2 + b 2 ))u 2 + I 2 b 2 c 2 + mW + n 2 a 2 b 2 = 0. If 1 = 0, that is, if the plane wave is parallel to the z-axis, we have The roots of this equation are w 1 = a, o> 2 = *Jm 2 c 2 + n 2 b 2 . This result can be represented by drawing lines in the yz-plane from the point (Fig. 118), which are proportional to the velocities of propaga- tion. The ends of these lines then lie on two curves, one of which is given by w 1 = a, and is a circle; the other is given by w 2 , and is an oval. If a > b > c, the minor semi-axis c of the curve given by o> 2 lies in the y-axis, and its major semi- axis b in the -axis. The relations of the " plane waves which are parallel to the y- and -axes respectively, are given in Figs. 119 and 120. The relation in the axz-plane is especially peculiar (Fig. 119). In that case, we have, for m = 0, 2 = v 7 /-c- + 7i 2 a 2 c FIG. 118. O tax. FIG. 119. and l 2 + n?=l. The direction of propagation in which the two velocities Wj and o> 2 are equal is given by /= J(a 2 -b 2 )/(a 2 -c 2 ) ; n =-- J(b 2 - c' 2 )/(a 2 - c 2 ). SECT, cm.] VELOCITIES OF PROPAGATION. 249 SECTION CIII. DISCUSSION OF THE VELOCITIES or PROPAGATION. If Wj and o> 2 represent the two velocities of propagation of the same plane wave, we have [CIL (g)] (a) 2 { Wl 2 . w 2 2 = We 2 + m 2 a?c 2 + n 2 a 2 b 2 , and ( Wl 2 - o> 2 2 ) 2 = (I 2 (b 2 + c 2 ) + m 2 (a 2 + c 2 ) + n 2 (a 2 + b 2 )) 2 - 4(/W + m 2 a 2 c 2 + n-a 2 b 2 ). If we multiply the last term on the right side of this equation by I 2 + m 2 + ri 2 = 1 we obtain 1 - c 2 ) 2 + m 4 (c 2 - a 2 ) 2 + 7i 4 (a 2 - b 2 ) 2 + 2m 2 n 2 (a 2 - b 2 )(a 2 - c 2 ) ' - c 2 )(b 2 - a 2 ) + 2l 2 m 2 (c 2 - a 2 )(c 2 - b 2 ). If a > b > c, it follows that (M^ - o> 2 2 ) 2 = l*(b 2 - c 2 ) 2 + m*(a 2 - c 2 ) 2 + n\a 2 - b 2 ) 2 + 2m 2 n 2 (a 2 - b 2 )(a 2 - c 2 ) - 2l 2 n 2 (b 2 - c 2 )(a 2 - b 2 ) + 2l 2 m 2 (a 2 - c 2 )(b 2 - c 2 ), or ( Wl 2 - o> 2 2 ) 2 = (I 2 (b 2 - c 2 ) + m 2 (a 2 - c 2 ) + n 2 (a 2 - b 2 )) 2 (b) ^ -^v-^xs 2 -^ 2 ), Hence the two velocities ^ and w 2 are equal for certain directions of the wave normals. This equality exists when m = and either Ijb 2 - c 2 + nja?^b' 2 = 0, or ijb 2 ^ 2 - W 2 - 6 s = 0. These conditions are satisfied by m = Q and Z/= J(a' 2 -b-)/(b 2 - c 2 ). These equations represent four directions, which are parallel to the rc^-plane and perpendicular to the axis of mean elasticity b. If we represent the cosines of the angles made by these directions with the coordinate axes by 1 , m , w , we have (c) m = 0, I =j(a 2 -b 2 )/(a 2 -c 2 ), n = J(b 2 - c 2 )/(a 2 -I 2 ). We call the directions in the crystal, defined by equations (c), the optic axes. There are two such axes, since each of these equations represents two opposite directions. If Oa and Oc (Fig. 121) represent the axes of greatest and least elasticity a and c, and if OA l is one of the directions in which w x and w 2 are equal, they are equal not only in the opposite direction OB but also in the directions OA 2 and OB 2 , if OA 2 makes the same angle with Oa as that made by OA r 250 REFRACTION OF LIGHT. We will now express the velocity of propagation in any direction in terms of the angles made by this direction with the optic axes OA l and OA 2 . The cosines of the angles which the direction of propagation of the plane wave makes with the axes are Z, m, ?<. We then have f cos E l = / . V(a 2 - 6*)/(tt 2 - c 2 ) + n . V(& 2 - c 2 )/(a 2 - c 2 ), (c') From this it follows that (21 = (cos E l + cos E z ) . >/(a 2 - c 2 )/(a 2 - 1 2 = (cos E l - cos E 2 ) . J(tf^ and 2 - c 2 ). FIG. 121. If we eliminate m from equation (a) by the help of the equation P + m 2 + n' 2 =l,w6 obtain o^ 2 + o> 2 2 = 2 + c- - l' 2 (a 2 - b-) + n-(b- - c 2 ), from which we obtain by use of equation (d), (e) 2 2 = a 2 + c 2 - (a 2 - c 2 ) . cos E 1 . cos ^ 2 . From equations (c') we obtain (a 2 - c 2 ) . sin 2 ^ = a 2 - c 2 - Z 2 (a 2 - ft 2 ) - w 2 (6 2 - c 2 ) - (a 2 - c 2 ) . sin 2 2 = a 2 - c 2 - / 2 (a 2 - 6 2 ) - n 2 (& 2 - c 2 ) + 2/n . Further, since Z 2 + m 2 + ri 2 = 1 , we also have Z 2 (6 2 - c 2 ) + m 2 (a 2 - c 2 ) + n 2 (rt 2 - i 2 ) = a 2 - c 2 - ^(a 2 - ft 2 ) - rc 2 (6 2 - c 2 ). By help of these relations we obtain from the first of equations (b) (f ) w/ 2 - w 2 2 = (a- - c 2 ) . sin E l . sin E 2 , and from (e) and (f ), / 2 Wl 2 = a 2 + c 2 - (a 2 - c 2 ) . cos(^ 1 - E 2 ), \ 2o> 2 2 = a 2 + c 2 - (a 2 - c 2 ) . cos (E^ + E. 2 ). The greatest value of the velocity of propagation is a and the least c. This follows if we set E l = E 2 and E l + E 2 = 7r. If the normal to the waves is parallel with one of the optical axes, for example OA lt we have E l = and cos \E. 2 = / , and hence Wl = o> 2 = b. The velocity of propagation is then equal to the axis of mean elasticity. SECT, civ.] THE WAVE SURFACE. 251 SECTION CIV. THE WAVE SURFACE. Suppose a plane wave to start from the origin of the system of coordinates, in the direction in which its normal makes angles with the axes whose cosines are I, m, n. After the lapse of a unit of time, the distance of the wave from the origin is co. If about each of the points of the plane wave we construct a wave surface as it would appear after the lapse of unit time, the plane wave thus propagated is the envelope of all the wave surfaces. If x, y, z are the coordinates of a point of the plane wave in its new position, we have (a) Ix + my + nz = (o, and co is determined by (b) Z 2 /(a 2 - co 2 ) + m 2 /(b 2 - co 2 ) + n 2 (c 2 - co 2 ) = 0. AVe have, further, (c) l 2 + m 2 + n 2 =l. If I, m, n, and co vary, the planes (a) envelope a surface, which is called the wave surface. Hence if we consider all possible plane waves passed through a point, and if we determine the position of the same waves after unit time, the wave surface is the envelope of all the plane waves thus determined. We will now investigate the equation of this wave surface. AVe obtain from (a), (c), and (b), (d) x . dl + y . dm + z . dn = rfto, (e) I . dl + m . dm + n . dn = 0, (f ) I . dl/(a 2 - co 2 ) + m . dm/(b 2 - co 2 ) + w . dn/(c 2 - to 2 ) + Fu . da> = 0, where (g) F= Pj(a z - co 2 ) 2 + m 2 /(& 2 - to 2 ) 2 + 7i 2 /(c 2 - co 2 ) 2 . If we eliminate e?co by means of (d) from equation (f), we have [Ftax + //(a 2 - a> 2 )]d/ + [F(mo) - y), n/(c 2 - to 2 ) = Fwtyw - z). If we square both sides of these equations, add them, and use equation (g), we have 1 = /V(co 2 - 2co(c + my + nz) + r 2 ), in which r- = : 252 REFRACTION OF LIGHT. [CHAP. xn. Further, by reference to (a) we obtain (k) F<^(r 2 - w 2 ) = l. F may now be eliminated from equations (i) by means of k, and we have a 1 - o> 2 ) = Z(Z> 2 - r 2 ), These equations enable us to determine the point of contact between the wave surface and the plane wave, and therefore the direction of propagation of the ray. The plane wave moves in the direction determined by Z, m, n. If we multiply both sides of equations (1) by x, y, z respectively, and add, we have z 2 (a 2 - w 2 )/(a 2 - r 2 ) + y z (b* - o> 2 )/(6 2 - r 2 ) + z 2 (c 2 - w 2 )/(c 2 - r 2 ) = o> 2 , since by (a) Ix + my + nz = w. This equation may be written in the abbreviated form 2z 2 (a 2 - w 2 )/(a 2 - r 2 ) = w 2 , or in the form Sr 2 (a 2 - a> 2 )/(a 2 - r 2 ) = 2* 2 (a 2 - r 2 + r 2 - a> 2 )/(a 2 - r 2 ) But since in this notation 2z 2 = x 2 + y 2 + z 2 = r 2 , we have finally (r 2 - w 2 )(l+2z 2 /(a 2 -r 2 )) = 0. ?%e equation of the wave surface is therefore - r-) + 1-0. But because Sa^/r 2 = 1 and 2(x 2 /(a 2 - r 2 ) + z 2 /r 2 ) = 2a 2 .c 2 /(a 2 - r 2 ) = we may also write the equation of the wave surface in the form (m) aV/(a 2 - r 2 ) + &y/(6 2 - r 2 ) + c% 2 /(c 2 - r 2 ) = 0. We can easily transform this equation into (n) (aV + jy + c 2 * 2 )r 2 -(6 2 + c>V- (a 2 + c 2 )&V-(a 2 + 6 2 )c 2 5 2 + 2 i 2 c 2 = 0. 7%g equation of the wave surface is therefoi'e of the fourth degree. In order to investigate this equation we set x=fr, y = gr, z = hr. By substitution of these values in the equation of the wave surface, it becomes 2,- 2 (a 2 / 2 + bY + c% 2 ) - [(fr 2 + c> 2 / 2 + (a 2 + c 2 )6 2 ^ 2 + (a 2 + 5 2 )c 2 A 2 ] = R, From which we get B* = [(a 2 - c 2 )&V 2 + (afJW^c* + x [(a 2 - c 2 )6V + (a/v/^^c 2 - Hence a straight line drawn from the origin of coordinates cuts the surface in two points, which coincide when R = Q or when (o) .9 = and f/h = c/a. J(a'-b' 2 )/(b' 2 -c 2 ). SECT. CIV.] THE WAVE SURFACE. 253 In this case f=e/b. V( 2 - & 2 )/(tt 2 -~c~ 2 ), h = ajb . *J(b 2 - c 2 )/(a 2 - c 2 ). There are therefore four such points in the wave surface, all of which lie in the o^-plane. Hence the wave surface is a surface of the fourth degree with two nappes. The four points which the two nappes have in common are called umbilical points. To exhibit the form of this surface we will determine the curves formed by the intersection of the wave surface, and the coordinate planes yz, xz, yx. If, for this purpose, we set in equation (n) z = 0, y = 0, 2 = successively, we obtain - Z> 2 c 2 ) = 0, 2 c 2) ^ Q, W) = 0, Hence the curves formed by the intersection of the wave surface with the coordinate axes are circles and ellipses, as represented in figures 122, 123, and 124. The curves in the o^-plane are of 9 FIG. 122. b FIG. 123. c b FIG. 124. special interest. The equation z z + x 2 = b 2 represents a circle of radius b. The equation c 2 2 + a?x 2 - a?c 2 = represents an ellipse whose semi-axes are a and c. On the assumption that a>b>c, the circle and the ellipse intersect at a point P, and this point is one of the umbilical points. Equations (1) and (h) serve to determine the coordinates of the point of contact between the wave surface and a plane wave which moves in a direction determined by I, m, n. The case in which the wave is propagated in the direction of one of the optic axes is of special interest. In this case [CIIL], the velocity equals b, and the direction of propagation is given by the equations since we here consider only that optic axis which lies between the positive directions of the z- and a:-axes. Equations (1) then become z(a 2 - b 2 ) = lb(a 2 - r 2 ), z(b* - c~) = nb(r 2 - c 2 ). 254 REFRACTION OF LIGHT. [CHAP. xn. If we introduce in these equations the values for / and n given above, we have (p) xj(a 2 - 6 2 )(a 2 - c 2 ) = b(a 2 - r 2 ), W(& 2 - c' 2 )(a* - c-) = b(r* - c 2 ). These equations represent two spheres, in whose lines of intersection lie the points of contact of the wave plane and the wave surface, therefore in this case the plane of the wave touches the wave surface in a circle. We may also obtain this result in the following way. By the use of equations GUI. (c), we give (p) the form (q) x = b(a*-x 2 -f-z*)/l (a 2 -c>-), z = b(x* + y 2 + z' 2 -c 2 )/n (a 2 -c 2 ). The curve represented by these two equations is a plane curve because (r) xl + zn = b. We now introduce a new system of coordinates with the same origin ; suppose the ?/-axis to coincide with the y-axis, while the axis coincides with the optic axis. To effect this, we set (s) z = ra + $o> y = >?, z= -S/o + fV The equation (r), which represents a plane, then becomes (t) =b, that is, the plane of the curve of intersection is perpendicular to the direction of the optic axis and passes through its end point. The first of equations (q), by the use of (s) and (t), takes the form (u) f + ^Vo(rt 2 -< ;2 )/Z' + '? 2 = This represents a circle, which passes through the point = 0, *7 = 0, and =&, or through the end point of the optic axis. The radius r of the circle is r = >J(b 2 - c 2 )(a 2 - b' 2 )j 2b, and the coordinates of its centre are =?', t] = 0, = b. Thus the circle is determined in which the plane perpendicular to one of the optic axes at its end point touches the wave surface. SECTION CV. THE WAVE SURFACE (continued). Let ON (Fig. 125) be the normal to a plane wave; the direction of the normal is determined by the cosines /, m, n. Let OP l and OP 2 be the two velocities of propagation of the wave considered. Let Q l and Q 2 represent the points of contact between the plane wave and the wave surface. We then have OQ l = r l and 0$ = ?V We represent the coordinates of the points Q l and Q 2 by x v y v z^ and x 2 , y 2 , z 2 respectively. If Q l P i =p l and Q.f^p^ are the per- pendiculars let fall from the points of contact on the directions of propagation, we have SECT. CV.] THE WAVE SURFACE. 255 A The connection between the direction of the normal and the points of contact is given by equations (1) CIV. We will investigate more particularly the directions of the lines p 1 and p 2 . The projection of PQ on the a-axis is ul-x. If we represent the cosines of the angles which p makes with the axes by A.', /*', v', we will have A' = (wZ - x)/p, p! = (p/(a? - w 2 ), fjf = mupf(b 2 - to 2 ), v' = n 2 2 ) = 0, we also have (w x 2 - w 2 2 ) . 2/ 2 /(a 2 - m ^)(a z - u> 2 2 ) = 0. Hence, if the values of co 3 and to 2 are different, we have cos(P l Q l P 2 Q 2 ) equal to zero, and the angle between P^ and P 2 Q 2 a right angle. But if w 1 equals o> 2 , the points P l and P 2 coincide, as we saw in CIV. In this case, there is an infinite number of points of contact which lie on a circle passing through the wave normals. If the lines P 1 T l and P 2 r 2 are drawn from P l and P 2 perpendicular to OQ l and OQ 2 respectively, and if we set P l T 1 = q l and P 2 7 T 2 = j 2 , we have q:p = a)-.r, and therefore q=p(ojr. Further, OT^aPjr. If A, /j., v are the cosines of the angles which q makes with the coordinate axes, we have A = (coZ - OT . x/r )/q, etc., and hence [CIV. (1)], (b) A(a 2 - co 2 ) = Ia 2 pjr, p.(b 2 co 2 ) = mb 2 p/r, v(c 2 - co 2 ) = nc 2 pjr. If we compare this result with the expressions in CIL (e), which determine the direction of the electrical force F, whose components are X, Y, Z, we see that the electrical force is parallel to q. If we introduce in the equation CIL (d) the values for A, /*, v given above, and notice that Ix + my + nz = to, we have cos 8 p/r. Since there are two directions of q, namely q l and q 2 , there are two direc- tions, q l and q 2 , of the force in any plane wave. These lie in two 256 EE FRACTION OF LIGHT. [CHAP. xn. planes perpendicular to the plane wave. There are two values for 8, namely, ^OQ 1 P 1 and L.OQ.,P Z ; these angles are equal to Ll^P-fl and t-T^P^O respectively. The electrical forces X, Y, Z cause an electrical polarization, whose variation may be looked on as an electrical current. The components of current [OIL (a) and (c)] are = KJtir . VX^i = Kj_ A/47T . 9 Upt, etc. If A. , /x , v are the cosines of the angles made by the axes with the direction of the current, we have A : /* : v = A/a 2 : /z/5 2 : v/c 2 . But we obtain, by the help of OIL (e), A : p Q : v = //(a 2 - w 2 ) : m/(b* - a> 2 ) : n/(c 2 - w 2 ). Hence the current has two directions, corresponding to the two values of w. From equation (a) the same ratio holds between the cosines of the angles which p makes with the axes as between the cosines determining the directions of the current. Hence the two directions of the current are parallel to p { and p. 2 respectively. In order to determine the direction of the electrical force and the current, we proceed in the following way. If a plane wave moves in the direction determined by the normal ON, we construct two planes which touch the wave surface and are parallel to the plane wave. These planes are those constructed at Q l and Q 2 . We then draw Q l P l and Q 2 P 2 perpendicular to the wave normal. The electrical currents, which are in the wave planes, are parallel to $iA and Q 2 P 2 . The corresponding velocities of propagation are OP l and OP 2 . There are two directions of current in every plane wave, which are perpen- dicular to each other. The electrical forces, which are connected with these directions of current, are parallel to P^T^ and P 2 T 2 , SECTION CVI. THE DIRECTION OF THE RAYS. When a plane wave is propagated in an isotropic medium, the direction of the normal to the wave coincides with the direction of the ray. In doubly refracting media, the direction of the ray is in general different from the direction of the wave-normals. We will now determine the direction of the ray. Let MN (Fig. 126) be the surface of a doubly refracting body on which the cylinder of rays KOPL falls perpendicularly. By Huygen's principle, the separate points in the bounding surface OP may be considered as centres of luminous disturbance. The luminous disturbance is propagated THE DIRECTION OF THE RAYS. 257 through the body in such a way that, after unit time, it reaches the wave surfaces which are constructed about the separate points of the bounding surface OP. Therefore, if the wave surfaces RA, SO, etc., are constructed about 0, P, and the intervening points, we obtain a plane ES which touches every wave surface and is congruent to and similarly situated with OP. The direction OR or PS is then the direction of the rays. If from the point we let fall a perpendicular OB on the plane US tangent to the wave surface RA, OB = u is the velocity of propagation of the wave. If /, m, n represent the direction cosines of the normal to the wave surface, o> is determined by equation GIL (f) (a) / 2 /(a 2 - " 2 ) + m 2 l(W - w 2 ) + n 2 /(c 2 - co 2 ) - 0. -fl M Li P JT J, , where = OB, and s is constant. The reciprocal wave surface is then the locus of the points determined by (c). This surface, like the wave surface, is a surface of two nappes. Its equation is obtained in the following way. If I, m, n 258 EEFRACTION OF LIGHT. [CHAP. xii. represent the direction cosines of OB = to, and x', y', zf the coordinates of the point B 1 , we have (d) x' = lr\ y' = mr', d = nr'. But because l 2 /(a? - w 2 ) + w 2 /(6 2 - w 2 ) + n 2 /(c 2 - w 2 ) = 0, it follows by (c) and (d) that x' 2 /(a 2 r' 2 - s 4 ) + 2rY(6V 2 - s 4 ) + *' 2 /(cV 2 - s 4 ) = 0. We set (e) a' = s 2 /a, b' = s 2 jb, c' = s 2 /c, and obtain the equation of the reciprocal wave surface in the form (f ) a' 2 x' 2 /(a' 2 - r" 2 ) + b' 2 y' 2 j(b' 2 - r' 2 ) + c'-Y 2 /(c' 2 - r' 2 ) = 0. This surface differs from the ordinary wave surface [cf. CIV. (m)] only in that its constants a', I', c' are the reciprocals of the constants a, b, c of the wave surface. If we draw through the point B' (Fig. 127) a plane tangent to the reciprocal wave surface FA', we can show that the plane B'R is perpendicular to the prolongation of OR, and that therefore OR is perpendicular to B'R. Further, if OR = u>, Ofi = r, we have (g) w' = s 2 /r. This follows by the same method by which we have passed from one surface to the other. We can also prove it directly. If the direction OR is determined by the cosines I', m', n', we have [CIV. (1)] (h) z'( a ' 2 -to' 2 ) = r to'(a' 2 -' /2 ) etc. The equations (h) determine w', I,' m', n'. Setting u/ = s' 2 /r and l' = xfr, m' = y/r, ri = z/r, and using equations (c), (d), (e), (g), equation (h) takes the form x(a? w 2 ) = Zo(a 2 - r 2 ), etc. Since these equations are identical with those in CIV. (1), it follows that the point of intersection of OR SECT, cvi.] THE DIRECTION OF THE RAYS. 259 and the wave surface is the point at which the tangent plane touches the wave surface. It follows further from (e) and (g) that (i) r'to = ra> or OB .OB' = OR .OR. In order to determine the direction of a ray from the reciprocal wave surface, we produce the wave normal until it cuts that surface. The direction of the ray is then perpendicular to the tangent plane at the point of intersection. SECTION CVII. UNIAXIAL CRYSTALS. If two of the constants a, b, c are equal, for example if b = c, the equations become much simplified. The bodies for which this relation holds are called uniaxial crystals. In order to find the velocities of propagation w 1 and o> 2 , we apply equation CII. (g) which is trans- formed into (a) to 4 - [b 2 + r-b 2 + (l- / 2 )a 2 > 2 + b 2 [l 2 b 2 + (1 - I 2 )a 2 ] = 0. From this equation we obtain (b) o^ 2 = b 2 , o> 2 2 = I 2 b 2 + (1 - l' 2 )a 2 . Hence the velocity w x is constant ; the velocity o> 2 depends on the direction of the wave normal, or on the angle which the wave normal makes with the axis of elasticity a. This axis is called the optic axis; it coincides with the principal axis of the crystal. In the direction of this optic axis there is only one wave velocity, arid therefore also only one ray velocity. If we designate the angle between the wave normal and the optic axis by e, we have (c) o> 2 2 = 2 sin 2 + 6 2 cos 2 e. Hence, a plane wave, on its passage from an isotropic to an uniaxial medium, is divided into two waves, one of which is propagated with a velocity o^, which is independent of the direction of the wave normal. This wave is called the ordinary wave. The velocity of the other or extraordinary wave changes with the direction of the wave normal. AVe obtain the equation of the wave surface for uniaxial crystals from CIV. (n), by setting b = c. We thus obtain b, the crystal is called negative. The sphere can enclose the ellipsoid or inversely ; crystals of the first kind are called positive, those of the second kind negative. Iceland spar is a negative crystal, quartz is a positive crystal. If we set o> = 5 in OIL (e), we obtain (e) \ 1 = Q and S I = ^TT, that is, the direction of the electrical force in the ordinary wave is perpendicular to the optic axis as well as to the wave normal : it is therefore perpendicular to the principal section. In order to obtain the direction of the electrical force in the extraordinary waves from CIL (e), we introduce in it the value for r f r , f r = F r . ew - (irx+ r y+r*)/n]. That the electrical force shall be perpendicular to the direction of the ray, we must have (e) X r l r + p, r m r + v r n r = 0. Finally, for the re- fracted ray, we have (f) A>A 6 / 6 , Y b = nJ b , Z b = v J M (g)f t = F^-**++W. w depends on l b , m b , n b , or on the direction of the propagation of the refracted wave. The boundary conditions are the same as those of isotropic bodies. We have everywhere in the bounding surface, for which x = 0, Y t +Y r =Y M or Since this equation must hold for all values of y and z, we have (h) sin a/0 = m r /0 = m b /o> and (i) = n r /l = 6 /w. By the last equation = 7i r = 7i M that is, the wave noi-mals of the reflected and refracted waves lie in the plane of incidence. It follows from (h) that w r = sina, that is, the angle of reflection is equal to the angle of incidence. Therefore the direction of the reflected ray is determined in the same way as in the case of reflection by an isotropic body. 262 REFRACTION OF LIGHT. [CHAP. xn. If (3 represents the angle of refraction, we have l b = - cos ft, m b = sin j3, n b = 0, therefore from (h) sin a/12 = sin /3/w. If we determine the direction of the wave normal by the cosines of the angles which it makes with the axes of elasticity, we have, to determine w, the equation (1) / 2 /(a 2 - a) 2 ) + TO 2 /(6 2 - w 2 ) + rc 2 /(c 2 - w 2 ) = 0. If (xa), (ya\ etc., denote the angles between the axes of elasticity and the coordinate axes, we have l = l t> cos(xa) + m b cos(ya) + n l> cos(za). Introducing here the values for l b , m t , etc., given above, we obtain c I = - cos (3 . cos(za) + sin /3 . cos(ya) (m) -j m = - cos (3 . cos(a$) + sin /3 . cos(yb) \. n= -cos/?, cos(zc) + sin ft. cos (yc). By the help of equations (m) and (1), w can be expressed in terms of (3. The equation thus obtained in connection Avith (k) determines the angle of refraction. In general we obtain two values for f3, one or both of which may be imaginary ; if this is the case the reflection is total. We can find the direction of the wave normal and that of the ray by a construction given by Huygens. About the point (Fig. 129) as centre construct the sphere PD, whose radius is OD = Q, where 12 denotes the velocity of light in air. If the incident ray is produced, it meets the sphere at the point D. The plane which the sphere touches at D cuts the refracting surface in a straight line, whose projection on the plane of the figure is Q. The plane QR containing this line is drawn tangent to the wave sur- 129. face FR, whose centre is at 0. The perpendicular OB = i is let fall from on the tangent plane QR. The normal to the refracted wave is then OB and L'OB = (3, if LOL' is the normal to the surface. Now OB= l sin(3, or fl/sina = w/sin/:J, so that equation (k) is also satisfied. OB is the direction of the wave normal of the refracted wave, and OR the direction of the corresponding ray. Since the wave surface in general has two nappes, two planes tangent to the wave surface can be SECT, cvni.] DOUBLE EEFEACTION OF A CEYSTAL. 263 drawn through Q. The construction therefore determines two wave normals and two ray directions. This construction really serves only as a representation of the refraction; it cannot be used for the determination of the direction of propagation so long as the construction is confined to the plane, because the point of contact R does not lie in the plane of incidence ; we can, however, obtain the direction of the wave normal by a construction in the plane of incidence given by MacCullagh. If we draw through D (Fig. 129) the line DE perpendicular to the refracting surface, the point of intersection B'. of DE and the wave normal OB is so situated that OB . OB = OD 2 , for we have OB = OQ.sinf3, OB' = OE/sin(3, and further, as may easily be seen from Fig. 129, OQ. OE=OD 2 . From this follows the relation OB.OB' = OD\ But we have OB = u, OD = Q, and if we set OB' = r', it follows that (n) r' = 12' 2 /w. Therefore the point B' lies on the reciprocal wave surface whose equation is [CVI. (f)] a' 2 x 2 /(a' 2 - r 2 ) + b' 2 f/(b' 2 - r 2 ) + c' 2 z 2 /(c' 2 - r 2 ) = if the coordinate axes are parallel to the axes of elasticity. In this equation a' = fl 2 /a, b' = ^ 2 /b, c' = 2 /c. If we set JV^fi/a, ^2 = ^/6, JV 3 = 0/c, and choose as the unit of length the velocity of light fi in the surrounding medium, it follows that ( o) N^KN^ - r 2 ) + N 2 y*l(N 2 - r 2 ) + N 3 2 z 2 /(N s 2 - r 2 ) = 0. This is the equation of the reciprocal wave surface. It follows further from the discussion of CVI. that the direction of the rays OR is perpendicular to the plane tangent to the reciprocal wave surface at the point B. We can therefore construct the wave normal in the following way. About the point as centre with unit radius we construct the circle PD ; about the same point we draw the curve of intersection between the plane of incidence and the surface (o). This curve is represented in (Fig. 29) by B'F'. W e then produce the incident ray to the point D lying on the circle, and draw the straight line DB' perpen- dicular to the refracting surface and cutting F'B' at B'. OB is then the direction of the wave normal, while the direction of the ray OE is perpendicular to the plane tangent to the surface F'B at the point B'. We can easily derive the condition for total reflection from this construction. It can also be applied to the reflection of light within the crystal itself. 264 REFRACTION OF LIGHT. [CHAP. xn. SECTION CIX. DOUBLE REFRACTION IN UNIAXIAL CRYSTALS. Using the same notation as in CVIIL, we have, to determine the angle of refraction of the wave normal, the equation (a) sin a/0 = sin /?/a>. In the case of uniaxial crystals w has the values w : and w 2 which are [CVII. (b), (c)] o) 1 2 = i' 2 , w 9 2 = a' 2 sin 2 e + > 2 cos 2 e. In the first case the angle of refraction /3 1 is obtained from the equation sin a = JV sin /^ where NQ = &/(O V the so-called ordinary index of refraction. The corresponding direc- tion of the electrical force is perpendicular to the principal section. If the second wave normal makes the angle /3 2 with the normal to the surface, we have sin a/fl = sin /? 2 /w.,. If we represent the angles made by the axis of the crystal with the coordinate axes by (xa), (ya), (za) and notice that TT - (3 y |z- - jS^ |TT, are the angles made by the refracted ray with the coordinate axes, we have cosc= -cos()cos/? 2 + cos(?/a)sin/3 2 . Hence, for the calculation of /3 2 , we have the equation fl 2 sin 2 /3 2 /sin 2 a = a 2 - (a 2 - Z> 2 )(cos(.ra)cos /3 2 - cos (ya)sin /3 2 ) 2 . The corresponding direction of the electrical force is parallel to the principal section. If the optic axis lies in the plane of incidence we set cos(:ra) = cos \f/, cos (ya) = sin ^, and then obtain J2 2 sin 2 2 /sin 2 a = a 2 - (a 2 - 6 2 ) . cos 2 (^ + /3 2 ). If ^ = a 2 sin 2 r/'+6 2 cos 2 ^, J B = a 2 cos 2 ^ + 6 2 sin 2 ^, (7=( 2 -6 2 )sin ^cos f, we have A-C 2 = aW 2 and I2 2 /sin 2 a = A cotg 2 /? 2 + 2(7 cotg /3. 2 + H. From this follows (b) A cotg /3 2 = - C + v If the axis of the crystal is perpendicular to the plane of incidence, we have (xa) = (ya) = ^TT, from which sin a = JV^sin /? 2 , where N e = Q.ja, is the extraordinary index of refraction. If a and b are expressed in terms of N, and N , we have from (b) r (A^sin 2 ^ + JV^cosV) cotg /3 2 = - (N 2 - N*) sin ^ cos $ I + JV JV eX /sin--'a( JV^sin 2 ^ + ^V e -'cos 2 ^) - 1 . In order to obtain the equation of the reciprocal wave surface, we set fl = l, and substitute N, for a, N for b, in the equation for the wave surface. Thus we obtain [CVII. (d)], (d) (r> - N*%N& + N Q % 2 + * 2 ) - NJN*] = 0, SECT, cix.] DOUBLE REFRACTION IN UNIAXIAL CRYSTALS. 265 FIG. 130. as the equation for the reciprocal wave surface referred to the axes of elasticity as coordinate axes. We obtain the same result from CVIIL (o), if we set N^N. and N 2 = N 3 = N . In Fig. 130, OP is the refracting surface, and OA the optic axis, supposed to lie in the plane of incidence ; AM l and AM. 2 are the curves in which the plane of incidence cuts the reciprocal wave surface. AM^ is a circle with radius N Q , AM 2 an ellipse whose semi-major axis OA equals N , and whose semi-minor axis OM 2 equals N e . We draw a circle of radius OD = 1 , which cuts at D the prolongation of the incident ray. The line ED, perpendicular to the refracting surface, cuts the reciprocal wave surface at the points B l and B. 2 . The normals to the refracted waves are then OB l and OB%. For the ordinary wave the direction of the ray coincides with the wave normal OB l ; for the extraordinary Avave it is perpendicular to the plane tangent to the ellipsoid at the point B. 2 . If the crystal is immersed in a fluid whose index of refraction is greater than that of the crystal, the circle PD is replaced by another circle of greater radius, for example PD'. If this circle cuts the prolongation of the incident ray at D', the directions of the wave normals are determined by the point of intersection between the reciprocal wave surface and the line UE', perpendicular to the refracting surface. In this case total reflection can occur. If HE' does not cut the reciprocal wave surface there will be no refraction; if UE' cuts only one curve, there is only one refracted ray. If, as in Fig. 130, D'E' touches the ellipse at a point C, refraction will occur ; the direction of the ray is parallel to the bounding surface OP. Our presentation of optics is based on Maxwell's conception of light as an electrical vibration. A more extended discussion on this same basis has been given by H. A. Lorenz. Glazebrook published a discussion of the most important optical theories in the Report for 1885 of the British Association for the Advancement of Science, von Helmholtz has lately given a theory of the dispersion of light in which he employs the electromagnetic theory of light. CHAPTER XIII. THERMODYNAMICS. SECTION CX. THE STATE OF A BODY. IF the particles of a system are in motion and exert force on one another, the system possesses a certain energy U. The energy of a system of discrete particles is made up of their kinetic and potential energies. The former depends on the velocities of the particles at any instant, the latter on their distances apart, or on the configuration of the system ; together they determine the state of the body. Thus the energy at any instant depends only on the state of the system at that instant, and is independent of its previous states. The principle of energy has been proved only for a system of discrete particles ; we make the assumption in the mechanical theory of heat, that the same principle or a corresponding one holds for all systems of particles. A certain amount of energy is inherent in every body. This we call its internal energy, since we take no account of that part of its energy which arises from its mutual actions with other bodies. By the possession of this internal energy the body is in a condition to do work ; thus variations occur in its form, volume, temperature etc. The energy is determined solely by the state of the body ; if the body in a certain state possesses the energy U, and if it is subjected to any variations of form, magnitude, etc., and finally returns to its original state, the internal energy will be again equal to U. To determine the internal energy of a body it is necessary to know the quantities which determine its state. From Boyle's and Gay-Lussac's laws the state of an ideal gas is completely determined by its pressure and volume. The temperature is given if these two quantities are known. Boyle's and Gay-Lussac's laws furnish an equation which expresses the relations between pressure, temperature, CHAP. xni. SECT, ex.] THE STATE OF A BODY. 267 aud volume ; we call it the equation of state of a gas, because it enables us to determine the state of an ideal gas under any con- ditions, if it is known under definite conditions, for instance, at C. and 760 mm. pressure. The behaviour of real gases cannot be accurately represented by an equation embodying Boyle's and Gay-Lussac's laws, but conforms to other equations which include these laws as a limiting case. The state of a fluid is in general determined by the same quantities ; it depends to some extent on the form of the surface and the nature of the bodies in contact with it. The actions of electrical and magnetic forces may come into play in both gases and fluids. As a rule the knowledge of a great number of quantities is required to express the state of a solid, especially if it is subjected to the action of forces. The equation which unites all quantities which determine the state of a body is called the equation of state. Since the state of a gas only depends on the pressure p and the volume v, it may be represented by a point in a plane with the coordinates p and v ; a series of such points, or a curve, represents a series of successive states. The t'-axis of this system (Fig. 131) is drawn horizontal ; and the ^?-axis vertical. We represent the volume and pressure of the gas in its original state by r and p ; its state is then given by the point A. If the gas ex- pands under constant pressure, its state is represented by a horizontal line AB, parallel to the v-axis. This is called the curve of constant pressure. The curves of constant volume are vertical straight lines. If heat is communicated to the gas whose original state is given by A, at constant volume r , the variation of the state of the gas is represented by the straight line AC, and its pressure increases. If the tem- perature of a gas remains constant during its successive states, we have, from Boyle's law, v.p = const. Hence the curves of constant temperature or the isothermal lines are rectangular hyperbolas whose asymptotes are the coordinate axes. In order to change the state of a gas in such a way that its temperature remains constant, there is required either compression with abstraction of heat or expansion J) B FIG. 131. 268 THERMODYNAMICS. [CHAP. xin. with communication of heat. If a gas whose original state is represented by A is subjected to compression with abstraction of heat, or to expansion with communication of heat, in such a way that its temperature remains constant, its successive states will be represented by the hyperbola DAE. We may suppose the gas enclosed in a receptacle put in connection with an infinitely great source of heat, whose temperature is equal to that of the gas at the point A. If we change the volume of the gas, the source of heat sometimes takes up heat and sometimes gives it out, but the gas retains the temperature of the source. If the gas is enclosed in an envelope through Avhich heat cannot pass, it is heated by compression so that its temperature rises, or cooled by expansion so that its tem- perature falls. In this case the changes of state are called adiabatic and the curve which represents them is called an adiabatic or isentropic curve. The state of a solid cannot in general be represented in a plane, since it depends on more than two variables. A series of changes by which the state of a body is altered in any manner, and which is such that the body finally returns to its original state, is called a cyclic process. If a body goes through a cyclic process the energy which it receives from surrounding bodies is equal to that which it gives up to them. The steam-engine is a system of bodies which periodically returns to the same state. It appears from the action of the steam-engine, that heat and work are similar or equivalent quantities, which can be transformed into one another, and are both, therefore, forms of energy. This conclusion has been established by accurate experiment. The quantity of energy produced in the one form is always proportional to that applied in the other form. This law of the equivalence of heat and energy was first formulated by K. Mayer (1842). The later observations of Joule and others have shown that the quantity of work which is equivalent to a unit of heat, or to the quantity of heat which will raise the temperature of a gram of water by 1 C. is equal to 4.2.10 7 absolute units of work (C.G.S.). This result is called the first law of thermodynamics. It may be thus stated : Heat and work are equivalent ; work can be obtained from heat and heat from work. The work equivalent or the mechanical equivalent of the unit of heat is designated by J. If the quantity of heat dQ is communicated to a body it receives the energy / . dQ. This goes partly to increase the internal energy U of the body, partly to do the work dW. We then have (a) J.d SECT. CX.] THE STATE OF A BODY. 269 This equation is called the first fundamental equation. We will apply it to the case in which the work dW is done by expansion against external pressure. We consider the body ABC (Fig. 132), which is subjected to the hydrostatic pressure p at every point on its surface. When the body expands its volume becomes A'B'C'. The normals AA', BB' are drawn from the surface- element AB = dS to the new surface. We set A A' = v and obtain for the work done by the body, \vp . dS=p^v . dSp . dv, where dv denotes the total increase in volume of the body. Equation (a) then becomes (b) J .dQ = dU+p .dv. If the state of a body is determined by the independent variables p and v, the definite values p^ FIG. 132. JL ' FIG. 133. and v l correspond to a point A (Fig. 133). Suppose the body to pass through a series of states represented by the curve ACB; the values p. 2 and v. 2 correspond to the point B. We then have from (b) (c) JQ=Ut-U 1 + p.dv. Q is the quantity of heat introduced during the change of state, /2 - ZTj the increase of the internal energy, and / p . dv the work done. The increase U 2 - U l is determined by the initial and final values of p and v, or by the position of the points A and B. The external work is measured by the area of the figure A' ABB' A ; this work therefore depends on the process by which the change from one state to the other is effected. This holds also for Q. Since U is a function of p and v, we obtain (d) J.dQ = 'dU/'dp.dp + ('dUj'dv+p)dv. If the function U is known, it is possible to find the quantity of heat necessary to produce any change in the state of the body. U is determined from equation (c), by measuring the quantity of heat received by the body and the quantity of work done by it. Our knowledge of the quantity U is still very limited. 270 THERMODYNAMICS. [CHAP. xm. SECTION CXI. IDEAL GASES. Clement and Desormes and subsequently Joule showed that the temperature of a gas which expands without overcoming resistance, that is, without doing work, remains unchanged.* The initial and final states of a gas which expands without doing work lie on the same isothermal, that is, the internal energy of a gas is a function of its temperature only, and is therefore independent of its volume if the temperature remains constant. If we take the temperature and the volume v of the gas as independent variables, we have J.dQ = 'dUJW .de + 'd Ufdv . dv +p . dr. Now 3 Ufdv = and therefore J.dQ = 'd U/W .d6+p. dv. If the mass of gas contained in the volume v is equal to unity then ?>U/'dO = Jc n where c, denotes the specific heat of the gas at constant volume, that is, the quantity of heat which must be communicated to its unit of mass in order to raise its temperature one degree in such a way that, while its pressure changes, its volume remains constant. If the specific heat of the gas at constant volume is constant, its internal energy must be a linear function of its temperature. For ideal gases the equation giving the relation between pressure, volume and temperature is pv = R6, where R is a constant. If and v are the independent variables of the gas, we have (a) J.dQ = Jc,.d6+p.dv. From the observations of Regnault, c, is independent of the pressure and temperature of the gas. If 6 and p are chosen as the independent variables, v must be considered as a function of them, so that dv = "dvfde . dO + 'dvj'dp . dp, -and substituting this in equation (a) we have /. dQ = (Jc, +p . 'dv/Wfie +p . "dvfdp . dp. From the equation pv = E6, it follows that p . 'dv/W - R and p . 'dv/dp = - v, and J.dQ = (Jc, + B)dO - v . dp. In order to obtain the specific heat c p at constant pressure, that is, the quantity of heat which must be communicated to the unit of mass of the gas to raise its temperature one degree, in such a way that, while its volume changes, its pressure remains constant, we set dp = and obtain c p = c, + ElJ, (b) J.dQ = Jc p .dO-v.dp. If p and v are *More exact measurements show that the gas, in these circumstances, is slightly cooled. From this it follows that there are attractive forces between its separate particles. SECT, cxi.] IDEAL GASES. 271 chosen as the independent variables, we have dO = 'dOj'dp . dp + Wfdv . dv. It follows from pv = R0 that R.o0/'dp = v, R.Wj?)v=p, R.d0 = v.dp+p.dv, and from (b) that (c) R.dQ = c,v.dp + c p p . dv. If therefore the specific heat c p and the constant R are known, the equation (c) enables us to determine the specific heat for any change of state in the vp-p\a.ne. The specific heat has an infinite number of values for a given state in the t^-plane, depending on the direction in which this change of state takes place. The expressions (a), (b), (c) show a noteworthy peculiarity. If one of them, say (a), is divided by 0, we obtain by the use of the fundamental equation pv = R9, J .dQ/0 = Jc,.d0/9 + R.dv/v. If, for example, the gas passes from the state A (Fig. 133) to the state B, and if the temperatures and volumes at these points are 6 V v l and , v 2 respectively, we have by integration, (d) /. \dQjO = J.c.. log^flj) + R . log(Vi)- Therefore, while the integral ^dQ depends on the path on which the gas passes from one state to another, the integral ^dQ/d does not depend on this path. Clausius called the quantity S = J .\dQjO the entropy. This concept is of great importance in the theory of heat. If a body passes from one state to another the change of the entropy is determined by the coordinates of the initial and final points. This theorem is here proved only for a gas, but holds also for all bodies. If the change of state of a gas occurs along an isothermal curve, we have from (a) J.dQ=p.dv. Using the equation of state and inte- grating, we obtain (e) JQ = p . dv = Rd . log(^ 1 ). All the heat communicated is therefore used in keeping the tempera- ture constant. If we set v 2 equal to fiv v fj?v v p?v v etc., in succession, where ^ is any number, the corresponding values for Q are Q = R0/J.logp, 2R6/J. logp, SRe/J.logp, etc. If the change of state occurs along an isothermal curve, and ii the quantities of heat introduced are in arithmetical progression, the volumes, according to equation (e), are in geometrical progression ; at the same time the pressure changes proportionally to the density. If the change of state occurs along an admbatic curve, we have from (c) c v logp + c p log v = c r Setting c p /c, = k we obtain (f ) pv* = c, where c is 272 THERMODYNAMICS. [CHAP. xm. constant. The equation (f) is the equation of the adiabatic curves. Com- bining this with the equation pv = B6, we have from (f) Bdv k ~ l = c. If we introduce in this formula the density 8 = M/v of the gas, where M denotes its mass, it follows that its temperature is proportional to the (&-1) power of the density when the state of the gas changes along an adiabatic curve. Further we obtain the relation [CX. (b)] fp.dv=U l - U. 2 . The work is therefore done at the expense of the internal energy, if the change of state is adiabatic. SECTION CXII. CYCLIC PROCESSES. A simple reversible cycle is one in which all changes occur in such a way that if reversed they may be effected under the same circum- stances. The body which performs the cycle is called the working body. In the performance of a simple reversible cycle the working body must be associated with two others, one which communicates heat to it, and another which receives heat from it. In a gas engine the working body is the gas in the cylinder ; in a steam engine it is the water or steam. The gases of the fire and the walls of the boiler give up heat, the water in the condenser receives heat. The gas or steam passes through a series of states and, at least in some machines, returns to its original state; it is then in condition to repeat the same process. Since the value of the internal energy U at the beginning and end of the process is the same, we have [CX. (a)] (a) JQ=W, where Q is the difference between the heat received and the heat given up. The quantity of heat received by the working body and not given up to the colder body is the equivalent of the work done. If the working body is a gas, we have for the cycle JQ = ^pdi: The entropy of a gas depends only on the coordinates and therefore has the same value at the beginning and end of the process. If S } denotes the entropy at the starting point, the entropy at any instant during the process is equal to S l + \dQjQ. If the integration is extended over the whole cycle, the entropy returns again to its value S v and we have therefore \dQ/B = 0. We will discuss more particularly a special case, the so-called Carnofs cycle, which is of great importance in the theory of heat. SECT. CXII.] CYCLIC PROCESSES. 273 Suppose a gram of gas to be in the state represented by the point B (Fig. 134) in the vp-plane. The curve representing the cycle is in this case composed of two isothermal curves BC and ED and of two adiabatic curves CD and BE. The gas first expands at the constant temperature O r This is accomplished by keeping it in contact with the infinitely great body M l at the temperature lt and by so regulating the external pressure on the gas that it passes to the state C along the path BC. During the change of state BC the quantity of heat Q l is absorbed and the work represented by the surface BCC'B 1 is done. The gas then expands adiabatically, in the manner represented by the adiabatic curve CD, and its temperature falls to 2 . Then the gas is brought in contact with an I - 4 i if FIG. 134. infinitely great body Me, at the temperature 2 and compressed ; during this process it gives up to M. 2 the quantity of heat Q. Its state is represented by E. Finally the gas is further compressed without communication of heat until it returns to the original state B. The integral ^p . dv, extended over the whole cycle, equals the area BCDE, and represents the work done by the gas. We have [CX. (a)] (c) J(Q l - Q 2 ) = W. When the gas expands from B to C, its entropy is increased by QI/OI ; it remains constant along the path from C to D, is diminished by Q.2/Q-2 along the path from D to E, and again remains constant along the path from E to B. Since the gas on its return to B has the same entropy as at the outset we have From (c) and (d) it follows that Therefore the work done by this cyclic process is proportional to the quantity of heat Q } absorbed and to the difference of temperature Q l - 6. 2 , and is inversely proportional to the absolute temperature O lt at which the heat is absorbed. The heat received from the source M l is not wholly transformed into work, but is divided into two parts, one of which is transformed into work, and the other transferred to M . The efficiency of the Carnot's cycle is the ratio of the heat trans- formed into work to that communicated to the gas. We have 274 THERMODYNAMICS. [CHAP. xin. Hence the efficiency depends only on the temperatures d l and 0. 2 of the sources of heat. If we consider the reversed cycle, the state of the gas first changes along BE ; along the path ED it receives a certain quantity of heat from M*, and has a certain quantity of external work done upon it along the path DC. The heat received and the work done transformed into heat are given up by the gas to the body M^. along the path CB. In the case of the cyclic process first considered heat is transformed into work ; in the reverse process work is trans- formed into heat. SECTION CXIII. CARNOT'S AND CLAUSIUS' THEOREM. It was shown in the preceding section that ^dQ/d = Q for any reversible cycle, when the body describing the cycle is a gas. We will now see if this theorem holds when any other body is used as the working body instead of a gas. Let us take the simple case in which the process is carried out along two isothermal lines BC and ED (Fig. 134), and two adiabatic lines CD and BE. Suppose the change of state to take place in the sense given by the letters BODE. If the body at the temperature O l expands from B to C, it receives the quantity of heat Q^ ; when it is compressed from D to E it gives up the quantity of heat Q. 2 . Along the paths CD and EB heat will neither be received nor rejected. In this process the total quantity of heat received by the body is Q l - $ Since it returns to its original state, the quantity of heat Q 1 - Q 2 is equivalent to the work done, which is therefore J(Q l - Q. 2 ). S. Carnot published, in 1824, a work on the motive power of heat, in which he proposed an important theorem on the connection between heat and work. He was of the opinion that heat was a fundamental substance whose quantity remained invariable in nature. Applying this view to explain the action of the steam-engine, he supposed that the steam gave up a quantity of heat Q 1 at the higher temperature 6 V that this heat was transferred to the condenser at the lower temperature 2 , and that the motive power of the heat was due to its passage from the higher to the lower temperature. The work thus done by this passage of heat from a higher to a lower temperature was considered analogous to that done by n falling fluid or by any falling body. This latter is proportional to the weight of the falling body and to the distance which it falls. Hence for the work done by the heat Carnot proposed the expression SECT, cxin.] CAKNOT'S AND CLAUSIUS' THEOREM. 275 KQ l (O l - 2 ) where K is a function of the absolute temperatures O l and 2 . This conclusion of Carnot was confirmed by experiment, but did not agree with the mechanical theory of heat in so far as it regarded heat as an invariable quantity. If for the present we disregard this error, we have for the cycle just described Since K must be independent of the nature of the body doing the work we have, if the body is a gas [CXIL (e)], (b) K^J/0^ It therefore follows that (Q l - Q^/^ - 2 ) = Q 1 /6 l and hence Q l /0 l = Q 2 /0 2 , that is, if a body traverses a Carnot's cycle any number of times, by being placed alternately in contact with two infinite sources of heat, the quantities of heat which it receives from one source and gives up to the other are in the same ratio as the temperatures of the sources. There can be no doubt that this theorem holds for a cycle of the kind considered. The application of the theorem in many departments of physics and chemistry has led to no results which are as yet contradicted by experiment. Several attempts were made to give a direct proof of the theorem, the first and most important of which is due to Clausius, whose method may be presented in the following way : Suppose a gas to traverse the cycle BCDE (Fig. 135) composed of the isothermal curves BC and DE, which correspond to the absolute temperatures O l and 2 , and of the adiabatic curves CD and BE. During its expansion from B to C, the gas takes the quantity of heat Q l from an infinitely great source M-^ whose temperature O l is constant. It then expands from C to D without communication of heat. It is then brought in communication with the infinitely great source of heat M.> whose temperature is constant, and by com- :// pression is made to give up to it the ' FIG 135 quantity of heat Q y Finally it is brought back to its original state B. During the cycle the gas has received from the source M l the quantity of heat Q v which is divided into two parts. One of these parts is transferred as a quantity of heat Q 2 to the source M. 2 , the other is transformed into work and is represented in amount by, the area BCDE. Suppose B'C' and E'D' (Fig. 135) to be the two isothermal curves corresponding to the temperatures 6 l and 9. 2 for another body, say 276 THERMODYNAMICS. [CHAP. xni. for water vapour. C'D' and B'E' are two adiabatic curves so chosen that the surface BC'D'E equals the surface BODE. If the water- vapour is subjected to a process similar to that just described for the gas, the heat which it will receive while in contact with the source M l is Q l + q, and during its passage from H to E', while in contact with the source M 2 , it gives up to that source the quantity of heat Q 2 '. The work done by the vapour is equal to that done by the gas, because the surface BCDE is equal to the surface B'C'DE', and we therefore have Q l + q - Q. 2 ' = Q l - Q 2 , and therefore Q 2 =Q. 2 + q. The vapour in expanding along B'(J receives the quantity of heat Q l + q, and gives up the quantity Q 2 + q along the path D'E'. The cycle described can also be performed in the opposite sense. For example, the water-vapour can expand along the isentropic curve B'E'; it may then be brought in contact with the source of heat M 2 , and expand from E' to D', during which expansion the quantity of heat Q 2 + q is taken from M 2 . It may then be compressed along the isentropic curve Z^C", and lastly along the path (7-6', while in contact with the source of heat M v During this compression it gives up to M l the quantity of heat Q l + q. To carry out this pro- cess, a quantity of work must be done which is equivalent to the heat this work is represented in Fig. 135 by the surface B'C'HE'. We consider finally two engines, one of which is a gas engine, in which the gas performs the cycle BCDE, and the other a steam- engine, in which the steam performs the reversed cycle B'C'D'E'. The work done by the one is equal to that supplied to the other, if we neglect friction and other resistances. The gas engine in each revolution takes from the source M l the quantity of heat Q l and gives up to the source M. 2 the quantity Q. 2 ; at the same time the steam-engine takes from M 2 the quantity of heat Q 2 + q and gives up to M^ the quantity Q l + q. Hence, in these circumstances, the source of heat at higher temperature receives during each revolu- tion the quantity of heat q, while the source at lower temperature M 2 gives up the same quantity of heat ; this transfer of heat q from the lower to the higher temperature being effected without the doing of work. By this process, therefore, heat can be transferred from a colder to a hotter body. This Clausius declares to contradict experience. While heat invariably tends to flow from hotter to colder bodies, in the process described above the opposite occurs. The objection has been raised to this conception of Clausius that a thermoelectric SECT, cxin.] CARNOT'S AND CLAUSIUS' THEOREM. 277 circuit, in which one junction is at the temperature 100 and the other at 0, can produce a current which will heat a platinum wire red hot, so that heat passes from a colder to a hotter body, that is, to the red hot platinum. Clausius answered this objection by asserting that this transfer of heat to a higher temperature is' compensated for by the heat generated at the points of contact. Clausius therefore proposed this theorem : Heat can never pass from a colder to a hotter body without the expenditure of work or the occurrence of some change of state. Hence, by this principle of Clausius, 2 = 0, and therefore for any cycle of the sort described, whatever body is used in it, we have (a) Q l /0 l = Q 2 /Q 2 . We can now show that a similar theorem holds for a cycle of any sort. Suppose that the change of state of a body proceeds from B along the curve EG (Fig. 13G). If the isothermal curve BD passes through B and the adiabatic curve CD through C, we may replace the path BC by the path BDC, that is, the body may first expand at constant temperature along BD and then at constant entropy, that is, without communication of heat, along DC. If BC is infinitesimal, BD and DC are so also, and the change of state BC may be replaced by the two changes BD and DC. On both paths the body receives FIG. 136, FIG. 137. the same increment dU of internal energy. The external work is in the one case BCC'B, in the other BDCC'B. But since B'C' = dv is infinitely small, while B'B=p remains constant, the surface BDC vanishes in comparison with the surface BCC'B. If dQ represents the quantity of heat supplied, we have J.dQ = dU+p.dv along the path BC, as well as along the path BDC. Let BCPDEQ (Fig. 137) be any cycle, Be, Cc', Ed, Dd' isothermal curves, and BE, CD, etc., adiabatic curves. Let the body receive 278 THERMODYNAMICS. [CHAP. xm. the quantity of heat dQ along EC, and give up the quantity dQ. 2 along DE. As we have already seen, the body would receive by a change of state along Be the same quantity of heat dQ v and give up along dE the same quantity dQ. 2 . Therefore we have for the cycle BcdE, dQ l /6 l = dQ. 2 /6 2 , if 6 l and 2 are the absolute temperatures corresponding to the isothermals Be and Ed. In the same way we have for Cc' and Dd', etc., dQ^/O^dQ^/0^ dQ 1 a /0 l " = dQ/l0 2 " i etc. If Q and P denote the points at which the cycle touches two isothermal curves, we have by addition (b) \dQ 1 /0 l = Je^/0 2 , where dQ 1 is the quantity of heat received along an element of QBCP and 6 l the corre- sponding temperature, dQ. 2 the quantity given up along an element of PDEQ and 0., the corresponding temperature. If the heat received is considered positive and that given up negative, tJie sum of all infini- tesimal quantities of heat received during the performance of a reversible cycle, each divided by the absolute temperature at which it is received, equal* zero, that is, (c) ^dQ/0 = 0. This is the second law of thermodynamics. The theorem (c) which Clausius first expressed in this form may be given in another way. Let ABCD (Fig. 138) represent a cycle, so that f dQ/6 = Q. We divide the integral into two parts, J A BCD A of which the first is extended from A over B to C. the second from C over D to Aj and have If, therefore, the body passes from the state A to the state C, the value of FlG 13g the integral \dQ/6 is independent of the path. If 6 l and i\ are the coordinates of the point A, and 6 2 and v 2 those of the point C, we have Clausius introduced a special symbol for the entropy by setting dS = J. dQ/e, from which jCdQ/6 = S 2 - S r The function S represents the entropy of the body ; it depends only on the state of the body at any instant, and is independent of all previous states. SECT, cxiv.] APPLICATION OF THE SECOND LAW. 279 SECTION CXIV. APPLICATION OF THE SECOND LAW. We have already obtained [CX. (a)] the equation (a) J.dQ = dU+p.dv. If the state of the body is determined only by the independent variables 6 and v, equation (a) may take the form (b) /. dQ = (VUfdff). . dO + ((3 Ufdv) e +p)dr, where the indices attached indicate that the quantities which they represent remain constant during differentiation. If S denotes the entropy, we have dS = J. dQ/6 = 1/6 . (3 UfiO), .dO+(l/e. (d Upv) g +p i 'B)dv. Since S is here a function of v and B, we may set (3S/30), = 1 JO . (3 7/30), ; (dSf'dv), = 1/6. (dUfdv), +p/0. But for the same reason we have also and further 3(3S/30),/3t;=l/0. 3(3 tf/30),/ 30, 3(3S/3) 9 /30= 1/0. 3(3C7/30) e /30- 1/0 2 . Whence it follows that (c) (3Z7/30) e = 2 . 3(p/0),/30, since U is also a function of and v only, and The internal energy must satisfy the differential equation (c). The second law furnishes the means of determining the internal energy. It follows from equations (c) and (b) that (d) J.dQ = (3//30X . d6 + () e = - 1 . If the pressure on the ends is increased by dp, so that the total pressure is A.dp = P, and if there is no communication of heat, the temperature of the body increases by d9 = 6PL^!JMc p , or, if m is the mass of unit length, by dd=B/3P/Jmc p . If the cylinder is stretched by the force P a corresponding cooling will occur. SECTION CXV III. VAN DER WAAL'S EQUATION OF STATE. The equation of state of an ideal gas is pv = B&, and its isothermal curve is therefore a rectangular hyperbola. Real gases, however, at low temperatures and under high pressures, do not conform to this equation. Suppose that a certain quantity of gas at a given temperature has the volume OC' (Fig. 140) and is under the pres- 284 THERMODYNAMICS. [CHAP. xni. PG sure CO'. If the pressure is increased while the temperature remains constant, the volume will be diminished. At last the space in which the gas is contained becomes saturated with it ; let the corresponding pressure be DD'. DD' is then the pressure of the saturated vapour or the vapour pressure at the given temperature. If the volume is still further diminished, the pressure remains constant, while a part of the vapour passes over into the liquid state. At last all the vapour is transformed into liquid; let the corresponding volume be OF. So long as the vapour and liquid are in the same space, the isothermal curve is a straight line parallel to the axis Ov. If the volume of the liquid is now diminished, the pressure increases very rapidly ; the corre- sponding isothermal curve is represented by FG (Fig. 140). Andrews found, by experimenting with carbon-dioxide, that, as the temperature rises, the line DF becomes shorter, and that, at a certain temperature, which he called the critical tempera- ture, it disappears altogether. If the temperature of the carbon- dioxide remains constant, its state changes along curves which are represented in Fig. 141. The abscissas represent volumes, the ordinates pressures. For example, let us examine the isothermal ^~> TV ~, curve ABCD, which corresponds to the temperature 15'1C. ; at the point A the carbon-dioxide is still in the gaseous state; at B it may be considered as saturated vapour. If the compression is continued, condensation begins, and the pressure remains constant until the substance has become liquid, that is, until its state is represented by C. From C on, the pressure increases very rapidly as the volume is diminished. At the tem- perature 21-5C. the condensation begins at B, and the horizontal part of the curve is shorter. At 31'1C. the horizontal part of the isothermal curve vanishes ; the critical temperature has now been reached. Isothermals corresponding to higher temperatures are continuous curves ; it is therefore impossible to reduce carbon-dioxide to the liquid state at a temperature higher than 31-1C. At higher temperatures than this there is no apparent difference between its liquid and gaseous states. The liquid, at the critical temperature, has the same density as the saturated vapour. A gas can be reduced SECT, cxvni.] VAN DEE WAAL'S EQUATION OF STATE. 285 to the fluid state by compression only when its temperature is lower than the critical temperature. James Thomson substituted for the isothermal curve here described a continuous curve CDHEJFG (Fig. 140); the part DHEJF cor- responds to an unstable state. It appears from various investigations on the relations between vapours and their liquids at the boiling point, that it is possible to obtain a vapour in the states represented FIG. 141. by DH and FJ, while the states represented by HEJ are always unstable, since in these states the pressure and volume change in the same sense. J. Clerk Maxwell called attention to an important peculiarity of these isothermals which may be deduced by applying the laws of thermo-dynamics. If a gas traverses the cycle FEDHEJF, in passing along the straight line from F to D it receives the quantity of heat 286 THERMODYNAMICS. [CHAP. xm. L, and in passing along the curve DHEJF gives up the quantity L' ; the temperature is the same along both paths. Since the gas has traversed a complete cycle, we have \dQ/0 = L/0 - L'/O = 0, and therefore L = L'. Therefore, since no heat is used in this cycle, no work can be done, and hence the surface FJE is equal to the surface DEE. Thus, if the isothermal curve CDHEJFG is given, the maximum pressure of the vapour can be determined, by determining the line FD so that the surfaces FJE and DHE are equal. Van der Waals has proposed an equation of state for gases, which represents, more exactly than the simple one, the behaviour of the gas and which permits the calculation of the critical temperature. The volume of a gas is determined not only by the external pressure but also by the attraction of its molecules ; we may think of this attraction as replaced by a pressure p added to the external pressure p. Since the attracting and attracted molecules approach one another as the density increases, p' must be directly proportional to the square of the density, and therefore inversely proportional to the square of the volume. Hence we set p' = a/v*, so that the total pressure of the gas is p + a/v 2 . Further, the molecules are not free to move everywhere in the region v, for they themselves occupy part of the region. Van der Waals assumed that the volume of a fluid cannot fall below a certain limit without the particles losing their freedom of motion. In place of the apparent volume v, he used, as the true or effective volume, v - b, where b is a very small quantity, though much greater (about 4-8 times) than the volume of all the molecules of the gas. We thus obtain the equation of state If the volume v is very great, this equation becomes the equation of state of ideal gases. The positions of the points H and / (Fig. 140), at which the tangents to the isothermal curves are parallel to the r-axis, are obtained from the equation dpfdv Q or (b) p + a/iP 2a(v fc)/t' 3 = 0. This is the equation of the curve which passes through all points at which the tangents to the isothermal curves are parallel to the axis Ov. All these isothermal curves correspond to temperatures at which the body can be either liquid or gaseous. When the two points coincide we reach the critical state. But since the two coin- cident points must have a line joining them parallel to the axis Oi; we introduce the condition for the critical state by setting dp/dv, obtained by differentiating the foregoing equation (b), equal to zero, or (Z/0)/c>0 = 1/0 . (3 UJW +p . Hence it follows that (h) U^ - U^(v^ - 2 ) . . 3p/30 -p(v 1 - 2 ), and thus the difference between the internal energy of the vapour and that of the fluid is determined. We obtain from equations (d) and (h), (i) JL = (v l -v z )B . 'dp/W. Now we know by observation that ^ > v , so that 3p/30 is positive. The boiling point is therefore higher, the higher t)ie pressure. We may also apply equation (i) to the process of melting. In that case i\ denotes the volume of the liquid, v z that of the solid. We must here distinguish between two kinds of substances, those 292 THERMODYNAMICS. [CHAP. xm. like wax, whose volume increases during melting, and those like ice, whose volume diminishes during melting. For the former i\ > r. 2 , and therefore 'dpI'dB is positive; for the latter i' l . By dissociation two molecules of hydriodic acid form one molecule of hydrogen and one of iodine, hence the ratio between dn, dn v dn. 2 is - 2 : 1 : 1. If we set generally dn : dn-^ : dn. 2 : ... = v : v l : v 2 : . . . , the condition of equilibrium becomes (g) v . oQ/dn + Vl . 3*73^ + v 2 . 3*/3 3 + . . . = 0. From equations (f) and (g) we obtain (h) 2[v(c + l)(log 6 - 1) - v log P - v log C+ vk - vh/ff] = 0. To simplify the calculation Planck assumed that the atomic heat is constant even in the compound gas, and that the molecular heat is equal to the sum of the atomic heats ; experiment shows that this is approximately true in all cases. If a, a 1? a 2 denote the number of atoms in the molecule of each gas, we may set C = ya, C t = ya v c. 2 = ya. 2 .... Since the whole number of atoms is unchanged by dissociation, the sum na + n l a l + n 2 a 2 + ... is constant, and hence a.dn + a l . dn^ + a 2 . dn. 2 + . . . = 0. Consequently also va + v^ + v 2 a 2 4- . . . = and vc + v l c l + v. 2 c. 2 + . . . = 0. Further if we set v + v l + v. 2 + v 3 + ... =V Q ; vh + v 1 h l + vji 2 + ... = it follows from (h) that SECT, cxxi.] DISSOCIATION. 297 Hence for hydriodic acid we have v = 0, and C l C z IC z = k .h. If no hydrogen or iodine is present except that set free by dis- sociation, we have n l = n 2 , and therefore C l = C. 2 and C 1 /C=-Jk .h 1 ' e . Now (7 1 /C'=n 1 M. Hence the degree of dissociation is independent of the pressure, but increases with the temperature. In this case, however, the dissociation can never become complete, since, for = o> , we have C l /C=Jk^. From equation (i) the pressure has no influence on the degree of dissociation if the total volume remains unchanged by the dissociation. This is the case when v = 0. If, on the other hand, the volume increases during the dissociation, any increase of the pressure will lessen the degree of dissociation. This occurs in the case of nitrogen- dioxide, N 4 , in which one molecule is broken up by the dissociation into two molecules NO 2 . Hence v= - 1, Vj = 2, and therefore c^/c=k .h^.e/p. This equation, together with C l + C=\, determines the degree of dissociation. In order to occasion the dissociation determined by the quantities dn, dn v dn 2 , at constant temperature and at constant pressure, the quantity of heat dQ is required, which is determined by or, from equations (b) and (c), by J . dQ = 2(c0 + h)dn + 6 . *2dn. Since -ir = 0, the quantity of heat required for the dissociation determined by the quantities v, v v v 2 ... is determined by (k) J.Q = vh + v 1 h l + vJi 2 +... + v 0=v 8-logh . We reach the same result from the equation J.dQ=B.dS together with the relations (d) and (h). CHAPTER XIV. CONDUCTION OF HEAT. SECTION CXXII. FOURIER'S EQUATION. IF the temperatures of the different parts of a body are different, a gradual change goes on until the temperature of all parts of the body is the same, that is, until equilibrium of temperature has been reached. In this statement it is assumed that the body neither receives heat from surrounding bodies nor gives up heat to them. The rate at which the condition of equilibrium is reached depends upon the facility with which the body conducts heat. Without making any assumptions on the nature of heat, we may say that heat flows in a body until a state of equilibrium is reached. We define the rate of flmv of heat in any direction, as that quantity of heat which passes in unit time through unit area perpendicular to that direction. Hence, if Q represents the rate of flow of heat through an area dS within the body, the quantity of heat which will pass through that area in the time dt is Q . (IS . dt. If Z7, V, W are the components of flow in the directions of the coordinate axes, the quantities of heat which pass through the elementary areas dy . dz, dx . dz, dx . dy, in the time dt, are U.dy.dz. dt, V . dx . dz . dt, and W.dx.dy.dt respectively. Using the general equations (XIV.) of fluid motion, we obtain for Q (a) Q = lU+mV+nll f ', where /, m, n are the direction cosines of the normal to the elementary area dS. Let 00' (Fig. H2) be a rectangular parallelepiped, whose edges OA = a, OB = b, and 00 = c are parallel to the coordinate axes. If U, V, W represent the components of flow at the point 0, those at A are U+'dU/'dx . a, F+^Vj'dx.a, W+'dWJ'dx.a respectively, supposing a, b, c so small that only the first terms in the expansion CHAP. xiv. SECT, cxxii.] FOUEIEE'S EQUATION. 299 need be retained. The parallelepiped receives the quantity of heat U.dt.bc in the time dt through the surface OB AC, and loses the quantity (U+'dU/'dx. a)bc. dt, which flows out through the surface AC'O'B' in the same time. The parallelepiped gains, on the whole, the quantity - 9 U/'dx . a.bc.dt. If we take account of the other surfaces, the quantity of heat which remains in the parallelepiped is -(Bl7/aa;+3F/3y+a#7a).a6e.& J or, if we set a.b.c = dv, - (d U/dx + "d Vj-dy + 3 Wfiz)dvdt. This quantity of heat raises the temperature of the parallelepiped by d6, which, if c denotes the specific heat of the body, and p its density, is determined by the following equation, (b) cp . dd = - This equation holds only if the heat received is used solely in causing change of temperature and does not produce any change in the state of aggregation or any chemical change. Sometimes, too, heat exists in the interior of a body which has not penetrated into it in the form of heat, but is produced by friction or by an electrical current in the body ; and to this the above equation does not apply. The components of flow U, V, W depend on the distribution of heat in the body and on the nature of the body. If the body conducts heat equally well in all directions, that is if it is isotropic, we may determine the rate of flow in the following way. Let A and B be two points within the body infinitely near each other, in which the temperatures are respectively 9 and &, If dv denotes the distance between the points A and B, and k the conductivity of the body for heat, the rate of flow of heat in the direction AB is 300 CONDUCTION OF HEAT. [CHAP. xiv. given by Q = k(0- &)/dv. Hence the condi.i<-tint>/ is the quantity of heat which flows in unit time through unit area of a surface in the body, parallel with and between two surfaces whose temperatures differ by 1, and which are distant from each other by one centimetre. Now since & = 6 + ddjdv . dv, we have also (c) Q= -k.dOldv. We obtain in like manner for the components of flow U, V, IV, the expressions (d) U= -k.W/Vx, V= -k.'dBj'dy, fT= -k.'dO/'dz. In actual cases the conductivity k is a function of but for the sake of simplicity we will assume that k is constant. We obtain from (b) and (d) (e) cp . Wfit = k(d-epx 2 + 3 2 0/9/ + 3 2 0/3z 2 ). This equation was first given by Fourier, and is therefore called Fourier's equation. The specific heat c, the density p, and the con- ductivity k are functions of 6; we will, however, consider them constant. Fourier's equation may also take the form (f) 30/ctf = K 2 (9 2 0/9z 2 + 3 2 0/3/ + 9 2 0/^ 2 ), where (g) K z = k/cp. In the following table the values of k and K for several metals at the temperatures and 100 C. are given from the experiments of L. Lorenz : N> *m *0 AT 10 Copper, . . 0,7198 0,7226 0,909 0,873 Tin, . . . 0,1598 0,1423 0,392 0,344 Iron, . . . 0,1665 0,1627 0,202 0,179 Lead, . . . 0,0836 0,0764 0,242 0,222. SECTION CXXIII. STEADY STATE. The state of the body with respect to heat is called steady, if the temperatures of the different parts of the body are different, but do not change with the time. In this case each particle gives up on the one side as much heat as it receives on the other, and the temperature is independent of the time t and dependent only on the coordinates x, y, z. For the steady state, equation CXXII. (f) becomes (a) 3 2 0/d.c 2 + 3 2 0/3/ + 3 2 ^ 2 = V 2 # = 0. The components of flow are expressed by equations CXXII. (d). Flow of lieat in a phite. We will consider a thin plate whose faces L and M are parallel to the yz-plane. The temperatures of the faces are respectively 6 l and 2 . This being so, the flow of heat is parallel to the .r-axis, and the temperature in the vicinity of the .T-axis SECT, cxxin.] STEADY STATE. 301 depends only on x, so that from (a) we have d 2 0jdx 2 = 0. Hence 0=px + q. If the distances of the faces L and M from the y^-plane are a and b respectively, we have 1 =pa + q, 2 =pb + q, and further 6 = (bO } - a0 z )f(b -a)- (6 1 - 8 2 )x/(b - a). If we represent the distance b - a between the faces by e, the rate of flow of heat U between them is (c) 17=^-0^/6. Every integral of equation (a) corresponds to a steady state of heat. If 8=f(x, y, z) is an integral of (a), 1 =f(x, y, z) and 2 =f(x, y, z) are the equations of two surfaces of constant temperatures, or of two isothermal surfaces, where O l and 2 are constant. If the body is bounded by the surfaces which are determined by l and 6. 2 , and if 6 is a temperature which lies between l and 2 , 0=f(x, y, z) is the equation of any isothermal surface. The flow of heat in a sphere. If m and c are constant, and if r 2 = x 2 + y 2 + z 2 , = m/r + c is a solution of (a). Therefore setting l = m/r l + c, 9 = m/r. 2 + c, we >-^-^ as the equation of the system of isothermal surfaces, which in this case are spheres. For the rate of flow of heat U in the direction r, we have (e) U= -k. d0/dr = k(0 l - 2 )r l r 2 /r 2 (r< i - 1\). The temperature and flow of heat in a hollow sphere, whose internal and external surfaces are at the temperatures l and 2 respectively, are also given by equations (d) and (e). The total quantity of heat which flows out through the hollow sphere is ^Trr 2 U=4irk(0 l - 0. 2 )r l r 2 /(r 2 - r a ). The flow of heat in a tube. If c and c are constants and if r 2 = x 2 + y-, we have, from XV., = clogr + c' as an integral of (a). Therefore, if we set 6 l = c.\ogr l + c', 2 = c . log r. 2 + c', we obtain (f ) 8 = (0 l - 2 )log /-/(log r, - log r z ) + (^log r 2 - 2 log r 1 )/(log r z - log r,}. The rate of flow U in the direction r is U=k(8 l - 8 2 )/r(\ogr 2 -\ogr l ). The quantity of heat which flows out through a unit length of the tube is (g) 2irrCT=2^(0 1 -0 2 )/(log3-logr 1 ). SECTION CXXIV. THE PERIODIC FLOW OF HEAT IN A GIVEN DIRECTION. If the temperature of the body depends only on one coordinate, say on x, Fourier's equation becomes (a) 'dO/'dt = K 2 . 'd 2 0/'dx 2 . We will hereafter investigate in what way this equation can be integrated. 302 CONDUCTION OF HEAT. [CHAP. xiv. For the present we will consider special integrals which correspond to simple yet important cases. The temperature of the earth changes during the year ; it rises and falls with the temperature of the air. The time at which the maximum or minimum temperature at any point is reached is later as the point lies further below the surface. In the following dis- cussion we will not take into account the internal heat of the earth. If the temperature at the earth's surface is given by (b) = sin at, we may express the temperature at a point in the interior of the earth by (c) 6 = P . sin at + Q . cos at, where P and Q are functions of the distance x of that point from the earth's surface. If we substitute for 9 in (a) the expression (c) we have Pa. . cos at-Qa. sin at = K 2 (sin at . d^P/^x 2 + cos at . d^Q/dx 2 ). Hence we must have *c 2 . d 2 P/dx 2 = - Qa and K*.d 2 Q/dx z = Pa. Now if we set e 2 = a/* 2 , we have (d), (e), d*P/dx* = - e 4 P and Q = - 1/c 2 . d 2 P/dx?. In order to integrate equation (d) we set P = Ae px , and obtain p = f$J ' - 1. The integral of equation (d) then takes the form Since must equal when x = GO , we have A=B = Q, and hence We obtain from equation (e) Q = (C- 1 + V ^* X 1 V *- J) e (-l- But from equations (b) and (c) we have P 1 and Q = when x = 0, and therefore C=D = . Hence we obtain p = e - ex/v/r ^ cos (^2) ; Q=-e- /^ . s i n and 6 = e- /^ . sin (at - ee,>/2). Substituting for e its value, we have 6 = e -*/ . sin (at - *//*). The difference between the highest and lowest temperatures at the depth x below the surface is therefore 2e~ xV W K . This difference depends on the value of a. The faster the temperature changes at the surface the smaller the influence of this change on the tem- perature in the interior. For example, if we set the temperature at the surface equal to = 8in(27r2/r), the difference between the highest and lowest temperatures is equal to '2e- x ^^l K , and this is very much greater when T is a year than when it is a day. SECT, cxxv.] A HEATED SURFACE. 303 The temperature relations within the earth are actually different from those here described, because the temperature at the surface cannot be expressed in any simple way. The main features of the phenomena, however, are similar to those deduced in this discussion. SECTION CXXV. A HEATED SURFACE. Let the temperature in an infinite body at the time t = be every- where zero, except in a plane in which each unit of area contains the quantity of heat o-. Fourier showed that at the time t the temperature 6 at a point at the distance x from the heated plane is given by ~ where k is the conductivity and K the quantity defined in CXXIL (g). We will now examine whether this expression for 6 satisfies all the conditions of the problem. We will first consider the differential equation W/dt = K 2 . 3 2 0/3x 2 . From (a) we obtain (b), (c), (d) It follows from (b) and (d) that the differential equation is satisfied. Since the function ze~^ approaches zero as its limit if z becomes infinitely great, it follows that for / = we have = 0, for all values of a-, with the exception of the value x = Q. If 6 is determined by the equation (a), we can further show that each unit of area of the heated surface S contains the quantity of heat cr at the time t = 0. The total quantity of heat which is present in the body is given by the expression But because (e) I e~**dq = the quantity of heat present at any time must be So-, and since this quantity is present on the infinite surface 8 at the time 2 = 0, the unit of surface at that time must contain the quantity o-. It follows from (a) that 6 = for t = as well as for t = ; there- fore there must be a certain time at which 6 is a maximum. This time is found from (f) = 0, which gives t = x z /2K 2 . The corre- sponding value of is (g) 6 = \/\/2ire . a-fcpx. It appears from equation 304 CONDUCTION OF HEAT. [CHAP. xiv. (a) that the heat is propagated with an infinite velocity, since is everywhere different from zero as soon as t has a finite value. We will now determine the temperature at any time in a region in which the original distribution of heat depends only on one of the coordinates. Let 0=f(a) when / = 0, where a is the distance from the y^-plane. The part S of the region which is bounded by two parallel planes for which x = a and x = a + da, contains the quantity of heat So- = S.da. pc ./() Therefore the quantity o- which is present in the unit of area of this sheet is a- = da . pc ./(). If the temperature of the rest of the region is zero, the heat flows out from this sheet on both sides, and at a point whose distance from the f/2-plane equals x, and which therefore is at the distance x-a from the shell, the temperature by (a) is ,70 -JL ^fr All other similar sheets emit heat according to the same law, and we therefore have If we set (i) q = (a-x)/2K,Jt, we have 6 = 1 / N /TT . [ + The expressions (h) and (i) contain the complete solution of the problem before us. If we set t = in equation (i) and make use of (e), it follows at once that 6=f(x). For example, if the initial temperature is constant and equal to within the portion determined by -I ) 2 +(e-c)W 2 < .y( a> ^ c )dadbdc. J QO J-oo J-oo This expression for B is an integral of the differential equation (b) Wf-dt = K 2 (3 2 0/3.r 2 + 3 2 0/3/ + 3 2 0/3z 2 ). We notice that the integration of this equation depends on that of the simpler equation (c) 'dX/'dt = K-3 2 JT/3x 2 . For if X is a function of x and t t which satisfies equation (c), and if Y and Z are functions of y, t and z, t respectively, which satisfy equations analogous to (c) for y and z, the equation = XYZ satisfies (b). We have YZX+ XZY+ XYZ= K\YZ . 9 2 JT/3z 2 + XZ . ^Y/^f + XY. ^Z/^). It follows from (c) and the analogous equations for y and z that this equation is satisfied, from CXXV. (a), by X= l/Jt. g -C*-W, hence the expression \IJt.e-b- W*^ .Ijjt.e-to- *> W .\IJt.e-to- c ^ K ' 2f is an integral of equation (b). Therefore, also, e = Cf / + "I + "T"{ 1 ^ 3 ' e- {(x - a ? +(>J - b ? +( *- c KU -/(a, b, c,)dadbdc is an integral of equation (b). C is a constant, and /(a, b, c) an arbitrary function of a, b, c. If we set a = (a-x)l-2K^t, /3 = (b-y)/2 K Jt, y = (c-z)!'2 K ,Jt, it follows that = ( 2K ) 3C f +X f^f^ e ~ at ~^-^f(x + Znajt, y + '2> /7r) 3 , and obtain The expressions (a) and (d) are identical, as may be shown by the substitution already employed. SECT, cxxvin.] THE FORMATION OF ICE. 307 SECTION CXXVIII. THE FORMATION OF ICE. Suppose that the temperature of a mass of water is everywhere = 0, and that the surface of the mass is in contact with another surface whose temperature is - . 6 may be either constant or variable, but must be always below zero. A sheet of ice will be formed under this surface, whose thickness e is a function of the time t. The temperature Q of the mass of ice is itself a function of t and of the distance x from the surface. For x = e, is always equal to zero. The equation (a) Wfftt**K*&Ofdx* holds everywhere within the mass of ice. New ice will form continually on the bound- ing surface of the ice and water. The quantity of heat which flows outward through unit area of the lowest sheet of ice is given by k~dd/'dx.dt. During the same time a sheet of ice, whose thickness is de, is formed, and the quantity of heat set free thereby is Lpde, where L represents the heat of fusion of ice arid p its density. When x = e, we have (b) kWfdx = Lpd/dt, or We may write for 6 the expression As may easily be seen, this expression satisfies equation (a). It also satisfies the condition that = when x = e. In order to find whether it satisfies the condition contained in (b), we differentiate (c) with respect to x, and obtain When x = e this becomes equation (b). Since, at the surface, 6= 6 , it follows from (c) that If the thickness of the sheet of ice is given as a function of the time t, # may be easily determined ; on the other hand, if is given, it is in general difficult to determine e. *This solution was communicated to the author by L. Lorenz. See also Stefan, Wied. Ann., Bd. XLIL, S. 269. 308 CONDUCTION OF HEAT. [CHAP. xiv. If is constant, the right side of equation (d) must also be con- stant. This condition is fulfilled if 2 //c 2 = 2p 2 t, where p is constant. From (d) we then obtain the equation which serves to determine p. In order to put the series in (e) into a finite form, we form from (e) and thus obtain d(cd /Lp)/dp = 1 + cBJL. The integral of this equation is (f) cejL^pTe-^-^^da. If the thickness of the sheet of ice increases in direct ratio with the time t, that is, if = JK/, where q is a new constant, it follows from (d) that rAft n 6fS ^o/ = ^+fVr 2 3 + "'' or (g) cOJL = f-\. If e is very small, we obtain from (d) c6 /L= 1/2/c 2 . d(?/dt, and hence (h) e 2 = 2k/ Lp . f 6 dt. This result also follows if we set the flow of heat upward equal to kdjt, in which, however, we assume that the temperature in the ice increases uniformly from its upper surface downward. On this assumption, the quantity of heat kO^.dtje. flows upward through the ice in the time (It. In the same time a sheet of ice, whose thickness is dt, is formed, and the quantity of heat Lp.de is set free. Hence we have kO Q . dt/t Lp . de. This equation leads to the result we have already obtained. If (} is con- stant, it follows that (i) f = j2 SECTION CXXIX. THE FLOW OF HEAT IN A PLATE WHOSE SURFACE is KEPT AT A CONSTANT TEMPERATURE. It is in general very difficult to determine the variations of tem- perature in a limited body. We will discuss a few cases in which it is possible to solve this problem. Suppose that the temperature in the interior of a plate bounded by parallel plane faces is 6 =/(), where x denotes the distance of the point considered from one of the faces of the plate. From the time t = on, the surfaces are supposed to be in contact with a mixture of ice and water, or to be so conditioned that their temperature is kept at zero. The law SECT, cxxix.] THE FLOW OF HEAT IN A PLATE. 309 according to which the temperature changes in the interior of the plate is to be determined. Designating the thickness of the plate by a, we have for t = 0, 0=f(x) ; for t = oo , = 0; for x = 0, = 0; for x = , = 0. The rate at which the temperature changes at the surface is infinitely great; just outside the surface it is equal to zero, while just within it, at one face, it is equal to /(O). At the other face the temperature outside the plate is also zero, and within it f(a). The function must satisfy not only these conditions, but also the differential equa- tion (b) 'dd/'dt = K 2 3 2 0/2te 2 . An integral of this equation is (c) 6 = e~ m ' K2t (A sinmx + Bcosrnx). From (a) B = 0, so that (d) = Ae~ mVt sin mx. This value of satisfies not only equation (b), but also vanishes for x = 0. Since is also zero when x = a, we must have sin ma = 0, and therefore ma= pir, where p is a whole number. Hence we have (e) = Ae-#****l<* . sin (pxirja). If we notice further that 6=f(x) when ^ = 0, we have (f ) f(x) = A sin (pTTX/a). In general, the function /(x) can not be represented by this expres- sion. To solve the problem we use the following method. Since the expression (e) is an integral of Fourier's equation, the complete integral is obtained by taking the sum of the similar expres- sions, which are obtained by giving p all values between 1 and oo . The terms which correspond to a negative value of p differ from those terms for which p is positive only in sign, and can therefore be considered as contained in the latter. Hence we set (g) 8 = A l sin fax/a) . e - **&!* + A 2 sin (2vx/a) . e ~ 2ViA/a* + _ When t = Q, we have =/(), so that for 0)]. Applying the above given summation formula, we find that this factor is equal to zero. In the same way the factors of A 3 , A v etc., vanish, and we obtain finally A l = 2/n. \_f(a/n) sin (77/71) +/(2a/n) . sin ( 2ir/n) + . . . +f((n- l)a/7i)sin((n- l)r/)]. In general we have, for 00 we obtain (f ) B m = 2/a . I a f(x) . cos (mirxla)dx and =l/a. ["/(zjdx. We obtain B Q by multiplying both sides of (e) by dx and integrating from to a. If f(x) is an even function, the series holds within the limits -a/o /(a)cos(m7r(2 - a)/aVo. m=l J -a Now setting wi7r/a = A, and therefore 7r/a = dX, it follows that (o) /() = I/TT . jf dx|_ + /(a)cos( A(a; - a))^, where -co < a; < ao . Instead of this equation we may often use one of the two which are obtained from (g) and (h). The general term in (g) is sin (mirx/a) . f /(a) sin (mira/a)da, and hence f(x) = 2/7T . IT I a . ^ sin (mirx/a) I /(a) sin (mTra/a)da. Now if we set mir/a = A, and therefore via = d\, we have for < x < oo , (P) /(*) = 2/7T . l"d\. . sm(Xx)j /(a)sin(AaXa. From (h) we obtain in the same way for < x < oo , (q) /(*) = 2/7T . {d\ cos(Aa;) . [7(a)cos(Aa)^a. SECT, cxxxi.] APPLICATION OF FOURIEK'S THEOREM. 315 SECTION CXXXI. THE APPLICATION OF FOURIER'S THEOREM TO THE CONDUCTION or HEAT. If the temperature in a certain region depends only on the x-co- ordinate, the temperature 6 must satisfy Fourier's equation (a) 30/3* = K 2 3 2 0/3a; 2 . From CXXIX. (c) 6 = e~ x * K2: (A sin Xx + Bcoskx) is an integral of equation (a) where A, A, B are constants. We may also give the expression for 9 the form 6 = e~^ K2t cos (\(x - a)) . /(a), where /(a) is an arbitrary function of a, and A. and a are constants, which may take all possible values. Any sum of such terms satisfies the equation, and as the integral (b) = I/TT . J^Ae-^'[ + 7(a)cos(A(z - a))da is such a sum, it will also satisfy the equation. But when / = 0, we have 6>=1/7T. {d\f*f(a)cos(\(x-a))da, from which, by comparison with CXXX. (o), we obtain 6=f(x). The formula (b) contains the solution of the problem, to determine the temperature in a body at any time t, when the temperature is given, at the time = 0, by 6=f(x). This problem has already been solved in another way in CXXV. (h) and (i). We proceed to show that the solution hero given is identical with the former one. Since (c) = 1 / . f + J(a)da /" V *** cos ( X(x - a))d\, we first determine the value of the integral If we develop cos (A(z- a)) in a series, this integral is represented by It follows by integration by parts that / V XU% A*V*A. = (2n - l)/2* 2 and by continued reduction 316 CONDUCTION OF HEAT. [CHAP. xiv. But because j e~ f dq = \,Jir t we obtain The value of the integral sought is therefore or -- 2*^ If we replace the integral in (c) by this value, we obtain This expression for 6 is identical with that given in CXXV. (h). We will now apply Fourier's theorem to find the law of penetra- tion of heat into a body. For this purpose we will consider the simple case in which the body is in contact over a plane bounding surface F with another body whose temperature is constant and given. Let the original temperature of the cold body be zero. If we proceed as before and use CXXX. (p), we obtain B = + 2/?r . I "dX sin ( Aa-) or t. 2 /t l = x 2 2 /x l 2 , that is, the times required for two points to attain the same temperature are proportional to the squares of the. distances of the points from the heated surface F. AVe will now determine the quantity of heat which flows into the cooler body through unit area in unit time. For this purpose we give the equation (f) the form = 20J,jTr. [f(ao ) -f(x/2K,Jt)], from which follows, by (f), - kW/'dx = kOJitj^t . e~ zt: ^. Setting x = 0, we find the quantity of heat U desired, (g) U = k6 / K-Jiri. By the help of equation (g) we may solve an important problem. Two bodies L arid L' are in contact over a plane surface, the tem- perature of one of these bodies being T, that of the other T. If the two bodies are brought in contact, one of them is heated and the other is cooled. We can also determine the temperature 2\ of the surface of contact. Assuming that T is constant, the quantity of heat which L receives in unit time is, from (g), given by U=k(T ~T)lKj*t, In the same time, L' receives the quantity of heat U'=k'(T,-T')/ K '^t, where k' and K' have the same meaning for L' as k and K for L. But since the infinitely thin bounding surface can contain no heat, U+U' must equal zero, or k/n . (T - T) = k'/K . (T - T ), from which follows (h) * T = (Tjkcp + T*JJMji)l(Jkcp + */kVfi). It is thus shown that the assumption is correct, that the temperature in the bounding surface between two bodies which meet in a plane surface is constant. Strictly speaking, the bodies in contact must both be infinitely large, * L. Lorenz, Lehre von der Warme. S. 178. Kopenhagen, 1877. 318 CONDUCTION OF HEAT. [CHAP. xiv. but the formula (h) may also be applied to small bodies if we only consider them shortly after they are brought in contact. We may show from (h) that the temperature of a heated solid is very little diminished by contact with the air ; this holds for the metals and for good conductors in general. It follows from equation (h) that If T represents the temperature of the solid, p is always very much greater than p. Hence T -T is very much greater than T - T , especially since k is also greater than k', while c and c' are not very different from each other. SECTION CXXXII. THE COOLING OF A SPHERE. Let us suppose that the temperature at a point in the interior of a sphere depends only on the distance of that point from the centre of the sphere. In this case [CXXVI. (c)] Fourier's equation takes the form (a) 3(r0)/a* = K 2 3 2 (r0)/3r 2 . If m, A, B are arbitrary constants, an integral of equation (a) is rO = e-" . (A sin (mr) + B cos (mr)). But since this equation leads to the conclusion that = oo when r = 0, B must equal zero, and we obtain as the integral of equation (a) ft . sin (mr). If E represents the coefficient of radiation, the quantity of heat which radiates in unit time from an element dS of the surface is dS. E6. We will assume that E is constant. The same surface- element dS receives from the interior of the sphere in the same time the quantity of heat -k.dS.dO/dr. Since the quantity of heat which dS receives must be equal to that which it emits, we have (i) -k.dO/dr = E6 or -d6jdr = h6, where, for brevity, we set h = E/k. Hence, for r = R, 2^ TO x. This represents a line such as oa/3 (Fig. 143) which cuts the curve at o, a, /3. The abscissas 0, x^, x 2 ', x 3 ' ... of these points are roots of equation (1), and we have QK , since the temperature of the sphere is O n at the time t = 0. SECTION CXXXIII. THE MOTION OF HEAT IN AN INFINITELY LONG CYLINDER. Let the cross-section S of the cylinder be so small that its tem- perature 6 is constant, and let A arid B be two cross-sections separated by the distance dx. The quantity of heat - Sk . Wfdx . dt flows through A in the time dt, and the quantity - SktfO/'dx + 3 2 0/3a; 2 . dx)dt flows through B in the same time. Hence the part of the cylinder between A and B receives the quantity of heat Sk . 3 2 0/3z 2 . dxdt. A SECT. CXXXIIL] THE MOTION OF HEAT. 323 part of this heat is given up to surrounding bodies by conduction or radiation. If P is the perimeter of the cylinder, E a constant, and if the temperature of the medium around the cylinder is 0, the heat given up by conduction or radiation is PEO . dxdt. Another portion of the heat received serves to heat the cylinder ; this portion is S . dx . cp . dO. Hence we obtain the equation Sk.-d*-efdx' 2 = Spc.'d6/-dt + PEe, or (a) 90/3/ = K 2 . c>20/3x 2 - A0, if we set K* = k/cp and h = PE/S P c. If the state of the cylinder or rod has become steady, we have 30/3/ = 0, and equation (a) takes the form * 2 . 3 2 0/3z 2 = h0. From this it follows that (b) 6 = A rfal 2/Cx/7T< -/O To simplify this expression, we set p = (a- x)/2i through its surface. 3. From the work done by the surface forces X^ Y^ ... on that part of the fluid which is situated on the surface of the element dw. 4. From the heat contained by that part of the fluid which flows through the surface-element ddt; e 1 is therefore the quantity of energy received by the unit of volume in the unit of time only through the influence of the accelerating forces. We will now investigate the values of e^ &>, . . . . We determine the work done by the accelerating forces in the time dt in the following way. The volume element contains the mass pdo> and moves in the time dt through the distance udt in the direction of the ar-axis. Thus the force X does the work pd receives heat by conduction. The com- ponents of flow of heat are [CXXIL] -Jk.VOfdx, -Jk.Wfiy, -Jk.-ddj-dz. If we set the quantity of energy thus received by da> in the time dt, equal to e^d^dt, and assume the conductivity constant, we have (h) e. a = J . SECT, cxxxiv.] CONDUCTION OF HEAT IN FLUIDS. 329 The increase in energy which d, and we therefore have (i) \ . d(fW)jdt + Jd( P Q)/dt = e 1 + e 2 + e, + e,+ e y Introducing in this equation the values found for e v e. 2 , e 3 , ...it follows that If internal friction exists in the fluid, we have from XLVII. (h) X x = -p + 2p. . 'duj'dx - ^(dufix + Zv/dy + "die fix) Z J , = fu(dwl'dy + 'dvl'dz), etc. By the help of these relations we may give equation (k) the form Jp(dQj r \ = - p(du/dx + dv + (dz;/^) 2 + (dw/-dz)* - I 2 + (du/oz For the determination of the motion and temperature of the fluid we have the five equations given under (a), (b), and (1). These five equations are not sufficient to determine the seven unknown quantities u, v, w, p, p, 9, and 6. We obtain two other equations in the follow- ing way. The total quantity of heat 9 contained by the unit of mass must depend on 0, and we assume that (m) 9 = c0, where c is the specific heat, a constant. If the fluid considered is gaseous, c denotes the specific heat of constant volume. The second equation must express the relation between density, pressure, and temperature. In the case of liquids, we may set approxi- mately (n) p = pJ(l+aB), where p Q is the density when = and a is a constant. But for gases, if V is the volume of the unit of mass at pressure p and temperature 6, V$ the volume of the same mass at pressure p and temperature 0, we have ^F"=j9 F (l +a0). Since Vp=\ and J> =1, we have (o) p/p =pJp Q . ( 1 + a0). The equation (o) in connection with (a), (b), (1), and (m) serves to deter- mine the unknown quantities. The complicated equations which determine temperature and motion in a fluid are very hard to integrate, so that up to this time no case has been completely solved. 330 CONDUCTION OF HEAT. [CHAP. xiv. SECTION CXXXV. THE INFLUENCE OF THE CONDUCTION OF HEAT ON THE INTENSITY AND VELOCITY OF SOUND IN GASES. We have the following equations [CXXXIV.] for the determina- tion of motion in a gas in which the temperature is variable : 1. The equation of continuity [CXXXIV. (a)], which may take the following form : 'dp/'dt + p(dufdx + 'dv/'dy + 'dw/'dz) + udpj'dx + vdpfiy + wdpfdz = 0. 2. The equations of motion [CXXXIV. (b)]. We replace in these equations the forces X a X y , ...by the values found in XL VII. (b) and (h), and obtain [cf. XLVIIL (a)] p(dufdt + uduj'dx + vdufdy + w'duj'dz) = pX- 'dpj'dx + ft V 2 ^ + If^^ufdx + 'dv/'dy + 'dwfdz)l'dx, and analogous equations for y and z. 3. The condition for the conservation of energy [cf. CXXXIV. (1)]. 4. The connection between the heat contained in the body and the temperature [cf. CXXXIV. (m)]. 5. The Boyle-Gay-Lussac law [cf. CXXXIV. (o)]. Let the velocity and change of temperature be very small quantities ; the same is then true of such differential coefficients as 'dp/'dx, 'dQ/'dx, etc., and we will therefore neglect the product of these quantities, that is, terms of the form udpj'dx, udu/~dx, itdQjdx, etc. The equa- tions 1 5 are then very much simplified. We obtain (a) 3(log p)j?>t + 'du/'dx + 'dv/'dy + 'dwj'dz = 0. .If we further set /t/p = //, equation (2), by use of (a), takes the form (b) 'du/'dt + 1 Ip . 'dp/'dx = // V 2 - /*' . 3 2 (logp)/ar3i5. Similar equations hold for u and v, if x is replaced by y and s respectively. Eliminating in equation CXXXIV. (1) by means of the relation Q = cO and introducing the heat equivalent A of the unit of work for l/J, it follows that (c) cp.Wj'dt-k\7-d = Apd(\op)l'dt. We have further the equation [CXXXIV. (o)], (d) p/p=p fp .(l + a8). We consider ///, k, and c as constants. We substitute p for p, if p or l/p occurs as a coefficient; we also substitute p for p in (c). In these substitutions we neglect only infinitely small quantities of the second order. Setting p = Po (l+ a -) ) we obtain (e) log p = log p + r > because is also a small quantity, (0 7>=/> SECT, cxxxv.] VELOCITY OF SOUND IN GASES. 331 Equation (c) now becomes Wfdi - k/cp . V 2 # = -dpo/ c Po d^/cM, and if we set K- = k/cp and 9 =^ cp Q B/Ap , we obtain from the last equation (g) Wfdt - K 2 V 2 = ftr/3*. By the use of (e) and (f) we may transform equation (b) into 'du/'dt +PQ/PQ . 'do-fa +p Q a/p . 'dB/'dx = /*' . V 2 - $/*' ^(r/'dt'dx. Introducing in place of 6 the quantity already defined, we have 'dv/'dt + b 2 . 'da-f'dy + (a 2 - The heat required to raise the temperature of a gram of air under constant pressure from 6 to + dO is equal to C . dd, if C is the specific heat at constant pressure. A part of this heat, namely c . dO, is used in raising the temperature, the other part is used in overcoming resistance during the expansion, by which the woAp*dP is done. We have therefore C.dO = e.dO + Ap.dT. It follows from the equation pF=p F' (l + ad), because p is here constant, that p.dV=pJP r c = b-JC/c. This value for the velocity of sound was found by Laplace. It differs from the value calculated in XXXV., which was originally found by Newton, and which in our present notation is b = Jp Q /p . The difference between the two formulas is due to the fact that in the first we have taken into account the heating of the air by compression and its cooling by expansion. Since the ratio C/c has been determined by direct experi- ment, the true velocity of sound in the air may be calculated. For atmospheric air at C,, C/c = 1,405 ; hence a = 33815 cm. This value agrees very well with experiment. Suppose that a plane-wave is propagated in the direction of the z-axis, and that K and /*' are not zero. The vibrations are parallel to the x-axis, so that v = Q and w = Q. Since u, 0, and cr are then functions of x and t alone, equations (k) become -dufit + 6 2 . "do-fix + (a 2 - V) 39/3* - /c 2 . 9 2 0/3.s 2 --= fa fit. The unknown quantities u, 0, and o- are periodic functions of t. We will represent by h a real magnitude, and by u', 0', and o-' three magnitudes which are functions of x alone. It is then admissible to make the assumptions (n) u = u' . e*", = 6'. e hit , a- = ar' . e*", where i = -J - 1. By the help of these equations we obtain from (m) hi