/!** ITY\ LIBRARY UMIVfltSI" OP I THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA GIFT OF Prof. G. C. Evans ELEMENTS OF THE THEOEY OF THE NEWTONIAN POTENTIAL FUNCTION Third, Revised and Enlarged Edition BY B. O. PEIRCE, PH.D. HOLLIS PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY IN HARVARD UNIVERSITY BOSTON, U.S.A. GINN & COMPANY, PUBLISHERS bcnatum J)tc00 1902 COPYRIGHT, 1886, 1902, BY GINN & COMPANY ALL BIGHTS RESERVED /MATH/ ST/AT PREFACE TO THIRD EDITION. THIS little text-book was written some years ago to accom pany the lectures in a short preparatory course on the New tonian Potential Function, especially intended for students who were afterwards to begin a systematic study of the Mathematical Theory of Electricity and Magnetism, with the help of some of the standard treatises on the subject. In preparing the present edition a few imperative changes have been made in the plates, some sections have been intro duced, and a large number of simple miscellaneous problems have been added at the end of the last chapter. The reader who wishes to get a thorough knowledge of the properties of the Potential Function and of its appli cations, is referred to the works mentioned in the list given below. Most of those that had then been published I con sulted and used in writing these notes, and from some which have appeared since the body of this book was electro- typed I have borrowed material for problems : many other problems I have taken from various college and university examination papers. I am indebted also to my colleagues, Professors Trowbridge, Byerly, E. H. Hall, Osgood, Sabine, M. Bocher, and C. A. Adams for valuable criticisms and suggestions. The slight use which I have made of developments in terms of Spherical Harmonics and Bessel s Functions is explained by the fact that students who use this book in Harvard Uni versity study at the same time Professor Byerly s admirable Treatise on Fourier s Series, and Spherical, Cylindrical dhid Ell ipso Ida I Ha nn on ics. iii 313 IV PREFACE TO THIRD EDITION. In the following pages the change made in a function u by giving to the independent variable x the arbitrary incre ment A#, and keeping the other independent variables, if there are any, unchanged, is denoted by A x w. Similarly, & y u and AgW represent the increments of u due to changes respectively in y alone and in z alone. The total change in u due to simultaneous changes in all the independent variables is sometimes denoted by Aw ; so that, if u =f(x, y, z), & x u \u & z u Aw = - Ax + -* - A?/ + - - &z + c, Ax Ay Az where e is an infinitesimal of an order higher than the first. The partial derivatives, -5-, -r-, -T-, are denoted, for conven- dx dy dz ience, by D x u, D y u, and D z u, and the sign = placed between a variable and a constant is used to show that the former is to be made to approach the latter as its limit. In those cases where it is desirable to draw attention to the fact that a cer tain derivative is total, the differential notation is used. dx It is tacitly assumed that the physical quantities under con sideration can be represented in the regions to which the theorems refer, by continuous point functions, having con tinuous derivatives of the orders which present themselves in the investigation in hand. In a few instances, as the reader will see, a theorem is predicated of analytic functions only, when so narrow a limitation is not required by the proof given. SHORT LIST OF WORKS ON THE POTENTIAL FUNCTION AND ITS APPLICATIONS. Bacharach : Abriss der Geschichte der Potentialtheorie. Bedell and Crehore : Alternating Currents. Betti : Teorica delle Forze Newtoniane e sue Applicazioni all Elettrostatica e al Magnetismo. Also W. F. Meyer s transla tion of the same work into German. Bocher: Reihenentwicklungen der Potentialtheorie. Boltzmann : Yorlesungen iiber Maxwell s Theorie der Elektricitat und des Lichtes. Burkhardt and Meyer : Die Potentialtheorie. Chrystal: The article "Electricity" in the Ninth Edition of the Encyclopaedia Britannica. Clausius : Die Potentialfunction und das Potential. Cumming : An Introduction to the Theory of Electricity. Dirichlet : Yorlesungen iiber die im umgekehrten Verhaltniss des Quadrats der Entfernung wirkenden Krafte. Edited by Grube. Drude : Physik des Aethers. Duhem : Lemons sur 1 Electricite" et le Magne"tisme. Ewing : Magnetic Induction in Iron and other Metals. Ferrers : Spherical Harmonics. Fleming: The Alternate Current Transformer. Franklin and Williamson: The Elements of Alternating Currents. Gauss: Allgemeine Lehrsatze in Beziehung auf die im verkehrten Yerhaltnisse des Quadrates der Entfernung wirkenden Anzie- hungs- und Abstossungskrafte. Also other papers to be found in Yolume V of his Gesammelte Werke. Gray: Absolute Measurements in Electricity and Magnetism. A Treatise on Electricity and Magnetism. VI SHORT LIST OF WORKS. Green : An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Harnack: Grundlagen der Theorie des logarithmischen Potentiales. Heaviside : Electrical Papers. Heine : Kugelfunctionen. Helmholtz : Wissenschaftliche Abhandlungen. Hertz : Gesammelte Abhandlungen. Jordan : Cours d Analyse. Joubert, Foster, and Atkinson : Elementary Treatise on Electricity and Magnetism. Kirchhoff : Gesammelte Abhandlungen. Yorlesungen liber mathe- matische Physik. Elektricitat und Magnetismus. Edited by Planck. Klein: Vorlesungen iiber die Potentialtheorie. Lame* : Legoiis sur les Coordonnees Curvilignes et leurs Diverses Appli cations. Mascart : Traite" d Electricite" Statique. Also Wallentin s translation of the same work into German, with additions. Mascart et Joubert : Lemons sur 1 Electricite" et le Magnetisme. Also Atkinson s translation of the same work into English, with additions. Mathieu : Theorie du Potential et ses Applications a 1 Electrostatique et au Magnetisme. Maxwell : An Elementary Treatise on Electricity. A Treatise on Electricity and Magnetism. Minchin : A Treatise on Statics. Neumann, C. : Untersuchungen iiber das logarithm! sche und New- ton sche Potential. Neumann, F. : Vorlesungen tiber die Theorie des Potentials und der Kugelfunctionen. Nipher: Electricity and Magnetism. Picard : Trait6 d Analyse. Poincare* : Electricite et Optique. Les Oscillations filectriqueg. Theorie du Potential Newtpnien. Riemann : Schwere, Electricitilt und Magnetismus. Edited by Hat- ten dorff. Routh : Analytical Statics. Schell : Theorie der Bewegung und der Kraf te. SHORT LIST OF WORKS. Vll Steinmetz : Alternating Current Phenomena. Tarleton : The Mathematical Theory of Attraction. Thomson, J. J. : Elements of Electricity and Magnetism. Recent Researches in Electricity and Magnetism. Thomson, W. : Reprint of Papers on Electrostatics and Magnetism. Thomson and Tait : A Treatise on Natural Philosophy. Todhunter : A History of the Mathematical Theories of Attraction and the Figure of the Earth. The Functions of Laplace, Bessel, and Lame. Turner : Examples on Heat and Electricity. Watson and Burbury : The Mathematical Theory of Electricity and Magnetism. Webster : The Theory of Electricity and Magnetism. Wiedemann : Die Lehre von der Elektricitat. Winkelmann : Handbuch der Physik. TABLE OF CONTENTS. CHAPTER I. THE ATTRACTION OF GRAVITATION. SECTION. PAGE. 1. The law of gravitation 1 2. The attraction at a point . . . ... . .1 3. The unit of force . . 2 4. The attraction due to discrete particles . . . . . - 2 5. The attraction, at a point in its axis, due to a straight wire . 3 6. The attraction, at any point, due to a straight wire ... 4 7. The attraction, at a point in its axis, due to a cylinder of revolution . . . . . . . 7 8. The attraction at the vertex of a cone of revolution, due to the whole cone and to different frusta . . . . .8 9. The attraction due to a homogeneous spherical shell ; to a solid sphere .11 10. The attraction due to a homogeneous hemisphere . . .13 11. Apparent anomalies in the latitudes of places near the foot of a hemispherical hill 15 12. The attraction due to any homogeneous ellipsoidal homoeoid is zero at all points within the cavity enclosed by the shell . 16 13. The attraction due to a spherical shell the density of which varies with the distance from the centre . . . . 18 14. The attraction at any point due to any given mass . . .19 15. The component in any direction of the attraction at a point P, due to a given mass, is always finite 21 16. The attraction between two straight wires . . . . 22 17. The attraction between two spheres .23 18. The attraction between any two rigid bodies . , . .24 Examples . .25 ix TABLE OF CONTENTS. CHAPTER II. THE NEWTONIAN POTENTIAL FUNCTION IN THE CASE OF GRAVITATION. SECTION. PAGE. 19. Definition of the Newtonian potential function . . .20 20. The derivatives of the potential function relative to the space coordinates are functions of these coordinates, which repre sent the components, parallel to the coordinate axes, of the attraction at the point (x, y, z) 30 21. Extension of the statement of the last section . . . .31 22. The potential function due to a given attracting mass is everywhere finite, and the statements of the two preced ing sections hold good for points within the attracting mass ........... 32 23. The potential function due to a homogeneous straight wire . 34 24. The potential function due to a homogeneous spherical shell . 35 25. Equipotential surfaces and their properties . . . .37 26. The potential function is zero at infinity . . . . .40 27. The potential function as a measure of work . . . .41 28. Laplace s Equation 44 29. The second derivatives of the potential function are finite at points within the attracting mass 45 30. The first derivatives of the potential function change continu ously as the point (x, y, z) moves through the boundaries of an attracting mass . . . . . . . .50 31. Theorem due to Gauss 52 32. Tubes of force and their properties ...... 55 33. Spherical distributions of matter and their attractions . . 56 34. Cylindrical distributions of matter and their attractions . . 60 35. Poisson s Equation obtained by the application of Gauss s Theorem to volume elements 61 36. Poisson s Equation in the integral form . . . .66 37. The average value of the potential function on a spherical surface. The potential function can have no maxima or minima at points of empty space ... . . .67 38. The equilibrium of fluids at rest under the action of given forces . . .- . . . . . . .70 Examples. . . . 71 TABLE OF CONTENTS. XI CHAPTER III. THE NEWTONIAN POTENTIAL FUNCTION IN THE CASE OF REPULSION. SECTION. PAGE. 39. Repulsion according to the Law of Nature . . . .75 40. The force at any point due to a given distribution of repelling matter . . . . . . . . . .76 41. The potential function due to repelling matter, as a measure of work v . . .78 42. Gauss s Theorem in the case of repelling matter . . .78 43. Poisson s Equation in the case of repelling matter ... 79 44. The coexistence of two kinds of active matter ... 80 Examples ... .. . . . . . .82 CHAPTER IV. THE PROPERTIES OF SURFACE DISTRIBUTIONS. GREEN S THEOREM. VECTORS. THE ATTRACTION OF ELLIPSOIDS. LOGARITHMIC POTENTIAL FUNCTIONS. 45. The force due to a closed shell of repelling matter ... 83 46. The potential function is finite at points in a surface distribu tion of matter . ... . .... .85 47. The normal force at any point of a surface distribution. The pressure on the surrounding medium ..... 88 48. Green s Theorem and its applications. Thomson s Theo rem. Dirichlet s Principle. Those properties of the potential function which are sufficient to determine the function . . ; " . .91 49. Surface distributions which are equivalent to certain volume distributions 109 50. Vectors. Stokes s Theorem. The derivatives of scalar point functions Ill 51. The attraction due to homogeneous ellipsoids . . . 117 52. Logarithmic potential functions . . ... . 120 Examples . . . . . . * . . .132 TABLE OF CONTENTS. CHAPTER V. THE ELEMENTS OP THE MATHEMATICAL THEORY OF ELECTRICITY. I. ELECTROSTATICS. SECTION. PAGE 53. Introductory ^5 54. The charges on conductors are superficial . . 146 55. General principles which follow directly from the theory of the Newtonian potential function . . . 143 56. Tubes of force and their properties . 150 57. Hollow conductors ...... 152 58. The charge induced on a conductor which is put to earth 156 59. Coefficients of induction and capacity . . . 157 60. The distribution of electricity on a spherical conductor . . 159 61. The distribution of a given charge on an ellipsoidal con ductor IQQ 62. Spherical condensers . . 161 63. Condensers made of two parallel conducting plates . . 164 64. The capacity of a long cylinder surrounded by a concentric cylindrical shell 166 65. The charge induced on a sphere by a charge at an outside point 167 66. The energy of charged conductors 171 67. Composite conductors .175 68. Specific inductive capacity. Apparent and real charges . 176 69. Polarized distributions. Magnets. Inductivity. Suscepti bility. Generalized induction. Polarized shells. Vector potential functions of the induction 185 II. ELECTROKINEMATICS. 70. Steady currents of electricity 222 71. Linear conductors. Resistance. Law of Tensions . . 226 72. Electromotive force 230 73. Kirchhoff s Laws. The Law of Divided Circuits. Wheat- stone s net 234 74. The heat developed in a circuit which carries a steady current 238 TABLE OF CONTENTS. PAGE. SECTION. 75. Properties of the potential function inside conductors which carry steady currents. Heterogeneous conductors . .241 76. Method of finding cases of electrokinematic equilibrium . 246 III. ELECTROMAGNETISM. 77. The electromagnetic field due to a straight current. Straight currents in cylindrical conductors ..... 251 78. Electromagnetic fields due to currents in closed linear circuits 259 79. The Law of Laplace. The mechanical action on a conductor which carries a current in a magnetic field. Electrokinetic energy ^62 80. The electrodynamic potential 81. Coefficients of self and mutual induction . . - .278 82. Maxwell s Current Equations. Solenoids. Ring Magnets. Hysteresis 281 IV. CURRENT INDUCTION. 83. Electromagnetic induction and its laws 291 84. Superficial induced currents 2 " 85. Variable currents in single circuits ... 86. Alternate currents in single circuits 312 87. Variable and alternate currents in neighboring circuits. Transformers 88. The general equations of the electromagnetic field . . .332 337 486 337 MISCELLANEOUS PROBLEMS INDEX THE NEWTONIAN POTENTIAL FUNCTION. CHAPTER I. THE ATTRACTION OP GRAVITATION, 1. The Law of Gravitation. Every body in the universe attracts every other body with a force which depends for mag nitude and direction upon the masses of the two bodies and upon their relative positions. An approximate value of the attraction between any two rigid bodies may be obtained by imagining the bodies to be divided into small particles, and assuming that every particle of the one body attracts every particle of the other with a force directly proportional to the product of the masses of the two particles, and inversely proportional to the square of the distance between their centres or other corresponding points. The true value of the attraction is the limit approached by this approximate value as the particles into which the bodies are supposed to be divided are made smaller and smaller. 2. The Attraction at a Point. By the attraction at any point P in space, due to one or more attracting masses," is meant the limit which would be approached by the value of the attraction on a sphere of unit mass centred at P if the radius of the sphere were made continually smaller and smaller while its mass remained unchanged. The attraction at .P is, then, the attraction on a unit mass supposed to be concentrated at P, 2 THE ATTRACTION OF GRAVITATION. If the attraction at every point throughout a certain region has a value other than zero, the region is called "a field of force " ; and the attraction at any point P in the region is called " the strength of the field " at that point. 3. The Unit of Force. It will presently appear that all spheres made of homogeneous material attract bodies outside of them selves as if the masses of the spheres were concentrated at their middle points. If, then, k be the force of attraction between two unit masses concentrated at points at the unit distance apart, the attraction at a point P due to a homogeneous sphere of radius a and of density p is k - r--A where r is the dis- 3 T tance of P from the centre of the sphere. In all that follows, however, we shall take as our unit of force the force of attrac tion * between two unit masses concentrated at points at the unit distance apart. Using these units, k in the expression given above becomes 1, and the attraction between two particles of mass mi and w 2 concentrated at points r units apart is * 2 2 - 4. Attraction due to Discrete Particles. The attraction at a point P, due to particles concentrated at different points in the same plane with P, may be expressed in terms of two components at right angles to each other. Let the straight lines joining P with \ ~x the different particles be denoted by r l9 r z> ?*3? > and the angles which these ms lines make with some fixed line Px, FIG. 1. by ait a z> ct 3 , . If, then, the masses * These are called "attraction units of force." When the attraction between two bodies is given in terms of absolute kinetic force units in any system, the corresponding value of k is sometimes called the " constant of gravitation." One dyne is equivalent to about 1.543 X 10 7 c.g.s. attrac tion units and one poundal to about 9.63 x 10 8 f.p.s. attraction units. For simple illustrative problems the reader may consult the Miscella neous Examples at the end of the book. THE ATTRACTION OF GRAVITATION. 3 of the several particles are respectively ??i x , ra 2 , ??i 3 , , the components of the attraction at P are Y- _ m l cos ftj ??i 2 cos a 2 _ ^^ m cos a , .? , _a rin ~^T~ ~Z*~^~ in the direction Px, and ^ _ W! sin Q! Wgsinas , _\^??i sin a ro -, ~ ~ ~~~ in the direction Py, perpendicular to Px. The resultant force at P is P=VX 2 +F 2 , [3] and its line of action makes with Px the angle whose tangent If the particles do not all lie in the same plane with P, we may draw through P three mutually perpendicular axes, and call the angles which the lines joining P with the different particles make with the first axis a n cu, a 3 , ; with the second axis, ft? ft? ft? I and with the third axis, y,, y 2 , y 3 , . The three components in the directions of these axes of the attraction at P due to all the particles are then m cos a v \^m cos/3 . 7 _X^ W cos 7 r/n 1> 5 * / ~> ^ 7 3 * L^J f t ~~^ / y^ ^^^ r The resultant attraction is R = VX 2 + F 2 +Z 2 , [5] and its line of action makes with the axes angles whose cosines are respectively 1 1 aQd I M 5. Attraction at a Point in the Produced Axis of a Straight Wire. Let p. be the mass of the unit of length of a uniform straight wire AB of length /, and of cross section so small that 4 THE ATTRACTION OF GRAVITATION. we may suppose the mass of the wire concentrated in its axis (see Fig. 2), and let P be a point in the line AB produced at a M FIG. 2. distance a from A. Divide the wire into elements of length Ax. The attraction at P due to one of these elements, M, whose nearest point is at a distance x from P, is less than ^ - and x 2 greater than ^ - - (8 + A*) 1 The attraction at P due to the whole wire lies between ^ - and 7 ^ X ^ ; but these quantities approach the x 2 L^ (x + Ax) 2 same limit as Ax is made to approach zero, so that the attrac tion at P is limit X>AOJ_ C" V-dx I 1 I 1-71 If the coordinates of P, A, and 5 are respectively (x, 0, 0), ?!, 0, 0), and (x t -f- Z, 0, 0) , this result may be put into the form r_i i -i [_x l x x l x -f- Ij 6. Attraction at any Point, due to a Straight Wire. Let P (Fig. 3) be any point in the perpendicular drawn to the straight wire AB at A, and let PA = c. AB = I. AM= x. and the angle ABP= 8. Let MN be one of the equal elements of mass (/xAx) into which the wire is divided, and call PJtf, r. The attraction at P due to this element is approximately equal to L^ and r - acts in some direction lying between PM and PN. This attrac tion can be resolved into two components whose approximate values are ^ x c i n the direction PA, and ^ x x i n the THE ATTRACTION OF GRAVITATION. direction PL. The true values of the components in these directions of the attraction at P, due to the whole wire, are, then, respectively : and P] FIG. 3. The resultant attraction is equal to the square root of the sura of the squares of these components, or and its line of action makes with PA an angle whose tangent is 1 - sin 8 1 - cosAPB 2sin 2 APB cos 8 sin./LPB ^ . That is, the resultant attraction at P acts in the direction of the bisector of the angle APB. From these results we can easily obtain the value of the attraction at any point P, due to a uniform straight wire B B (Fig. 4) . Drop a perpendicular PA from P upon the axis of the wire. Let AB = /, AB = V , PA = c, ABP=&, AB P = 8 , BPB =6. The component in the direction PA of the attrac tion at P is [9] ii - (cosS + cos 8 ), 6 THE ATTRACTION OF GRAVITATION, and that in the direction PL is so that the resultant attraction is FIG. 4. The line of action PK of R makes with PA an angle < such that sin S f sin 8 and . .B ! PK=-- -) = - It is to be noticed that PK bisects the angle 0, and does not in general pass through the centre of gravity or any other fixed point of the wire. Indeed, the path of a particle moving from rest under the attraction of a straight wire is generally curved ; for if the particle should start at a point Q and move a short distance on the bisector of the angle BQB to Q , the attraction of the wire would now urge the particle in the direction of the bisector of the angle BQ B , and this is usually not coincident with the bisector of BQB . THE ATTRACTION OF GRAVITATION. 7 If q is the area of the cross section of the wire, and p the mass of the unit volume of the substance of which the wire is made, we may substitute for p in the formulas of this section its value qp. If instead of a very thin wire we had a body in the shape of a prism or C3"linder of considerable cross section, we might divide this up into a large number of slender prisms and use the equations just obtained to find the limit of the sum of the attrac tions at an} point due to all these elementary prisms. This would be the attraction due to the given body. 7. Attraction at a Point in the Produced Axis of a Cylinder of Revolution. In order to find the attraction due to a homo geneous cylinder of revolution at any point P (Fig. 5) in the axis of the cylinder produced, it will be convenient to imagine the cylinder cut up into discs of constant thickness Ac, by means of planes perpendicular to the axis. Let p be the mass of the unit of volume of the cylinder, and a the radius of its base. Consider a disc whose nearer face is at a distance c from P, and divide it into elements by means of B B A A FIG. 5. radial planes drawn at angular intervals of A0 and concentric cylindrical surfaces at radial intervals of Ar. The mass of any element J/ whose inner radius is r is equal to pAc- A0[rA?* 4- -J(Ar) 2 ], and the whole attraction at P due to M is approximately p J in a line joining P with some point of M. The component of this attraction in the direction PC is found by multiplying the expression just 8 THE ATTRACTION OF GRAVITATION. ^ given by - , the cosine of the angle CPS, so that the Vc 2 4- 1* attraction at P in the direction P<7, due to the whole disc, is approximately If the ]>ases of the cylinder are at distances c and c -f 7i from P, the true value of the attraction at P in the direction PC, due to the cylinder QQ , is dc Vc 2 -h a 2 = 2 7rp[7i + Vc () 2 + a 2 - V(c + /0 2 +a 2 ] . [1 5] This is evidently the whole attraction at P due to the cylin der, for considerations of symmetry show us that the resultant attraction at P has no component perpendicular to PC. [14] gives the attraction due to the elementary disc ABA B , on the assumption that the whole matter of the disc is concen trated at the face ABC. The actual attraction at P due to this disc may be found by putting c = c and li = Ac in [15]. If a, the radius of the cylinder, is very large compared with h and c , the expression [15] for the attraction at P due to the cylinder approaches the value 2-Trph. 8. Attraction at the Vertex of a Cone. The attraction due to a homogeneous cone of revolution, at a point at the vertex of the cone, may be found by the aid of [14]. If Fig. 6 represents a plane section of the cone taken through the axis, and if PM= c, MM = Ac, and MB = r, the attraction at P due to the disc ABCD is approximately 27rpAc 1 C \= 27rpAc(l COS a), Vc 2 + r 2 J THE ATTRACTION OF GRAVITATION. and the attraction due to the whole cone is 2^(1 - COSa) AC = 2irp(l - COSa) = 277/3(1 -COSa) -PL. [16] The attraction at P due to the frustum ABKN is found by subtracting the value of the attraction due to the cone ABP from the expression given in [16]. The result is 2irp(l - cosa) (PL -P3/) = 27rp(l-cosa)3/L, [17] and it is easy to see from this that discs of equal thickness cut out of a cone of revolution at different distances from the vertex by planes perpendicular to the axis exert equal attractions at the vertex of the cone. FIG. 6. It follows almost directly that the portions cut out of two concentric spherical shells of equal uniform density and equal thickness, by any conical surface having its vertex at the common centre P of the shells, exert equal attraction at this centre ; but we may prove this proposition otherwise, as fol lows : Divide the inner surface of the portion cut out of one of the shells by the given cone into elements, and make the perimeter of each of these surface elements the directrix of a conical surface having its vertex at P. Divide the given shells into elementary shells of thickness Ar by means of concentric spheri cal surfaces drawn about P. In this way the attracting masses will be cut up into volume elements. Let ML (Fig. 7) represent one of these elements, whose inner surface has a radius equal to r ; then, if the elementary 10 THE ATTRACTION OF GRAVITATION. cone APB intercept an element of area Aw from a spherical sur face of radius unity drawn around P, the area of the surface element at MM is ?~Au>, and that at LL is (r + A?*) 2 Aw. The attraction at P in the direction PM, due to the element ML 1 , is approximately p and the component of this in any direction Px, making an angle a with PJf, is approximately p Aw A? cos a. The attraction at P in the direction Px, due to the whole shell EDFG, is, then, ^-N X = lim p A? Aw cos a, where the sum is to include all the volume elements which go to make up the shell. If PF=r , PG = i\, PP =r , PG = r 1 , and jL = FG = X= I pdr I cosadw = p/A I The attraction at P in the same direction, due to the shell E D F G , is X = p I l dr I cosadw = pp f cos adw. But the limits of integration with regard to <o are the same in both cases ; .-. X= X , which was to be proved. If the shells are of different thicknesses, it is evident that they will exert attractions at P proportional to these thick nesses. THE ATTRACTION OF GRAVITATION. 11 The area of the portion which a conical surface cuts out of a spherical surface of unit radius drawn about the vertex of the cone is called u the solid angle " of the conical surface. 9. Attraction of a Spherical Shell. In order to find the attraction at P, any point in space, due to a homogeneous spherical shell of radii r and ?* 1? it will be best to begin by dividing up the shell into a large number of concentric shells of thickness Ar, and to consider first the attraction of one of these thin shells, whose inside radius shall be r. Let p be the density of the given shell, that is, the mass of the unit of volume of the material of which the shell is com posed. Join P (Fig. 8) with by a straight line cutting the inner surface of the thin shell at JV, and pass a plane through PO cutting this inner surface in a great circle NLSL , which FIG. 8. will serve as a prime meridian. Using N as a pole, describe upon the inner surface of the thin shell a number of parallels of latitude so as to cut off equal arcs on NLSL . Denote by A0 the angle which each one of these arcs subtends at 0. Through PO pass a numbei of planes so as to cut up each parallel of latitude into equal arcs. Denote by A< the angle between any two contiguous planes of this series. By this means the inner surface of the elementary shell will be divided into small quad rilaterals, each of which will have two sides formed of meridian arcs, of length r-A4, and two sides formed of arcs of parallels of latitude, of length rsin0-A< and /-sin((9 -f A0) A</>, where 12 THE ATTRACTION OF GRAVITATION. 9 is the angle which the radius drawn to the parallel of higher latitude makes with ON. The area of one of these quadri laterals is approximately rs mO- A0- A<, and the thickness of the shell is Ar, so that the element of volume is approxi mately r 2 sin#- Ar- A(9 A^>. Let PM=i/, then the attrac tion at P, due to an element of mass which has a corner at A i pr 2 sin 6 Ar > . ,, ,. M, is approximately --, in the direction PM. This force ma}* be resolved into three components : one in the direction PO, the others in directions perpendicular to PO and to each other ; but it is evident from considerations of symmetry that in finding the attraction at P due to the whole shell we shall need only that component which acts in PO. This . , - cosKPM .,, 7 ^ is approximately - - ; or, if PO = c, *J P r 2 sin0(c f The attraction at P due to the whole elementary shell is, then, approximately (truly on the assumption that the whole mass of the shell is concentrated at its inner surface) , Ar f rpr 2 smO(c-rc J J if and the true value at P of the attraction due to the given shell is C^ Jr Xdr. [20] If in the expression for X we substitute for its value in terms of 2/, we have, since if = (? -f- r 2 2 GTCOS0, and hence 2ydy = 2cr sin dO, THE ATTRACTION OF GRAVITATION. 13 In order to find the limits of the integration with regard to y, we must distinguish between two cases : I. If P is a point in the cavity enclosed by the given shell, y = r c and y-^ = r + c ; ~~ ^ + + c ) 2 ? ~ ~ G ~ + (* ~ )n = Q, [22] r + c r c f ri JXar==0; [23] ^r so that a homogeneous spherical shell exerts no attraction at points in the cavity which it encloses. II. If P is a point without the given shell, y Q =c r and y l = c + r ; 2 + ( c + r ) 2 r 2 c 2 +(c r) - and c H- r c r and f ri Xdr = - ^ (r^- ?- 3 ) . [25] i O^^ 1 " LJ From this it follows that the attraction due to a spherical shell of uniform density is the same, at a point without the shell, as the attraction due to a mass equal to that of the shell con centrated at the shell s centre. If in [25] we make r = 0, we have the attraction, due to a solid sphere of radius i\ and density p. at a point outside the sphere at a distance c from the centre. This is 10. Attraction due to a Hemisphere. At any point P in the plane of the base of a homogeneous hemisphere, the attraction of the hemisphere gives rise to two components, one directed toward the centre of the base, the other perpendicular to the plane of the base. We will compute the values of these com ponents for the particular case where P lies on the rim of the hemisphere s base, and for this purpose we will take the origin 14 THE ATTRACTION OF GRAVITATION. of our s} stem of polar coordinates at P. because by so doing we shall escape having to deal with a quantity which becomes infinite at one of the limits of integration. Denote the coordi nates of any point L in the hemisphere by r, 0, </>, where (Fig. 9) XPN= </>, IPL = 0, and PL = r. FIG. 9. If TI be the radius of the hemisphere, PT= PNcosNPT = PXcos XPN- co = 2i\ sin0 cos<. cos . , = snip sin <. The mass of a polar element of volume whose corner is at L is approximately p- IL\<j>- PLM- A?- or p^sin^ArA^A^, and this divided by r 2 is the attraction at P in the direction PL of the element, supposed concentrated at L. The components of this attraction in the direction PX and PFare respectively /3sin0ArA0A<cosXP.L and p siii0Ar A0A0COS/SP.L. The component in the direction Py of the attraction at P due to the whole hemisphere is, then, f* n ,< 7 j sn 2d</> I d6 \ p sin 2 sin cos 4> [27] THE ATTRACTION OF GRAVITATION. 15 and the component in the direction Px is J (** /2 Tj sin cos <f> 2 d(j> ( dO \ psm 2 0cos<j>dr = f w/>?v [28] This last expression might have been obtained from [26] by making c equal to r and halving the result. 11. Attraction of a Hemispherical Hill. If at a point on the earth at the southern extremity of a homogeneous hemispheri cal hill of densit}* p and radius i\ the force of gravity due to the earth, supposed spherical, is g, the attraction due to the earth and the hill will give rise to two components, g ^pi\ down wards, and f Trpi\ northwards. The resultant attraction does not therefore act in the direction of the centre of the earth, but o makes with this direction an angle whose tangent is FIG. 10. Let < (Fig. 10) be the true latitude of the place and (< a) the apparent latitude, as obtained by measuring the angle which the plumb-line at the place makes with the plane of the equator. Let a be the radius of the earth and o- its average density. Then tena= [29] 16 THE ATTRACTION OF GRAVITATION. The radius of the earth is very large compared with the radius of the hill, and a is a small angle, so that approximately a = --, and the apparent latitude of the place is < 2 a<r 2 ao- If fa is the true latitude of a place just north of the same hill, its apparent latitude will be fa + -^-^ , and the apparent differ- ence of latitude between the two places, one just north of the hill and the other just south of it, will be the true difference plus -!. If there were a hemispherical cavity between the two dor places instead of a hemispherical hill, the apparent difference of latitude would be less than the true difference. 12. Ellipsoidal Homceoids. A shell, thick or thin, bounded by two ellipsoidal surfaces, concentric, similar, and similarly placed, shall be called an ellipsoidal homosoid. It is a property of every such shell that if any straight line cut its outer surface at the points S, S (Fig. 11) and its inner surface at Q, Q , so that these four points lie in the order SQQ S , the length SQ will FIG. 11. be equal to the length Q S * We will prove that the attraction of a homogeneous closed * The section of the homceoid made by a plane which passes through the centre and the secant line, is bounded by two concentric, similar, and similarly placed ellipses. This figure may be regarded as an orthogonal projection of two concentric circles cut by a straight line. THE ATTRACTION OF GRAVITATION. 17 ellipsoidal hoimroid, at any point P in the cavity which it shuts in, is zero. Make P the vertex of a slender conical surface of two nappes, A and B, and suppose the plane of the paper to be so chosen that PQ is the shortest and PM the longest length cut from any element of the nappe A by the inner surface of the homoeoid. Draw about P spherical surfaces of radii PQ. PM, PS, and PO, and imagine the space between the inner most and outermost of these surfaces filled with matter of the same density as the homoeoid. The nappe A cuts out a portion from this spherical shell whose trace* on the plane of the paper is QLOT. Let us call this, for short, "the element QLOT." The attraction at P, due to the element QMOS which A cuts out of the homoeoid, is less than the attraction at the same point due to the element QLOT, and greater than that due to the element whose trace is KMNS. But the attraction at P, due to the first of these elements of spherical shells, is to the attraction due to the other as the thickness of the first shell is to that of the other, or as Q7 7 is to KS. (See Section 8.) The limit of the ratio of QT to KS, as the solid angle of the cone is made smaller and smaller, is unity ; therefore the limit of the ratio of the attraction at P due to the element QMOS, to the attraction due to the element of spherical shell whose trace is QLNS, is unity. By a similar construction it is easy to show that the limit of the ratio of the attraction at P, due to the element which B cuts out of the homoeoid, to the attraction due to the portion of spherical shell whose trace is Q L N S , is unity. But the attractions at P, due to the elements Q L N S and QLNS, are equal in amount (since their thicknesses are the same) and opposite in direction, so that if for the elements of the homoeoid these elements were substituted, there would be no resultant attraction at P. In order to get the attraction at P in any direction due to the whole homoeoid we may cut up the inner surface of the homoeoid into elements, use the perimeter of each one of these elements as the directrix of a conical sur- 18 THE ATTRACTION OF GRAVITATION. face having its vertex at P, and find the limit of the sum of the attractions due to the elements which these conical surfaces cut from the homoaoid. Wherever we have to find the finite limit of the sum of a series of infinitesimal quantities, we may without error substitute for any one of these another infinitesimal, the limit of whose ratio to the first is unity. For the attractions at P due to the elements of the homoeoid we may, therefore, substi tute attractions due to elements of spherical shells, which, as we have seen, destroy each other in pairs. Hence our proposition. A shell bounded by two concentric spherical surfaces gives a special case under this theorem. 13. Sphere of Variable Density, The density of a homo geneous body is the amount of matter contained in the unit volume of the material of which the body is composed, and this may be obtained by dividing the mass of the body by its volume. If the amount of matter contained in a given volume is not the same throughout a body, the body is called heterogeneous, and its density is said to be variable. The average density of a heterogeneous body is the ratio of the mass of the body to its volume. The actual density p at any point Q inside the body is defined to be the limit of the ratio of the mass of a small portion of the body taken about Q to the volume of this portion as the latter is made smaller and smaller. The attraction, at any point P, due to a spherical shell whose density is the same at all points equidistant from the common centre of the spherical surfaces which bound the shell but dif ferent at different distances from this centre, may be obtained with the help of some of the equations in Article 9. Since p is independent of and <, it may be taken out from under the signs of integration with regard to these variables, although it must be left under the sign of integration with re gard to r. Equations 19 to 24 inclusive hold for the case that we are now considering as well as for the case when p is constant, THE ATTRACTION OF GRAVITATION. 19 so that the attraction at all points within the cavity enclosed by a spherical shell whose density varies with the distance from the centre is zero. If P is without the shell, the attraction is ^1/(rVrdr. or, if P =/(r), The mass of the shell is evidently limit ^ ViVr 2 -f(r)dr = TT Cf(r) U ^^r Jr [30] [31] and [30] declares that a spherical shell whose density is a function of the distance from its centre attracts at all outside points as if the whole mass of the shell were concentrated at the centre. If r Q = 0, we have the case of a solid sphere. 14. Attraction due to any Mass. In order to find the attrac tion at a point P (Fig. 12), due to any attracting masses Jf, we may choose a system of rectangular coordinate axes and divide FIG. 12. M up into volume elements. If p is the average density of one of these elements (Ar ), the mass of the element will be pAv . Let Q, whose coordinates are a? , y , z , be a point of the ele- 20 THE ATTRACTION OF GRAVITATION. mcnt, and let the coordinates of P be x, ?/, z. The attraction at P in the direction PQ due to this element is approximately 2, and the components of this in the direction of the coordi nate axes are -cos/8 , and ^- cos/, [32] PQ" PQ~ PQ where a , /3 , y r are the angles which PQ makes with the positive directions of the axes. It is easy to see that PL x -x and, similarly, that y< y cos ft = " -J ? and cos y = PQ PQ Moreover, and this we will call r 2 . The true values of the components in the direction of the coordinate axes of the attraction nt P, due to all the elements which go to make up M , are, then, /(V a;) P (x -x)dx dy dz C C C P J J J [( - x-) limit p&v (y y} = f f f P (y -y}dx dy dz f33 -, J J J [(x -a;) 2 +(y-2/) 2 +(^-^) 2 ]^ _ limit V^pA?/(2 z) r r r P (z -z)dx dy dz . r33 ., J J J [( - xy+ (y - y y+ (z>- zyy* THE ATTRACTION OF GRAVITATION. 21 where p is the density at the point (V, // , z ), and where the integrations with regard to a; , .?/, and z are to include the whole of M . The resultant attraction at P, due to J/% is fF 2 + Z 2 ; [34] and its line of action makes with the coordinate axes angles whose cosines are The component of the attraction at the point (x, y, z) in a direction making an angle e with the line of action of R is R cose. If the direction cosines of this direction are A , /u , v , we have COS e = AA -f /Jifji -f- vv . 15. The quantities X, F, Z, and 7?, which occur in the last section, are in general functions of the coordinates &, i/, and z of the point P. Let us consider X, whose value is given in [33 A ] . x - x If P lies without the attracting mass 3/ , the quantity - is finite for all the elements into which 3/ is divided. Let L be the largest value which it can have for an}* one of these elements, then X is less than L \ \ \ pdx dy dz , or L-M 1 , and this is finite. If P is a point within the space which the attract ing mass occupies, it is easy to show that, whatever physical meaning we may attach to X, it has a finite value. To prove this, make P the origin of a system of polar coordinates, and divide M up into elements like those used in Section 10. It will then be clear that X= ( ( C P aw*0co8<t>drd8d<t>, [3(5] where the limits are to be chosen so as to include all the at tracting mass. Since sin :> ^cos</) can never be greater than 22 THE ATTRACTION OF GRAVITATION. unity, X is less than J J jp(2r<20cty, which is evidently finite when p is finite, as it always is in fact. The corresponding expressions, Y= C C C P siu 2 Osin<l>drdOd<f>, [37] and Z = C f C P sin cos Odrd6d<j>, [38] can be proved finite in a similar manner ; and it follows that X, y, Z, and consequently R, are finite for all values of cc, ?/, and z. As a special case, the attraction at a point P within the mass of a homogeneous spherical shell, of radii r and r 1? and of den sity p, is f 3\ [39] where r is the distance of P from the centre of the shell. 16. Attraction between Two Straight Wires. Let AK and B K (Fig. 13) be two straight wires of lengths I and V and of line-densities x and ! ; and let KB = c. Divide AK into K M M FIG. 13. elements of length A#, and consider one of these MM , such that AM =x. The attraction of BK 1 on a unit mass concen trated at M would be (Sections 2 and 5), /* -- [ If, therefore, the whole element MM whose mass is /xA# were con centrated at 3f, the attraction on it, due to BK , would be THE ATTRACTION OF GRAVITATION. The actual force, due to the attraction of BK , with which the whole wire AK is urged toward the right, is Jo (p-(l + l +c) ~ X-(1+C)J 17. Attraction between Two Spheres. Consider two homo geneous spheres of masses M and J/ (Fig. 14), whose centres C and C" are at a distance c from each other. Divide the sphere M into elements in the manner described in Section 9. The attraction due to 3/at any point P outside of this sphere is, as we have seen PC. and its line of action is in the direction FIG. 14. Let P =(r, 0,0) be any point in the sphere 3f f , and let CP = y. The attraction of M in the direction PC on an element of mass p^^sin^A/* A0A0 supposed concentrated at P is and the component of this parallel to the line VC is for<je 24 THE ATTRACTION OF GRAVITATION. which the whole sphere M 1 is urged toward the right by the attraction of M is, then, ^rcos0) ; where the integration is to be extended to all the elements which go to make up M . It is proved in Section 9 that the M value of this triple integral is 5-, so that the force of attraction C" . MM between the two spheres is - 18. Attraction between any Two Rigid Bodies. In order to find the force with which a rigid body M is pulled in any direc tion (as for instance in that of the axis of x) by the attraction of another body M , we must in general find the value of a sextuple integral. Let M be divided up into small portions, and let Aw be the mass of one of these elements which contains the point (#,?/, z). The component in the direction of the axis of x of the attrac tion at (x, ?/, z) due to M is p (x -x)dx dy dz and this would be the actual attraction in this direction on a unit mass supposed concentrated at (x, y, z}. If the mass Am were concentrated at this point, the attraction on it in the direc tion of the axis of x would be Am f f f _ p(x -x)dx dy M_ [43] JJJ [( X -x)*+(y -y)*+(z -z)*]t The actual attraction in the direction of the axis of x of M upon the whole of M is, then, liniit p(* -x)dx dy tl* THE ATTRACTION OF GRAVITATION. 25 If p 1 is the density at the point (x, y, z) , and if the elements into which M is divided are rectangular parallelepipeds of di mensions A#, A#, and Az, the expression just given may be written P P( X> - x)dxdydzdx dyW (-45-1 * where the integrations are first to be extended over J/ and then over J/. EXAMPLES. 1. Find the resultant attraction, at the origin of a system of rectangular coordinates, due to masses of 12, 16, and 20 units respectively, concentrated at the points (3, 4), ( 5, 12), and (8, 6). What is its line of action ? 2. Find the value, at the origin of a system of rectangular coordinates, of the attraction due to three equal spheres, each of mass m, whose centres are at the points (a, 0, 0), (0, &, 0), (0, 0, c) . Find also the direction-cosines of the line of action of this resultant attraction. 3. Show that the attraction, due to a uniform wire bent into the form of the arc of a circumference, is the same at the centre of the circumference as the attraction due to any uniform straight wire of the same density which is tangent to the given wire, and is terminated by the bounding radii (when produced) of the given wire. 4. Show that in the case of an oblique cone whose base is any plane figure the attraction at the vertex of the cone due to any frustum varies, other things being equal, as the thickness of the frustum. 5. Find the equation of a family of surfaces over each one of which the resultant force of attraction due to a uniform straight wire is constant. 6. Using the foot-pound-second system of fundamental units, and assuming that the average density of the earth is 5.6, com pare with the poundal the unit of force used in this chapter. 26 THE ATTRACTION OF GRAVITATION. 7. If in Fig. 2 we suppose P moved up to A, the attraction at P becomes infinite according to [7], and yet Section 15 asserts that the value, at any point inside a given mass, of the attraction due to this mass is always finite. Explain this. 8. A spherical cavity whose radius is r is made in a uniform sphere of radius 2 r and mass m in such a way that the centre of the sphere lies on the wall of the cavity. Find the attraction due to the resulting solid at different points on the line joining the centre of the sphere with the centre of the cavity. 9. A uniform sphere of mass m is divided into halves by the plane AB passed through its centre C. Find the value of the attraction due to each of these hemispheres at P, a point on the perpendicular erected to AB at (7, if CP = a. 10. Considering the earth a sphere whose density varies only with the distance from the centre, what may we infer about the law of change of this density if a pendulum swing with the same period on the surface of the earth and at the bottom of a deep mine ? What if the force of attraction increases with the depth at the rate of -th of a dyne per centimetre of descent? n 11. The attraction due to a cylindrical tube of length h and of radii E Q and R, at a point in the axis, at a distance c from the plane of the nearer end, is 27rp[Vc 2 + ^ 1 2 -V^+^ 2 +V(c + /0 2 + ^o 2 -V(c + /0 2 + ^ 1 2 ]. [Stone.] 12. A spherical cavity of radius b is hollowed out in a sphere of radius a and density p, and then completely filled with matter, of densit}^ /> . If c is the distance between the centre of the cavity and the centre of the sphere, the attraction due to the composite solid at a point in the line joining these two centres, at a distance d from the centre of the sphere, is 4 [pa 3 6 3 (p -p)1 3 [_d 2 (dc) 2 j 13. The centre of a sphere of aluminum of radius 10 and of density 2.5, is at the distance 100 from a sphere of the same THE ATTRACTION OF GRAVITATION. 27 size made of gold, of density 19. Show that the attraction due to these spheres is nothing at a point between them, at a distance of about 26.6 from the centre of the aluminum sphere. [Stone.] 14. Show that the attraction at the centre of a sphere of radius r, from which a piece has been cut by a cone of revolution whose vertex is at the centre, is irpr siira, where a is the half angle of the cone. 15. An iron sphere of radius 10 and density 7 has an eccentric spherical cavity of radius 6, whose centre is at a distance 3 from the centre of the sphere. Find the attraction due to this solid at a point 25 units from the centre of the sphere, and so situated that the line joining it with this centre makes an angle of 45 with the line joining the centre of the sphere and the centre of the cavity. [Stone.] 16. If the piece of a spherical shell of radii r and r lf inter cepted by a cone of revolution whose solid angle is w and whose vertex is the centre of the shell, be cut out and removed, find the attraction of the remainder of the shell at a point P situated in the axis of the cone at a given distance from the centre of the sphere. If in the vertical shaft of a mine a pendulum be swung, is there any appreciable error in assuming that the only matter whose attraction influences the pendulum lies nearer the centre of the earth, supposed spherical, than the pendulum does ? 17. Show that the attraction of a spherical segment is, at its vertex, where a is the radius of the sphere and h the height of the segment. 18. Show that the resultant attraction of a spherical segment on a particle at the centre of its base is _ o \Cl nj 28 TIIK ATTRACTION OF GRAVITATION. 19. Show that the attraction at the focus of a segment of a paraboloid of revolution bounded by a plane perpendicular to the axis at a distance b from the vertex is of the form T a 4- & 4 IT pa log - 20. Show that the attraction of the oblate spheroid formed by the revolution of the ellipse of semiaxes a, 6, and eccen tricity e, is, at the pole of the spheroid, e w (. e j and that the attraction due to the corresponding prolate spheroid is, at its pole, e 2 I2e 1-e 21. Show that the attraction at the point (c, 0, 0), due to the homogeneous solid bounded by the planes x a, x = b, and by the surface generated by the revolution about the axis of x of the curve y=f(x}, is 22. Prove that the attraction of a uniform lamina in the form of a rectangle, at a point P in the straight line drawn through the centre of the lamina at right angles to its plane, is ab sn where 2 a and 2 b are the dimensions of the lamina and c the distance of P from its plane. [Answers to some of these problems and a collection of additional prob lems illustrative of the text of this chapter may be found near the end of the book.] THE NEWTONIAN POTENTIAL FUNCTION. 29 CHAPTER II. THE NEWTONIAN POTENTIAL FUNCTION IN THE CASE OF GEAVITATION, 19. Definition. If we imagine an attracting body M to be cut up into small elements, and add together all the fractions formed by dividing the mass of each element by the distance of one of its points from a given point P in space, the limit of this sum, as the elements are made smaller and smaller, is called the value at P of " the potential function due to M" If we call this quantity F, we have V= lim ^ ^~, [46] where A??i is the mass of one of the elements and r its distance from P, and where the summation is to include all the elements which go to make up J/. If we denote by p the average density of the element whose mass is A??i, and call the coordinates of the corner of this ele ment nearest the origin a; , y , z , and those of P, #, y, z, we may write and r =CC C JJJ where p is the density at the point (V, # , z ) , and where the triple integration is to include the whole of the attracting mass M. As the position of the point P changes, the value of the quan tity under the integral signs in [47] changes, and in general V is a function of the three space coordinates, i.e., V=f(x,y,z}. To avoid circumlocution, a point at which the value of the 30 THE NEWTONIAN POTENTIAL FUNCTION potential function is V n is sometimes said to be " at potential F ." From the definition of Fit is evident that if the value at a point P of the potential function due to a system of masses MI existing alone is Fi, and if the value at the same point of the potential function due to another system of masses M 2 exist ing alone is F 2 , the value at P of the potential function due to M! and M 2 existing together is V FI + F 2 . 20. The Derivatives of the Potential Function. If P is a point outside the attracting mass, the quantity which enters into the expression for V in [47], can never be zero, and the quantity under the integral signs is finite every where within the limits of integration ; now, since these limits depend only upon the shape and position of the attracting mass and have nothing to do with the coordinates of P, we may dif ferentiate Fwith respect to either x, 2/, or z by differentiating under the integral signs. Thus : = //JW p (x 1 x] dx dy dz where the limits of integration are unchanged by the differen tiation. The dexter integral in this equation is (Section 14) the value of the component parallel to the axis of x of the attraction at P due to the given masses, so that we may write, using our old notation, AF=X, [49] and, similarly, D y V=Y, [50] D Z V=Z. [51] The resultant attraction at P is 2 vy, [52] IN THE CASE OF GRAVITATION. 31 arid the direction-cosines of its line of action are : and cos 7 = ^-^. [53] Ji H H It is evident from the definition of the potential function that the value of the latter at any point is independent of the par ticular system of rectangular axes chosen. If, then, we wish to find the component, in the direction of any line, of the attraction at any point P, we may choose one of our coordinate axes parallel to this line, and, after computing the general value of F, we may differentiate the latter partially with respect to the coordinate measured on the axis in question, and substitute in the result the coordinates of P. 21. Theorem. The results of the last section may be summed up in the words of the following THEOREM. To find the component at a point P, injciny direction PK, oj the attraction due to any attracting mass 3/, ice may divide the difference between the values of the potential function due to M at P (a point beticeen P and K on the straight line PK) and at P by the distance PP . The limit approached by this fraction as P approaches P is the component required.* We might have arrived at this theorem in the following way : If X, Y, Z are the components parallel to the coordinate axes of the attraction at any point P, the component in any direction PK whose direction -cosines are A, /u,, and v, is XX -f n Y+ rZ = \D X V+ ^D y V+ vD z V. [54] Let x, y, z be the coordinates of P, and x + A.r, y -+- Ay, z-\-\z those of P , a neighboring point on the line PK. * If the force is required in absolute kinetic units, the result thus obtained must be multiplied by k, the proper gravitation constant. The reciprocal of A: is equal to 1.543 x 10" in the c.g.s. system and to 9.63 x 10 8 in the f.p.s. system. 32 THE NEWTONIAN POTENTIAL FUNCTION If V and V are the values of the potential function at P and P respectively, we have, by Taylor s Theorem, V = V + Ax I) x V + Ay D v V + A* D z V + e, where e is an infinitesimal of an order higher than the first. - V pp, but therefore, r I- r > K 1 J + and this (see [54]) is the component in the direction PK of the attraction at P : which was to be proved. 22. The Potential Function everywhere Finite. If P is a point within the attracting mass, the integrand of the expres sion which gives the value of the potential function at P becomes infinite at P. That V is not infinite in this case is easily proved by making P the origin of a system of polar coordinates as in Section 15, when it will appear that the value of the potential function at P can be expressed in the form r p =CCC P rBw6drd$d4>; [57] and tliis is evidently finite. Although V P is everywhere finite, yet when we express its value by means of equation [47], the quantity under the integral signs becomes infinite within the limits of integration, when P is a point inside the attracting mass. Under these circum stances we cannot assume with out further proof that the result obtained by differentiating with respect to x under the in tegral signs is really D x V. It is therefore desirable to com- ft r/y, \ // \L > >^ V FIG. 15. IN THE CASE OF GRAVITATION. 83 putc the limit of the ratio of the difference (A^F) between the values of 1 at the points /"= (.r -+- Ax, y, z) and P=.(x, y, z), both within the attracting mass, to the distance (Ax) between these points. For convenience, draw through P (Fig. lo) three lines parallel to the coordinate axes, and let Q = (x . ?/ , z ). Then r 2 = ?- -h (Ax) 2 2 r Ax cos i^, where cos = x x ) A* pdx dy dz Ax Therefore = CCC( **-* J J J ^r r + rr -, C C* /Y2? Ax cosi/^ (A.r) 2 \ pdx dy dz = JJJr Ax ^ j^__ limit f^x* = rrrirco^ fa , , , JJJ 2i = C C rpdx dy dz cost, This last integral is evidentl}* the component parallel to the axis of x of the attraction at P, so that the theorem of Article 21 ma} be extended to points within the attracting mass. It is to be noticed that p is a function of x , ?/ , and z , but not a function of x, y, and z, and that we have really proved that the derivatives with regard to x, y, and z of 34 THE NEWTON FAN POTENTIAL FUNCTION where F is any finite, continuous, and single-valued function of a; , y f , and 2 , can always be found by differentiating under the integral signs, whether (x,y,z) is contained within the limits of integration or not. 23. The Potential Function due to a Straight Wire. Let p. be the mass of the unit length of a uniform straight wire AB (Fig. 1C) of length 21. Take the middle point of the wire for the origin of coordinates, and a line drawn perpendicular to the wire at this point for the axis of x. y fy 7 6 FIG. 16. The value of the potential function at any point P (x, y) in the coordinate plane is, then, according to [47], If r = AP = J2 whence y = and r = BP = Vor 4:1 express V P in terms of r and r . Thus : + y)*, I , we may eliminate x and y from [59] and It is evident from [60] that if P move so as to keep the sum of its distances from the ends of the wire constant, V P will IN THE CASE OF GRAVITATION. 35 remain constant. P s locus in this case is an ellipse whose foci are at A and B. From [59] we get x{_ r r J r -i = 1 cos 8 1 cos8 f xl J eosS l, sS + cosS and this (Section 6) is the component in the direction of the axis of x of the attraction at P. 24. The Potential Function due to a Spherical Shell. In order to find the value at the point P of the potential function due to a homogeneous spherical shell of density p and of radii r and r 1? we may make use of the notation of Section 9. f rrpi**in0drdOd<l> _ C C C [61] If P lies within the cavity enclosed by the shell, the limits of y are (r c) and (r-f-c), whence F=27rp(r 1 2 -r 2 ). [62] If P lies without the shell, the limits of y are (c r) and (c + v) ? whence o C If P is a point within the mass of the shell itself, at a dis tance c from the centre, we may divide the shell into two parts THE NEWTONIAN POTENTIAL FUNCTION by means of a spherical surface drawn concentric with the given shell so as to pass through P. The value of the potential func tion at P is the sum of the components due to these portions of the shell ; therefore V*rp(rt-f)+\&(t-rf) [G4] If we put these results together, we shall have the following table : F= 3c 2 rf-rf) If we make F, D C F, and Z) c 2 Fthc ordinates of curves whose abscissas are c, we get Fig. 17.* Here LNQS represents F, and it is to be noticed that this curve is everywhere finite, continuous, and continuous in direc tion. The curve OABC represents D C V. This curve is every where finite and continuous, but its direction changes abruptly when the point P enters or leaves the attracting mass. The three disconnected lines OA, DE, and FG represent D C 2 V. If the density of the shell instead of being uniform were a function of the distance from the centre [/a =/(?)], we should have at the point P, at the distance c from the centre of the sphere, [65] * See Thomson and Tait s Treatise on Natural Philosophy. Notice that A D = 4 Ttp and that EF = 4 Ttp. For values of c greater than n, F, D C F, and D C 2 F are respectively equal to M/c, M/c* 2 , and where M is the mass of the shell. IN THE CASE OF GRAVITATION. 37 From this it follows, as the reader can easily prove, that the value of the potential function due to a spherical shell whose density is a function of the distance from the centre only is constant throughout the cavity enclosed by the shell, and at all outside points is the same as if the mass of the shell were concentrated at its centre.* 25. Equipotential Surfaces. As we have already seen, Fis, in general, a function of the three space coordinates [ V = f(x, y, )], and in any given case all these points at which the potential function has the particular value c lie on the surface the equation of which is V =f(x, y, z] = c. Such a surface is called an " equipotential " or " level " sur face. By giving to c in succession different constant values, the equation V = c yields a whole family of surfaces, and it is always possible to draw through any given point in a field of force a surface at all points of which the potential function has the same value. The potential function cannot have two differ ent values at the same point in space, therefore no two differ ent surfaces of the family V = c, where V is the potential func tion due to an actual distribution of matter, can ever intersect. * If the outer radius of the shell be unchanged while the radius of the cavity approaches zero, the values of V and D e Fat O approach as limits the corresponding values at the centre of a solid, homogeneous sphere of density p and radius TV The value of D 2 r, however, does not approach as a limit the value of Dj 2 V at the centre of such a sphere. 38 THE NEWTONIAN POTENTIAL FUNCTION THEOREM. If there be any resultant force at a point in space, due to any attracting masses, this force acts along the normal to that equi- potential surface on which the point lies. For, let F=/(#, y, z) = c be the equation of the equipotential surface drawn through the point in question, and let the coordi nates of this point be # , y , z . The equation of the plane tangent to the surface at the point is and the direction-cosines of any line perpendicular to this plane, and hence of the normal to the given surface at the point Oo, 2/o, Zo), are cos a = Dx * V , [66 A ] cos/3 = ^ [66.] V(A* F) 2 +(A, F) 2 +CD, F) 2 and cosy = - ^Q F _ - __. [66 ] But if we denote the resultant force of attraction at the point (# , 2/ , z ) by R, and its components parallel to the coordinate axes by X, F, and Z, these cosines are evidently equal to X Y Z , , and respectively, so that a, /?, and y are the direction- R R R angles not only of the normal to the equipotential surface at the point (a? , 2/07 ZG) i but also [35] of the line of action of the re sultant force at the point. Hence our theorem. Fig. 18 represents a meridian section of four of the system of equipotential surfaces due to two equal spheres whose sec tions are here shaded. The value of the potential function due to two spheres, each of mass Jf, at a point distant respectively TI and r 2 from the centres of the spheres, is IN THE CASE OF GRAVITATION. 39 and if we give to V in this equation different constant values, we shall have the equations of different members of the system of equipotential surfaces. Any one of these surfaces ma} be easily plotted from its equation by finding corresponding values FIG. 18. of rj and ? 2 which will satisfy the equation ; and then, with the centres of the two spheres as centres and these values as radii, describing two spherical surfaces. The intersection of these surfaces, if they intersect at all, will be a line on the surface required. If 2 a is the distance between the centres of the spheres, V = - gives an equipotential surface shaped like an hour- CL glass. Larger values of V than this give equipotential sur faces, each one of which consists of two separate closed ovals, one surrounding one of the spheres, and the other the other. 9 ~\r Values of Y less than _ give single surfaces which look more and more like ellipsoids the smaller Y is. Several diagrams showing the forms of the equipotential surfaces due to different distributions of matter are given at 40 THE NEWTONIAN POTENTIAL FUNCTION the end of the first volume of Maxwell s Treatise on Electricity and Magnetism. 26. The Value of V at Infinity. The value, at the point P, of the potential function due to any attracting mass M has been defined to be v _ limit ~Am = . Let r () be the distance of the nearest point of the attracting mass from P, then [67] The fraction has a constant numerator, and a denominator ? o which grows larger without limit the farther P is removed from the attracting masses ; hence, we see that, other things being equal, the value at P of the potential function is smaller the farther P is from the attracting matter ; and that if P be moved away indefinitely, the value of the potential function at P approaches zero as a limit. In other words, the value of the potential function at "infinity" is zero. About 0, any fixed point near the attracting mass, as centre, imagine a spherical surface, S, drawn, of fixed radius, r , so large that $ shall just include all the distribution. Then, if P is any distant point without S, and if OP = r, M M rM rM or <rV P < r-r r + r r - Since -- - r -7 = 1, V so vanishes at infinity r -j- r T PO that the limit of (r- V P ), as r increases without limit, is M. Since cos it is easy to see that ^limit ^ Dr Y)^-M and that /^ <V 2 D X V) = - M cos (a;, r), where (x, r) denotes the angle between the axis of x and OP. IN THE CASE OF GRAVITATION. 41 27. The Potential Function as a Measure of Work. The amount of work required to move a unit mass, concentrated at a point, from one position, P 1? to another, P 2 , by any path, in face of the attraction of a system of masses, J/, is equal to FIG. 19. Vi V 2 , where V\ and F 2 are the values at P l and P 2 of the potential function due to M. To prove this, let us divide the given path into equal parts of length As, and call the average force which opposes the motion of the unit mass on its journey along one of these elements AB (Fig. 19), F. The amount of work required to move the unit mass from A to B is PAs, and the whole work done by moving this mass from P 1 to P 2 will be limit As As is made smaller and smaller, the average force opposing the motion along AB approaches more and more nearly the actual opposing force at A, which is D S V: therefore limit As=C It is to be carefully noticed that the decrease in the potential function in moving from P l to P 2 measures the work required to move the unit mass from Pj to P 2 . If P 2 is removed farther and farther from J/, F 2 approaches zero, and FI F 2 approaches F! as its limit, so that the value at any point P 1? of the poten tial function due to any system of attracting masses, is equal to the work which would be required to move a unit mass, sup posed concentrated at P 1? from P l to " infinity" by any path. 42 THE NEWTONIAN POTENTIAL FUNCTION The work ( W") that must be done in order to move an attract ing mass M against the attraction of any other mass M, from a given position by any path to " infinity/ is the sum of the quantities of work required to move the several elements (Am ) into which we may divide M , and this may be written in the form w = limit VAm f f f pdxdydz Am -O^ J J J ^-.^^-(y -y^ + ^-zYJ C C C C C C pp dxdydzdx dy dz JJJJJJ[_(x -x) 2 + (y -y) 2 + (z -zry W is called by some writers " the potential of the mass M with reference to the mass M" ; by others, the negative of W is called " the mutual potential energy of M and M ." In many of the later books on this subject, the word "po tential " is never used for the value of the potential function at a point, but is reserved to denote the work required to move a mass from some present position to infinity. If V is the value of the potential function at a point P, at which a mass m is supposed to be concentrated, m V is the potential of the mass m. If we could have a unit mass concentrated at a point, the potential of this mass and the value of the potential function at the point would be numerically identical. Imagine any given distribution of attracting matter which has the potential function V, divided into elements, of volume AT I? Ar 2 , Ar 3 , , of density p x , p 2 , PS? > and of mass &m ly Ara 2 , Ara 3 , . If the density at every point in the distri bution were X times what it now is (X being any positive constant), the potential function would be XV, and, since the volume occupied by each element would be unchanged, the mass of the ^?th element would be XAm^. To change X to X 4- AX, the mass of every element must be increased and to the pih element must be brought up the mass-increment AX- Am p . If this quantity were brought up from an infinite distance, the attraction of the existing distribution would do upon it an amount of work represented by XF-AX-Aw^, so IN THE CASE OF GRAVITATION. 43 that the work done on the additions to the whole mass would be X AA- limit \^ V Am. The work done by the attractive forces while A. was being changed from A to A x would be limit^ V Am J \ d\. To find the work done by the attrac tion for one another of its own parts, while the given distri bution is constructed by bringing together its particles from infinite dispersion, we may put X = 0, AI = 1, and get where the summation is to extend over the whole distribution. This quantity, the negative of which (when the matter is attracting) is sometimes called " the intrinsic energy " of the distribution, is given by the formula in attraction units of work. In absolute kinetic work units, The potential function inside a homogeneous sphere of radius a and density p, at a distance r from the centre, being 2 trp (a 2 ^r 2 ), the intrinsic energy of the sphere is /*<* -j a q jif2 Trp (a 2 - i r 2 ) 4 Trpr 2 dr or - A 2 a 5 or - /i 15 5 a attraction units of work. If the c.g.s. system has been used throughout, this is equivalent to go ~^ 00) ergs. If V and V are the potential functions due to two neighbor ing distributions, M and M , if AJ/ and A M are mass elements of the two distributions, and P and P points in A M and A J/ respectively, the mutual potential energy of M and J/ may be found by integrating * " - over both distributions, and, since the order of integration is immaterial, the result may be written CvdM or Cv dM. J J The intrinsic energy of M and M considered as a single dis tribution is to be found by integrating % ( V + V) over both 44 THE NEWTONIAN POTENTIAL FUNCTION masses. This gives \ CvdM % Cv dM CvdM or the sum of the intrinsic energies of M and M and the mutual energy of the two. If M and M were made up of matter every particle of which repelled every other particle according to the Law of Nature, the intrinsic potential energy of M would be + \ ( VdM and the mutual potential energy of M and M would be CvdM , or + Cv 28. Laplace s Equation. We have seen that the value of the potential function, and the component in any direction of the attraction at the point P, are always finite functions of the space coordinates, whether P is inside, outside, or at the sur face of the attracting masses. We have seen also that by dif ferentiating V at any point in any direction we may find the always finite component in that direction of the attraction at the point. It follows that D X V, D y V, D Z V are everywhere finite, and that, in consequence of this, the potential function is everywhere continuous as well as finite. If P is a point outside of the attracting masses, the quan tity under the integral signs in [48], by which dx dy dz is multiplied, cannot be infinite within the limits of integration, and we can find D*V by differentiating the expression for D x V under the integral signs. In this case p te dy d* , [69] and similarly, -- -V^ W [70] /* r r JJJ 3 (z 1 - [71] IN THE CASE OF GRAVITATION. 45 Whence, for all points exterior to the attracting masses, /V V + DiV + D z 2 V = 0. [72] This is Laplace s Equation. For the operator (D x 2 + D; + A 2 ), the symbols 8, A, A 2 , V 2 , V 2 , and V 2 have been used by dif ferent authors, and [72] may be written V 2 r = 0. [73] The potential function, due to every conceivable distribu tion of matter, must be such that at all points in empty space Laplace s Equation shall be satisfied.* 29. The Second Derivatives of the Potential Function are Finite at Points within the Attracting Mass. If the point P lies within the attracting mass, V and D X V are finite, but the quantity under the integral signs in the expression for D X V becomes infinite within the limits of integration, and we can not assume that DV may be found by differentiating D X V under the integral signs. In order to find D x 2 V under these circumstances, it is convenient to transform the equation for D X V. Let us choose our coordinate axes so as to have all the attracting mass in the first octant, and divide the projection of the contour of this mass on the plane yz into elements (dy ds 1 ). Upon each one of these elements let us erect a right prism, cutting the mass twice or some other even number of times. Consider one of the elements dy dz the corner of which next the origin has the coordinates 0, y , and z 1 . The prism erected on this element cuts out elements ds l9 ds 2 , ds s , ds, ds 2n from the surface of the attracting mass, and that edge of the prism which is perpendicular to the plane yz at (0, y , 2 ) cuts into the surface at points whose distances from the plane of ijz are a D a s> a s> a 2n-v an d out of the surface at points whose dis tances from the same plane are 2 > *> e> 2- At every one * If a function, continuous with its first derivatives within a region, T, satisfies Laplace s Equation at every point of the region, it is sometimes said to be harmonic in T. 46 THE NEWTONIAN POTENTIAL FUNCTION of these points of intersection draw a normal towards the inte rior of the attracting mass, and call the angles which these normals make with the positive direction of the axis of x, a 1? a 2? a s? v n * -J-fc i g to be noticed that a 1? a 3 , a 5 , a 2n _ l are all acute, and that a 2 , a 4 , a 6 , a 2n are all obtuse. The element dy dz may be regarded as the common projection of the sur face elements ds^ ds 2 , ds s , ds 2n , and, so far as absolute value is concerned, the following equations hold approximately : dy dz = ds l cos a x = ds 2 cos a 2 = ds s cos a 3 = = ds 2n cos a 2)l . But dy dz , ds l} ds^, ds 3 , etc., are all positive areas, and cos a 2 , cos a 4 , cos a 6 , etc., are negative, so that, paying attention to signs as well as to absolute values, we have dy dz =+dsi cos a 1 =ds 2 cos a 2 =+ ds s cos a 3 = ds 4 cos a 4 = etc. FIG. 20. Now p (x -x)dx di/ dz _ ~ and in order to find the value of this expression by the use of the prisms just described, we are to cut each one of these prisms into elementary rectangular parallelepipeds by planes parallel to the plane of yz ; we are to multiply the values of every one of these elements which lies within the attracting IN THE CASE OF GRAVITATION. 47 mass by the value of p D x ( -- j at its corner next the origin [i.e., at (x 1 , ij, 2 )] ; and we are to find the limit of the sum of these as dx is made smaller and smaller. We are then to compute a like expression for each of the other prisms, and to find the limit of the sum of the whole as the bases of the prisms are made smaller and smaller and their number corre spondingly increased. Wherever the function is a continuous function of x\ we have hence, if the elementary prisms cut the surface of the attract ing mass only twice, ; [75] or =al and, in general, c c ri + JJJr D f dx = lim / ( cos ai dsi + cos a 2 ds 2 + cos a s ds 3 + * \ r l T 2 r 3 \ \ X* >^* /** I [77] where is the value of the quantity - at the point where the line y = y , z = z cuts the surface of the attracting mass for the kth time, counting from the plane yz. In order to find the value of the limit of the sum which occurs in this expression, it is evident that we may divide the entire surface of the attracting mass into elements, multiply 48 THE NEWTONIAN POTENTIAL FUNCTION the area of each element by the value of - at one of its points, and find the limit of the sum formed by adding all these products together ; but this is equivalent to the surface integral of * - taken all over the outside of the attracting mass, so that [78] where the first integral is to be taken all over the surface of the attracting mass and the second throughout its volume. This expression for D X V is in some cases more convenient than that of [48]. We have proved this transformation to be correct, however, only when is finite throughout the attracting mass. If P r is a point within the mass, is infinite at P. In this case surround P by a spherical surface of radius e small enough to make the whole sphere enclosed by this surface lie entirely FIG. 21. within the attracting mass. This is possible unless P lies exactly upon the surface of the attracting mass. Shutting out the little sphere, let V z be the potential function due to the rest (T 2 ) of the attracting mass ; then, since P is an out side point, with regard to T 2 , we have, by [78], D x Y 2 =f^cosa-ds + f^cosads+fff^dx dy dz , [79] where the first integral is to be extended over the spherical IN THE CASE OF GRAVITATION. 49 surface, which forms a part of the boundary of the attracting mass to which V z is due ; the second integral is to be taken over all the rest of the bounding surface of the attracting mass ; and the triple integral embraces the volume of all the attracting mass which gives rise to V z . As c is made smaller and smaller, V<> approaches more and more nearly the potential function J", due to all the attract ing mass. /-i cos a ds j cos a can never be greater than 1 nor less than 1, so that if p is the greatest value of p on the surface of the sphere, the absolute value of the integral must be less than ^ I els or 4 -rrp t, and the limit of this as approaches zero is zero. The second integral in [79] is unaltered by any change in e. If we make P the origin of a system of polar coordinates, it is evident that the triple integral in [79] may be written D x p r sin drdOd<j>, [SO] and the limit which this approaches as e is made smaller and smaller is evidently finite, for, if ? = 0, the quantity under the integral sign is zero. Therefore, , r, = D, l = cos a d, + dx dy ds , [81] and [79] is true even when P lies within the attracting mass. Under the same conditions we have, similarly, D 9 T= | - cos (3 da and D Z V=\ ^cosyrfs-f- J I I -^dx dy dz . [S3] Observing that in these surface integrals r can never be zero, since we have excluded the case where P lies on the surface of the attracting mass, and that the triple integrals belong to 50 THE NEWTONIAN POTENTIAL FUNCTION the class mentioned in the latter part of Section 22, we will differentiate [81], [82], and [83] with respect to x, y, and z respectively, by differentiating under the integral signs. If the results are finite, we may consider the process allowable. Performing the work indicated, we have * V= p cos a - D x ds+ D*V=: Cp cosyDA-}ds+ C C CD Z (- and by making P the centre of a system of polar coordinates and transforming all the triple integrals, it is easy to show that the values of 7> X 2 F, D/F, D?V here found are finite, whether P is within or without the attracting mass, if the derivatives of the density are finite. This result* is important. 30. The Derivatives of the Potential Function at the Surface of the Attracting Mass. Let the point P lie on the surface of p FIG. 22. the attracting mass, or at some other surface where p is discontinuous. Make P the centre of a sphere of radius e, * Lejeune Dirichlet, Vorlesungen iiber die im umgekehrten Verhdltniss des Quadrats der Entfernung wirkenden Krdfte. Riemann, Schwere, Electricitat, und Magnetismus. It is to be noticed that while the integral in the second member of [48] represents D X V even at points within the attracting mass, the integral, I, obtained by differentiating this expression for D X V under the signs of integration represents D X 2 F only at outside points. Within the mass I is infinite, while DJV is finite. IX THE CASE OF GRAVITATION. 51 and call the piece which this sphere cuts out of the attracting mass Tj and the remainder of this mass T 2 . Let V l and V z be the potential functions due respectively to T and T 2 , then v = v, + r a , D X v = D X v, + D X r t , and the increment [A (Z^ T)] made in /),. V by moving from P to a neighboring point P , inside Tj, is equal to the sum of the corresponding increments \_\(D X V^) and A(Z>,. r 2 )] made in D,F! and D X V* With reference to the space T 2 , P is an outside point, so that the values at P of the first derivatives of T 2 with respect to x, y, and z are continuous functions of the space coordinates Let dai be the solid angle of an elementary cone whose vertex is at any fixed point in J\ used as a centre of coordinates. The element of mass will be pr*dudr. The component in the direction of the axis of x of the attraction at due to I\ is the limit of the sum taken throughout T t of > where a is the cosine of the angle which the line joining with the element in question makes with the axis of x. The difference between the limits of w is not greater than 4 TT, and the differ ence between the limits of r is not greater than 2 e. If, then, K is the greatest value which pa has in T ly (l>.F,) <8w* It follows from this that if P is a point within T t so that PP < e, the change made in D x V^ by going from P to P is far less than 16 TTACC ; but this last quantity can be made as small as we like by making e small enough, so that limit PP^ whence limit A / 7-) v \ _ limit A , n rr \ limit zb o A (^ y ) ~ pp = o and D X V varies continuously in passing through P. In a similar manner, it may be proved that D y V and D z V are 52 THE NEWTONIAN POTENTIAL FUNCTION everywhere, even at places where the density is discontinuous, continuous functions of the space coordinates. The results of the work of the last two sections are well illustrated by Fig. 17. We might prove, with the help of a transformation due to Clausius,* that the second derivatives of the potential function are finite at all points on the surface of the attracting matter where the curvature is finite, but that the normal second derivatives generally change their values abruptly whenever the point P crosses a surface at which p is discontinuous, as at the surface of the attracting masses. The fact, however, that this last is true in the special case of a homogeneous spherical shell suffices to show that we cannot expect all the second derivatives of V to have definite values at the boundaries of attracting bodies. 31. Gauss s Theorem. If any closed surface S drawn in a field of force be divided up into a large number of surface \ FIG. 23. elements, and if each one of these elements be multiplied by the component, in the direction of the interior normal of the force of attraction at a point of the element, and if these products be added together, the limit of the sum thus obtained is called the " surface integral of normal attraction over S." If any closed surface S be described so as to shut in com pletely a mass m concentrated at a point, the surface integral * Die Potentialfunction und das Potential, 19-24. IN THE CASE OF GRAVITATION. 53 of normal attraction due to m, taken over S, is 4 irm ; and, in general, if any closed surface S be described so as to shut in completely any system of attracting masses J/, the surface integral over S of the normal attraction due to M is 4 nM. In order to prove this, divide S up into surface elements, and consider one of these ds at Q. The attraction at Q in the direction QO, due to the mass m concentrated at 0, is = : The component of this in the direction of the QO 2 * in> interior normal is cos a, and the contribution which ds yields to the sum whose limit is the surface integral required is m cos ads . . Connect every point or the perimeter or as with by a straight line, thus forming a cone of such size as to cut out of a spherical surface of unit radius drawn about an element c?o>, say. If we draw about a sphere of radius r = OQ, the cone will intercept on its surface an element equal to r 2 - da>. This element is the projection on the spher ical surface of ds; hence dscosa = r 2 rfa>, approximately, and the contribution of the element ds to our surface integral is mdu>. But an elementary cone may cut the surface more than once ; indeed, any odd number of times. Consider such a cone, one element of which cuts the surface thrice in S lt S 2 , and $ 3 . Let OS^ OS 2 , and OS 3 be called r l5 r 2 , and r 3 respec tively, and let the surface elements cut out of S by the cone be ds lf ds z , and ds s , and the angles between the line S 3 O and the interior normals to S at S lf S 2 , and S s be a 1? a 2 , a 3 . It is to be noticed that when the cone cuts out of S, the corresponding angle is acute, and that when it cuts in, the corresponding angle is obtuse, a! and a 3 are acute, and a 2 obtuse. If we draw about three spherical surfaces with radii r lt r, and >- 3 respectively, the cone will cut out of these the elements r^t/co, ?v</oj, and r s 2 da). In absolute size, ds l = i\ 2 d<j) sec^, ds 2 = r<fd& seca 2 , and ds s = r 3 2 da> seca s , approximately, but ds 2 and r 2 da) are both positive, being areas, and seca 2 is negative. Taking account of sign, then, 54 THE NEWTONIAN POTENTIAL FUNCTION ds 2 = r 2 du seca 2 , and the cone s three elements yield to the surface integral of normal attraction the quantity , . m ; - H -- , - H -- =5 = 1 = m (d<* \ V r<? ?v / However many times the cone cuts S 9 it will yield mdv to the surface integral required : all such elementary cones will yield then in N dw = m 4 TT, if S is closed, and, in general, m, where is the solid angle which S subtends at 0. If, instead of a mass concentrated at a point, we have any distribution of masses, we may divide these into elements, and apply to each element the theorem just proved ; hence our general statement. If from a point without a closed surface S an elementary cone be drawn, the cone, if it cuts S at all, will cut it an even number of times. Using the notation just explained, the con tribution which any such cone will yield to the surface integral taken over S of a mass m concentrated at is / ds l cos a x ds z cos 0.0 ds s cos a s ds cos a 4 \ m\ - - -- 1 -- 5 -- 1 -- -) -- 1 -- 5 -- r I V V V *v *v J = m ( dm + dw da> + dot ) = m = 0, and the surface integral over any closed surface of the normal attraction due to any system of outside masses is zero. The results proved above may be put together and stated in the form of a THEOREM DUE TO GAUSS. If there le any distribution of matter partly within and partly without a closed surface S 9 and if M be the sum of the masses which S encloses, and M the sum of the masses outside S, the surface integral over S of the normal attraction N toward the interior, due to both M and M , is equal to 4 irM. If V be the potential function due to both M and M , we have IN THE CASE OF GRAVITATION. 55 It is easy to see that if a mass M be supposed concentrated on any closed surface S the curvature of which is everywhere finite, the surface integral of normal attraction taken over S will be 2 TrM; for all the elementary cones which can be drawn from a point P on the surface so as to cut S once or some other odd number of times, lie on one side of the tangent plane at the point, and intercept just half the surface of the sphere of unit radius the centre of which is P. From Gauss s Theorem it follows immediately that at some parts of a closed surface situated in a field of force, but en closing none of the attracting mass, the normal component of the resultant attraction must act towards the interior of the surface and at some parts toward the exterior, for otherwise the limit of the sum of the intrinsically positive elements of the surface, each one multiplied by the component in the direction of the interior normal of the attraction at one of its own points, could not be zero. In other words, the potential function, the rate of change of which measures the attraction, must at some parts of the surface increase and at others decrease in the direction of the interior normal. 32. Tubes of Force. A line which cuts orthogonally the dif ferent members of the system of equipotential surfaces cor responding to any distribution of matter is called a "line of force," since its direction at each point of its course shows the direction of the resultant force at the point. If through all points of the contour of a portion of an equipotential surface lines of force be drawn, these lines lie on a surface called a FIG. 24. tube of force." We may easily apply Gauss s Theorem to a space cut out and bounded by a portion of a tube of force and two equipotential surfaces ; for the sides of the tube do not con- 56 THE NEWTONIAN POTENTIAL FUNCTION tribute anything to the surface integral of normal attraction, and the resultant force is all normal at points in the equipotential surfaces. If w and w are the areas of the sections of a tube of force made by two equipotential surfaces, and if F and F are the average interior forces on w and w , we have Fu+F u> = [87] if the tube encloses ernpt}" space, and F<-\-F w =7rm [88] when the tube encloses a mass in of attracting matter. 33. Spherical Distributions. In the case of a distribution about a point in spherical shells, so that the density is a function of the distance from this point only, the lines of force are straight lines whose directions all pass through the central point. Every tube of force is conical, and the areas cut out of different equipotential surfaces by a given tube of force are pro portional to the square of the distance from the centre. Consider a tube of force which intercepts an area ^ from a spherical surface of unit radius drawn with as a centre, and apply Gauss s Theorem to a box cut out of this tube by two equipotential surfaces of radii r and (r-j-Ar) respectively. Let AOB (Fig. 25) be a section of the tube in question. The area of the portion of spherical surface <o which is repre sented in section at ad is ?* 2 ^, and the area of that at be is (r-|- Ar) 2 \l/. If the average force acting on w toward the inside of the box is F. the average force acting on w toward the inside of the box will be (F + A r 7^), and the surface integral of normal attraction taken all over the outside of the box is ) (r + Ar) V = -^ . \(F- r 2 ) [89] IN THE CASE OF GRAVITATION. 57 If the tube of force which we have been considering be ex tended far enough, it will cut all the concentric layers of matter, traverse all the empty regions between the layers, if there are such, and finally emerge into outside space. If we choose r so that the box shall contain no matter, the surface integral taken over the box must be zero. In this case, therefore, F= - 2 , [90] and V=--+IL. [91] From this it follows that in a region of empty space, either included between the two members of a system of concentric spherical shells of density depending only upon the distance from the centre, or outside the whole system, the force of attrac tion at different points varies inversely as the squares of the distances of these points from the centre. Suppose that the box (abed) lies in a shell whose densit} is constant ; then the surface integral of normal attraction taken over the box is equal to 4?r times the matter within the box. In this case the quantity of matter inside the box is -r ]!- or p^Ar + e, 4?r where e is an infinitesimal of an order higher than the first. Therefore, whence F=--+, [92] O ?*"" and F=---^7r /0 r + />t. [93] 58 THE NEWTONIAN POTENTIAL FUNCTION If the box lies in a shell whose density is inversely propor tional to the distance from the centre, we shall have whence F=- >Tr\ + , [95] and V= - - - 2 7r\r -f p. [96] In general, if the box lies in a shell whose density is/(r), we shall have [97] whence F= - - ~f(r)^ dr. [98] In order to learn how to use the results just obtained to de termine the force of attraction at any point due to a given spherical distribution, let us consider the simple case of a single shell, of radii 4 and 5, and of density [Ar] proportional to the distance from the centre. At points within the cavity enclosed by the shell we must have, according to [90] and [91], F=^ and F= -+/*; r r But the force is evidently zero at the centre of the shell, where r is zero, so that c must be zero everywhere within the cavity, and there is no resultant force at any point in the region. The value, at the centre, of the potential function due to the shell is evidently 244 TT\ roon - , [99] O and it has the same value at all other points in the cavity. In the shell itself it is easy to see that we must have at all points, IN THE CASE OF GRAVITATION. 59 In order to determine the constants in this equation, we may make use of the fact that F and V are even-where continuous functions of the space coordinates, so that the values of .F and V obtained by putting r = 4, the inner radius of the shell, in [100], must be the same as those obtained by making r = 4 in the expressions which give the values of F and V for the cavity enclosed by the shell. This gives us ^~ n T , 500 TrA c = 2o67rA and //. = - , o so that for points within the mass of the shell we have and F= -TrAr . [101] For points without the shell we have the same general expres sions for F and V as for points within the cavity enclosed by the shell, namely, F=\ and F=-- + m, [103] but the constants are different for the two regions. Keeping in mind the fact that F and V are continuous, it is easy to see that we must get the same result at the boundary of the shell, where r 5. whether we use [103], or [101] and [102]. This gives k = 3G9 TrA and m = ; so that for all points outside the shell we have and V= . [105] These last results agree with the statements made in Section 13, for the mass of the shell is 369 TrA. The values, at every point in space, of the potential function and of the attraction due to any spherical distribution may be 60 THE NEWTONIAN POTENTIAL FUNCTION found by determining, first, with the aid of Gauss s Theorem, the general expressions for F and V in the several regions ; then the constants for the innermost region, remembering that there is no resultant attraction at the centre of the system ; and finally, in succession (moving from within outwards) , the con stants for the other regions, from a consideration of the fact that no abrupt change in the values of either F ov Vis made by crossing the common boundary of two regions. This method of treating problems is of great practical im portance. 34. Cylindrical Distributions. In the case of a cylindrical distribution about an axis, where the density is a function of the distance from the axis only, the equi potential surfaces are concentric cylinders of revolution ; the lines of force are straight lines perpendicular to the axis ; and every tube of force is a wedge. If we apply Gauss s Theorem to a box shut in between two equipotential surfaces of radii r and r -f Ar, two planes perpen dicular to the axis, and two planes passing through the axis, FIG. 26. we have, if \j/ is the area of the piece cut out of the cylindrical surface of unit radius by our tube of force, o> = r-i/A, to = (r-\- Ar) i/ , and for the surface integral of normal attraction taken over the box, F-h>V = -^-A r (r.jF f ). [10G] If our box is in empty space, A r (r.F) = 0, so that F={ lind V = clo IN THE CASE OF GRAVITATION. 61 If the box is within a shell of constant density p, so that F=- ~2-pr and Y=c\o^r [108] 35. Poisson s Equation. Let us now apply Gauss s Theorem to the ease where our closed surface is that of an element of volume of an attracting mass in which p is either constant or a continuous function of the space coordinates. We will consider three cases, using first rectangular coordinates, then cylinder coordinates, and finally spherical coordinates. I. In the first case, our element is a rectangular parallelepiped (Fig. 27). Perpendicular to the axis of x are two equal sur faces of area Ay - Az, one at a distance x from the plane yz, and one at a distance x + A# from the same plane. The average force perpendicular to a plane area of size AyAz, parallel to the plane yz, and with edges parallel to the axes of y and z, is evi dently some function of the coordinates of the corner of the element nearest the origin. That is, if P=(x, ?/, z), the average force on PP 4 parallel to the axis of x is X=f(x. y, z), and the average force on P 1 P 7 in the same direction is /(x + Aa;, y, z) = X + A X X, so that PP 4 and P 1 P 7 yield towards the surface integral of interior-normal attrition taken over the element, the quantity A-t-AyAz A.r . Similarly, the other two pairs of elementary surfaces yield 62 THE NEWTONIAN POTENTIAL FUNCTION A#A?/Az 2 and A# A?/ Az , and, if p is the average A?/ Az density of the matter enclosed by the box, we have s. [100] This equation is true whatever the size of the element A# A?/ Az. If this element is made smaller and smaller, the average nor mal force [X] on Pl\ approaches in value the force \_D X V~\ at P in the direction of the axis of x ; Y and Z approach respec tively the limits D y V and D Z V , and p approaches as its limit the actual density \_p] at P. Taking the limits of both sides of [109], after dividing by AxA?/Az, we have A 2 V+D*V+D,*V=-* P , or V 2 F=-47T /3 , [110] which is Poisson s Equation. The potential function due to any conceivable distribution of attracting matter must be such that at all points within the attracting mass this equation shall be satisfied. For points in empty space p = 0, and Poisson s Equation degenerates to Laplace s Equation. II. In the case of cylindrical coordinates, the element of vol ume (Fig. 28) is bounded by two cylindrical surfaces of revo- lution having the axis of z as their common axis and radii r and / -f- Ar, two planes perpendicular to this axis and distant Az IX THE CASE OF GRAVITATION. 63 from each other, and two planes passing through the axis and forming with each other the diedral angle A0. Call Jf?, 0, and Z the average normal forces upon the elemen tary planes PP 6 , PP 2 , and PP 3 respectively, then the surface integral of normal attraction over the volume element will be A0Az A r (r- R} \r\z\Q A0 \_r\r + ^(A/-) 2 ]A C Z = 47rp (vol. of box) ; [111] whence, approximately, 1 A /~z?\ 1 x ^ v 7 vol. of box ril9 -. Ar / A<9 Az The force at Pin direction PP 5 is DJ 7 , in direction PP 4 is D Z V, and perpendicular to LP in the plane PLP l is - D Q T 7 , so that if the box is made smaller and smaller, our equation approaches tlieform lA(r.AF)+iD/F+A F=-4^. [113] FIG. 29. III. In the case of spherical coordinates, the volume element is of the shape shown in Fig. 29. Let OP=r, ZOP=0, and 64 THE NEWTONIAN POTENTIAL FUNCTION denote by </> the diedral angle between the planes ZOP and *ZOX. Denote by 7t, , and <l> the average normal forces on tlie faces jP/e, /V s, and Pl\ respectively ; then the surface integral of normal attraction over the elementary box is approximately whence - - + = 47r/v(vol. of box) ; ** - A?- r sin 6 A</> r sin A0 vol. of box m -n = 47TA, - -- 1*15 1 - The force at Pin the direction PP S is 7) r F, in the direction PP l is I -DA T 7 , and in the direction PP 4 is i-D a F; there- 9 sin r fore, as the element of volume is made smaller and smaller, our equation approaches the form sin - D r (r-D r V) + D ^ V + D e (sin D e V) This equation, as well as that for cylinder coordinates, might have been obtained by transformation from the equation in rectangular coordinates. We may devote the rest of this section to the stating of some general results which will be intelligible only to those readers who are familiar with the theory and the use of curvilinear coordinates. If , v, w are any three analytic functions of x, y, z which define a set of orthogonal curvilinear coordinates, and if h w * = (^D x w)* -\- (I) y w)* + (D z w)* 9 it is possible to show that Poisson s Equation may be written in either of the forms D M 2 V- li* + D* V- h* + &* V- lij + D U V. v 2 ^ + D V- V 2 ^ + IN THE CASE OF GRAVITATION. 65 h...h t .-h.A DA . By giving to c in the equation u = c, where u is a given function of (x, y, s), different values in succession we may get the equations of any number of surfaces on each of which u is constant. These surfaces may or may not be the equipo- tential surfaces of a possible distribution of matter. If they are, it must be possible to find a potential function which changes only when u changes and is, therefore, a function of u only. We may in this case consider u as one of a set of three orthogonal curvilinear coordinates (u, v, w), and since, by hypothesis, D v V = 0, and D w V = 0, we may write Laplace s Equation in the form D*V- h* + D U V V 2 * = 0, or If now the ratio of V 2 ^ to h* is expressible as a function of u only, the equation is an ordinary differential equation the solution of which gives the most general solution of Laplace s Equation which is a function of u only. If, however, the ratio of \~a to h u 2 is not expressible as .a function of u only, V, which by hypothesis involves u only, must satisfy a differ ential equation which involves besides u one or both of the other coordinates v and ir, so that we infer that no solution of Laplace s Equation exists which is everywhere a function of u only. A set of confocal ellipsoidal surfaces forms a possible set of equipotential surfaces, while a family of concentric, sim ilar, and similarly placed ellipsoidal surfaces cannot be the level surfaces in empty space of any distribution of matter. Two concentric, similar, and similarly placed ellipsoidal sur faces, S l and S 21 may be equipotential but, in this case, the level surfaces between S l and S 2 will not be ellipsoidal sur faces similar to them. 66 THE NEWTONIAN POTENTIAL FUNCTION 36. Poisson s Equation in the Integral Form. In [109] A may be regarded as a function of x, ?/, z, A//, and Az, which ap proaches Z^Fas a limit when A?/ and Az are made to approach zero, and it may not be evident that the limit, when A.T, Ay, and A ~v Az are together made to approach zero, of the fraction -^-- is Z>/F. For this reason it is worth while to establish Poisson s P^quation by another method. It is shown in Section 29 that the volume integral of the quantity D x ( - 1, taken throughout a certain region, is the sur face integral of ^cosa taken all over the surface which bounds r the region. In this proof we might substitute for - any other function of the three space coordinates which throughout the region is finite, continuous, and single-valued, and state the results in the shape of the following theorem : If T is an} closed surface and U a function of oj, ?/, and z which for every point inside T has a finite, definite value which changes continuously in moving to a neighboring point, then f f fDJJ-dxdydz =- Cu cos ads, [117] f f ( D y U dxdydz=- Cucosflds, [118] and f f CD Z U- dx dy dz = - Cucos y ds, [119] where a, /?, and y are the angles made by the interior normals at the various points of the surface with the positive direction of the coordinate axes, and where the sinister integrals are to be extended all through the space enclosed by T, and the dexter integrals all over the bounding surface. If we apply this theorem to an imaginary closed surface which shuts in any attracting mass of density either uniform or vari able, and if for 7 in [11 7], [118], and [119] we use respectively IN THE CASE OF GRAVITATION. 67 D x V, I> y V, and D z V, and add the resulting equations together, we shall have * 2 V + P* V + A O dxdydz ~j( D * v cos a + D, f V cos ft 4- A V cos 7 ) <&. [120] The integral in the second member of this equation is evi dently (see [56]) the surface integral of normal attraction taken over our imaginary closed surface, and this by Gauss s Theorem is equal to 4 TT times the quantity of matter inside the surface, so that V+ D i V+ />/ H dxdydz [121] Since this equation is true whatever the form of the closed surface, we must have at every point For if throughout any region V 2 Fwere greater than -47rp, we might take the boundary of this region as our imaginary surface. lu this case every term in the sum whose limit gives the sinister of [121] would be greater than the corresponding term in the dexter, so that the equation would not be true. Similar reason ing shuts out the possibility of V 2 F s being less than 47rp. 37. The Average Value of the Potential Function on a Spheri cal Surface. If, in a field of force due to a mass m concentrated at a point P, we imagine a spherical surface to be drawn so as to exclude P, the surface integral taken over this surface of the value of the potential function due to m is equal to the area of the surface multiplied by the value of the potential function at the centre of the sphere. To prove this, let the radius of the sphere be a and the dis tance [OP] of P from its centre c. Take the centre of the 68 THE NEWTONIAN POTENTIAL FUNCTION sphere for origin and the line OP for the axis of x. Divide the surface of the sphere into zones by means of a series of planes cutting the axis of x perpendicularly at intervals of Ax. The area of each one of these zones is 2 TTCL dx, so that the surface i m integral of -- is /+ a m 2 ira dx |~2 irma Va 2 + c 2 2 cx~\ - Va 2 + c 2 - 2 ex L G J-a and the value of this, since the radical represents a positive . . ... quantity, is -- > which proves the proposition. The surface integral of the potential function taken over the sphere, divided by the area of the sphere, is often called " the average value of the potential function on the spherical surface." If we have any distribution of attracting matter, we may divide it into elements, apply the theorem just proved to each of these elements, and, since the potential function due to the whole distribution is the sum of those due to its parts, assert that: The average value on a spherical surface of the potential func tion due to any distribution of matter entirely outside the sphere is equal to the value of the potential function at the centre of the sphere. If a function, U, of the space coordinates attains a maxi mum (or a minimum) value at a point, Q, it is possible to draw about Q as centre a spherical surface, S, of radius so small that the value of U at every point of S shall be less (or greater) than the value of U at Q. It follows, therefore, from the theorem just stated that : The potential function due to a finite distribution of matter cannot attain either a maximum or a minimum value at any point in empty space. We may infer from the first of the theorems just stated that, if the potential function is constant within any closed surface, S, drawn in a region, T, which contains no matter, it IN THE CASE OF GRAVITATION. 69 will have the same value in those parts of T which lie outside S. For, if the values of the potential function at points in empty space just outside S were different from the value in side, it would always be possible to draw a sphere of which the centre should be inside S, and which outside S should in clude only such points as were all at either higher or lower po tential than the space inside S ; but in this case the value of the potential function at the centre of the sphere would not be the average of its values over its surface. A more satisfac tory proof can be given with the help of Spherical Harmonics. The value of the potential function cannot be constant in unlimited empty space surrounding an attracting mass J/, for, if it were, we could surround the mass by a surface over which the surface integral of normal attraction would be zero instead of 4 -n-M. The average value on a spherical surface of the potential function [ V~\, due to any distribution [J/] of attracting matter wholly within the surface, is the same as if M were concen trated at the centre of the space which the surface encloses. For the average values [ F and F + A,.F ] of V on con centric spherical surfaces of radii r and r + Ar may be written I Vds (or I Fc?<o, if do) is the solid angle of an ele- 47r> */ \irJ meutary cone with vertex at O, which intercepts the element ds 1 C from the surface of a sphere of radius ?), and - I (F+ A r V)dw ; 47T*/ whence A r F = -- ( A, F- efo>, and Z> P F = \ir Now \ D r V-a?du is the integral of normal attraction taken over the spherical surface, whence, by Gauss s Theorem, and 7r?- since V U = Q, for r=oo. 70 THE NEWTONIAN POTENTIAL FUNCTION 38. The Equilibrium of Fluids at Rest under the Action of Given Forces. Elementary principles of Hydrostatics teach us that when an incompressible fluid is at rest under the action of any system of applied forces, the hydrostatic pressure p at the point (x, ?/, z) must satisfy the differential equation dp = P (Xdx + Ydy + Zdz) , [122] where X, F, and Z are the values at that point of the force applied per unit of mass to urge the liquid in directions parallel to the coordinate axes. For, if we consider an element of the liquid [Ax A?/ As] (Fig. 27) whose average density is p and whose corner next the origin has the coordinates (x, y, z) , and if we denote by p x the average pressure per unit surface on the face PP%PP^ by p x + & x p x the average pressure on the face P 1 P 5 P 7 P 6 , and by X n the average applied force per unit of mass which tends to move the element in a direction parallel to the axis of x, we have, since the element is at rest, -f p X Ax A?/Az = (p x + A x p x ) A?/ As, If the element be made smaller and smaller, the first side of the equation approaches the limit pX, and the second side the limit D x p, where p is the hydrostatic pressure, equal in all direc tions, at the point P. This gives us D x p = p X. [123] In a similar manner, we may prove that D,P = pY, and D z p = p Z ; whence dp = D x p dx -f D y p dy + D z p dz If in any case of a liquid at rest the only external force applied to each particle is the attraction due to some outside mass, or to the other particles of the liquid, or to both together, X, F, and Z are the partial derivatives with regard to x, ?/, and IN THE CASE OF GRAVITATION. 71 sots, single function V, and we may write our general equation in the form dp = p (D X V- dx + D V V> dy + D z V-dz) = P -dV, whence, if p is constant, p = p V + const., [I 24 ] and the surfaces of equal hydrostatic pressure are also equi- potential surfaces. According to this, the free bounding surfaces of a liquid at rest under the action of gravitation only are equipotential. EXAMPLES. 1. Prove that a particle cannot be in stable equilibrium under the attraction of any system of masses. [Earnshaw.] 2. The earth s potential function expressed in the common, kinetic, centimetre-gramme-second units is 981 2 /r, for points above the surface. 3. Prove that if all the attracting mass lies within an equi potential surface on which V = C, then in all space outside S the value of the potential function lies between C and 0. 4. The source of the Mississippi River is nearer the centre of the earth than the mouth is. What can be inferred from this about the slope of level surfaces on the earth ? 5. If in [59] x be made equal to zero, V becomes infinite. How can you reconcile this with what is said in the first part of Section 22? 6. Are all solutions of Laplace s Equation possible values of the potential function in empty space due to distributions of matter ? Assume some particular solution of this equation which will serve as the potential function due to a possible dis tribution and show what this distribution is. 7. If the lines of force which traverse a certain region are parallel, what may be inferred about the intensity of the force within the region ? 8. The path of a material particle starting from rest at a point P and moving under the action of the attraction of a given 72 THE NEWTONIAN POTENTIAL FUNCTION mass Mis not in general the line of force due to M which passes through P. Discuss this statement, and consider separately cases where the lines of force are straight and where they are curved. 9. Draw a figure corresponding to Figure 17 for the case of a uniform sphere of unit radius surrounded by a concentric spherical shell of radii 2 and 3 respectively. 10. Draw with the aid of compasses traces of four of the equipotential surfaces due to two homogeneous infinite cylinders of equal density whose axes are parallel and at a distance of 5 inches apart, assuming the radius of one of the cylinders to be 1 inch and that of the other to be 2 inches. 11. Draw with the aid of compasses meridian sections of four of the equipotential surfaces due to two small homogeneous spheres of mass m and 2m respectively, whose centres are 4 inches apart. Can equipotential surfaces be drawn so as to lie wholly or partly within one of the spheres? What value of the potential function gives an equipotential surface shaped like the figure 8? Show that the value of the resultant force at the point where this curve crosses itself is zero. 12. A sphere of radius 3 inches and of constant density ^ is surrounded by a spherical shell concentric with it of radii 4 inches and 5 inches and of density /xr, where r is the distance from the centre. Compute the values of the attraction and of the potential function at all points in space and draw curves to illustrate the fact that V and D r V are everywhere continuous and that D r 2 Fis discontinuous at certain points. 13. A very long cylinder of radius 4 inches and of constant density ^ is surrounded by a cylindrical shell coaxial with it and of radii 6 inches and 8 inches. The density of this shell is inversely proportional to the square of the distance from the axis, and at a point 8 inches from this axis is JJL. Use the Theo rem of Gauss to find the values of F, D r V, and D r 2 V at differ ent points on a line perpendicular to the axis of the cylinder at its middle point. If the value of the attraction at a distance of 20 inches from the axis is 10, show how to find p. IN THE CASE OF GRAVITATION. 73 14. Use Dirichlet s value of D X V, given by equation [78], to find the attraction in the direction of the axis of x at points within a spherical shell of radii r and r x and of constant den sity p. 15. Are there any other cases except those in which the density of the attracting matter depends only upon the dis tance from a plane, from an axis, or from a central point, where surfaces of equal force are also equipotential surfaces ? Prove your assertion. 16. Show that the second derivative with respect to x, of the potential function due to a homogeneous sphere of density p and radius , with centre at the origin, is ^ -rrpr for inside points, and f wpa 3 (r 2 3x 2 ) / r b for points without the sur face. Similar expressions give the values of the second deriv atives with respect to y and z. Show that the normal second derivative of V is irp just within the surface and + f irp just without. Show that the tangential second derivatives are continuous at the surface. 17. Two uniform straight wires of length I and of masses m^ and ??i 2 ai *e parallel to each other and perpendicular to the line joining their middle points, which is of length y^. Show that the amount of work required to increase the distance between the wires to y 2 by moving one of them parallel to itself is 2m 1 ??l 2 f .-5 , -\Jl- 4- u 2 r\y=y* r *r- -i -^-b-VJ 2 + y-nog-r-*L. - I [Mmchm.] L y Jy=yi 18. Show that if the earth be supposed spherical and covered with an ocean of small depth, and if the attraction of the par ticles of water on each other be neglected, the ellipticity of the ocean spheroid will be given by the equation, 2 _ The centrifugal force at the equator g 19. A spherical shell whose inner radius is r contains a mass m of gas which obeys the Law of Boyle and Mariotte. Find the law of density of the gas, the total normal pressure on the inside of the containing vessel, and the pressure at the centre. 74 THE NEWTONIAN POTENTIAL FUNCTION 20. If the earth were melted into a sphere of homogeneous liquid, what would be the pressure at the centre in tons per square foot ? If this molten sphere instead of being homo geneous had a surface density of 2.4 and an average density of 5.6, what would be the pressure at the centre on the sup position that the density increased proportionately to the depth ? 21. A solid sphere of attracting matter of mass ra and of radius r is surrounded by a given mass M of gas which obeys the Law of Boyle and Mariotte. If the whole is removed from the attraction of all other matter, find the law of density of the gas and the pressure on the outside of the sphere. 22. The potential function within a closed surface S due to matter wholly outside the surface has for extreme values the extreme values upon S. 23. If the potential functions V and V due to two systems of matter without a closed surface have the same values at all points on the surface, they will be equal throughout the space enclosed by the surface. 24. The potential function outside of a closed surface due to matter wholly within the surface has for its extreme values two of the following three quantities : zero and the extreme values upon the surface. [Answers to some of these problems and a collection of additional prob lems illustrative of the text of this chapter may be found near the end of the book.] IN THE CASE OF REPULSION. 75 CHAPTER III. THE POTENTIAL FUNCTION IN THE CASE OP EEPULSION, 39. Repulsion, according to the Law of Nature. Certain physical phenomena teach us that bodies ma} acquire, by electrification or otherwise, the property of repelling each other, and that the resulting force of repulsion between two bodies is often much greater than the force of attraction which, ac cording to the Law of Gravitation, every body has for every other bod} 7 . Experiment shows that almost every such case of repulsion, however it may be explained physically, can be quantitatively accounted for by assuming the existence of some distribution of a kind of" matter," every particle of which is supposed to repel every other particle of the same sort according to the " Law of Nature," that is, roughly stated, with a force directly propor tional to the product of the quantities of matter in the particles, and inversely proportional to the square of the distance between their centres. In this chapter we shall assume, for the sake of argument, that such matter exists, and proceed to discuss the effects of different distributions of it. Since the law of repulsion which we have assumed is, with the exception of the opposite direc tions of the forces, mathematically identical with the law which governs the attraction of gravitation between .particles of pon derable matter, we shall find that, bj r the occasional intro duction of a change of sign, all the formulas which we have proved to be true for cases of attraction due to gravitation can be made useful in treating corresponding problems in repulsion. T6 THE POTENTIAL, FUNCTION 40. Force at Any Point due to a Given Distribution of Repelling Matter. Two equal quantities of repelling matter concentrated at points at the unit distance apart are called " unit quantities" when they are such as to make the force of repulsion between them the unit force. If the ratio of the quantity of repelling matter within a small closed surface supposed drawn about a point P, to the volume of the space enclosed by the surface, approaches the limit p when the surface (always enclosing P) is supposed to be made smaller and smaller, p is called the "density" of the repelling matter at P. In order to find the magnitude at any point P of the force due to any given distribution of repelling matter, we may suppose the space occupied by this matter to be divided up into small elements, and compute an approximate value of this force on the assumption that each element repels a unit quantity of matter concentrated at P with a force equal to the quantity of matter in the element divided by the square of the distance between P and one of the points of the element. The limit approached by this approximate value as the size of the elements is diminished indefinitely is the value required. FIG. Let Q (Fig. 30), whose coordinates are a; , ?/ r , z , be the corner next the origin of an element of the distribution. Let p be the density at Q and Aa; A//Az the volume of the element; then the force at P due to the matter in the element is approxi- IN THE CASE OF REPULSION. 77 mately equivalent to a force of magnitude p X ^ acting in the direction QP, or a force of magnitude ^ ._ acting ]_)/ \~ in the direction PQ. If the coordinates of P are #, y, z, the component of this force in the direction of the positive axis of x is aud thc forcc atP llel 22 to the axis of x due to the whole distribution of repelling matter is \-= - - ri . )5 -, JJJ[(. l - -x) 2 +(2/ -</) 2 +(2 -^) 2 ]i where the triple integration is to be extended over the whole space filled with the repelling matter. For the components of the force at P parallel to the other axes we have, similarly, and - C C C P (y -y)dx dy dz ri9M J J J [(* - a O+(y -jO*+(* -z)]r = -CCC P (z -z)dMyW ri251 J J J [(.v-.x ) 2 +(y-2/) 2 +(^-^)^j^ If we denote by V the function pdx dy dz 1 OT* [126] which, together with its first derivatives, is everywhere finite and continuous, as we have shown in the last chapter, it is easy to see that X=-D X V, Y=-D y V, Z = -D Z V, [127] T 2 , [128] and that the direction-cosines of the line of action of the re sultant force at P are ( O THE POTENTIAL FUNCTION It follows from this (see Section 21) that the component in any direction of the force at a point P due to any distribution M of repelling matter is minus the value at P of the partial derivative of the function V taken in that direction. The function Fgoes by the name of the Newtonian potential function whether we are dealing with attracting or repelling matter. In the case of repelling matter, it is evident that the resultant force on a particle of the matter at any point tends to drive that particle in a direction which leads to points at which the poten tial function has a lower value, whereas in the case of gravita tion a particle of ponderable matter at any point tends to move in a direction along which the potential function increases. 41. The Potential Function as a Measure of Work. It is easy to show by a method like that of Article 27 that the amount of work required to move a unit quantity of repelling matter, concentrated at a point, from P x to P 2 , in face of the force due to any distribution M of the same kind of matter, is F 2 VH where FI and F 2 are the values at P x and P 2 respec tively of the potential function due to M. The farther P 1 is from the given distribution, the smaller is Pi, and the less does F 2 FI differ from F 2 . In fact, the value of the potential function at the point P 2 , wherever it may be, measures the work which would be required to move the unit quantity of matter by any path from " infinity" to P 2 . 42. Gauss s Theorem in the Case of Repelling Matter. If a quantity m of repelling matter is concentrated at a point within a closed oval surface, the resultant force due to m at any point on the surface acts toward the outside of the surface instead of towards the inside, as in the case of attracting matter. Keeping this in mind, we may repeat the reasoning of Article 31, using repelling matter instead of attracting matter, and sub stituting all through the work the exterior normal for the in terior normal, and in this way prove that : IN THE CASE OF REPULSION. 79 If there be any distribution of repelling matter partly within and partly without a closed surface T, and if M be the whole quantity of this matter enclosed by T 7 , and M 1 the quantity out side T, the surface integral over T of the component in the di rection of the exterior normal of the force due to both M and M is equal to 4 irM. If V be the potential function due to M and W, we have 43. Poisson s Equation in the Case of Repelling Matter. If we applv the theorem of the last article to the surface of a volume element cut out of space containing repelling matter, and use the notation of Article 35, we shall find that in the case of rectangular coordinates the surface integral, taken over the element, of the component in the direction of the exterior normal is A* Ay A* + - + = 4 irft, A* Ay A*, [130] where X is the average component in the positive direction of the axis of x of the force on the elementary surface A?/Az, and where T and Z have similar meanings. It is evident that if the element be made smaller and smaller, X, 1% and Z will approach as limits the components parallel to the coordinate axes of the force at P. These components are D X V, D y V. and D Z V , so that if we divide [130] by A#A?/Az and then decrease indefinitely the dimensions of the element, we shall arrive at the equation V 2 F=-47rp. [131] B}* using successively cylinder coordinates and spherical co ordinates we may prove the equations = - 47rp, [132] and sin0 - D,(i*D r V) + - + D (sin6 - D 9 V) [133] 80 THE POTENTIAL FUNCTION so that Poisson s Equation holds whether we are dealing will: attracting or repelling matter. 44. Coexistence of Two Kinds of Active Matter. Certain physical phenomena may be most conveniently treated mathe matically by assuming the coexistence of two kinds of "matter" such that any quantity of either kind repels all other matter of the same kind according to the Law of Nature, and attracts all matter of the other kind according to the same law. Two quantities of such matter may be considered equal if, when placed in the same position in a field of force, they are subjected to resultant forces which are equal in intensity and which have the same line of action. The two quantities of matter are of the same kind if the direction of the resultant forces is the same in the two cases, but of different kinds if the directions are opposed. The unit quantity* of matter is that quantity which concentrated at a point would repel with the unit force an equal quantity of the same kind concentrated at a point at the unit distance from the first point. It is evident from Articles 2, 14, and 40 that m units of one of these kinds of matter, if concentrated at a point (#, ?/, z) and exposed to the action of m 1? w 2 , m^ ... m k units of the same kind of matter concentrated respectively at the points (o^, 2/1,^1), (Eg, ?/ 2 , z 2 ), Ov ?/. z 3 ), ... Ov ?/,, Z A ), and of m k+l , m k+2 , ... m n units of the other kind of matter concentrated respectively at the points (x k + l , y kJt 1: z A + 1 ), (aj M . 2 , y k + 2 , z k + 2 ), ... (x n , y n , Z H ), will be urged in the direction parallel to the positive axis of x with the force fe^, [134] i=l t=*+l where r t is the distance between the points (#, y, z) and * With this definition of the unit of quantity, the repulsion and attrac tion force unit is identical with the absolute kinetic force unit. IN THE CASE OF REPULSION 81 If we agree to distinguish the two kinds of matter from each other by calling one kind " positive " and the other kind " neg ative," it is easy to see that if every m which belongs to positive matter be given the plus sign and every m which belongs to negative matter the minus sign, we may write the last equation in the form -=^- [135] The result obtained by making m in [135] equal to unity is called the force at the point (x, y, z). In general, m units of either kind of matter concentrated at the point (#, y, z) , and exposed to the action of any continuous distribution of matter, will be urged in the positive direction of the axis of x b the force iii this expression, p, the density at (x , ?/ , z ) , is to be taken positive or negative according as the matter at the point is positive or negative : m is to have the sign belonging to the matter at the point (x, y, z) : and the limits of integration are to be chosen so as to include all the matter which acts on m. With the same understanding about the signs of m and of p, it is clear that the force which urges in any direction s, m units of matter concentrated at the point (#,?/, z) is equal to m-D 4 F, where F is the everywhere finite, continuous, and single-valued function pdx di/ dz ///IB= and that mV measures the amount of work required to bring up from u infinity" by any path to its present position the m units of matter now at the point (#, ?/, 2) . If we call the resultant force which would act on a unit of positive matter concentrated at the point P "-the force at P" 82 THE POTENTIAL FUNCTION. it is clear that if any closed surface T be drawn in a field of force due to any distribution of positive and negative matter so as to include a quantity of this matter algebraically equal to Q, the surface integral taken over T of the component in the direc tion of the exterior normal of the force at the different points of the surface is equal to 4=irQ. It will be found, indeed, that all the equations and theorems given earlier in this chapter for the case of one kind of repelling matter may be used unchanged for the case where positive and negative matter coexist, if we only give to p and m their proper signs. It is to be noticed that Poisson s Equation is applicable whether we are dealing with attracting matter or repelling mat ter, or positive and negative matter existing together. EXAMPLES. 1. Show that the extreme values of the potential function outside a closed surface S, due to a quantity of matter algebrai cally equal to zero within the surface, are its extreme values on. 2. Show that if the potential function due to a quantity of matter algebraically equal to zero and shut in by a closed sur face S has a constant value all over the surface, then this con stant value must be zero. GREEN S THEOREM. 83 CHAPTER IV. SUKFAOE DISTRIBUTIONS. -GREEK S THEOREM, 45. Force due to a Closed Shell of Repelling Matter. If a quantity of very finely-divided repelling matter be enclosed in a box of any shape made of indifferent material, it is evident from [127] and from the principles of Section 38 that if the vol ume of the box is greater than the space occupied by the repel ling matter, the latter will arrange itself so that its free surface will be equipotential with regard to all the active matter in existence, taking into account any there may be outside the box as well as that inside. It is easy to see, moreover, that we shall have a shell of matter lining the box and enclosing an empty space in the middle. That any such distribution as that indicated in the subjoined diagram is impossible follows immediately from the reasoning of Section 37. For ABC and DEF are parts of the same equipotential free surface of the matter. If we complete this surface by the parts indicated by the dotted lines, we shall enclose a space void of matter and having therefore throughout a value of the potential function equal to that on the bounding 84 SURFACE DISTRIBUTIONS. surface. But in this case all points which can be reached from by paths which do not cut the repelling matter must be at the same potential as 0, and this evidently includes all space not actually occupied by the repelling matter ; which is absurd. Let us consider, then (see Fig. 82), a closed shell of repelling matter whose inner surface is equi potential, so that at every point of the cavity which the shell shuts in, the resultant force, due to the matter of which the shell is composed and to any outside matter there may be, is zero. Let us take a small portion w of the bounding surface of the cavity as the base of a tube of force which shall intercept an FIG. 32. area <o on an equipotential surface which cuts it just outside the outer surface of the shell, and let us apply Gauss s Theorem to the box enclosed by w, <o , and the tube of force. If F is the average value of the resultant force on o> , the only part of the surface of the box which yields anything to the surface integral of normal force, we have F u = 4 urn, where m is the quantity of matter within the box. If we multi ply and divide by o>, this equation may be written F , = 4*m m _ t [137] to 0> If to be made smaller and smaller, so as alwa} T s to include a given point A, to 1 as it approaches zero will always include a point B on the line of force drawn through ^4, and F will ap proach the value F of the resultant force at B. The shell may be regarded as a thick layer spread upon the GREEN S THEOREM. 85 inner surface, and in this case the limit of -- may be consid er ered the value at A of the rate at which the matter is spread upon the surface. If we denote this limit by <r, we shall have If B be taken just outside the shell, and if the latter be very thin, ^^Q f~] evidently differs little from unity; and we see that the resultant force at a point just outside the outer sur face of a shell of matter, whose inner surface is equipotential, becomes more and more nearly equal to 4 TT times the quantity of matter per unit of surface in the distribution at that point as the shell becomes thinner and thinner. The reader may find out for himself, if he pleases, whether or not the line of action of the resultant force at a point just out side such a shell as we have been considering is normal to the shell. It is to be carefully noticed that the inner surface of a closed shell need not be equipotential unless the matter composing the shell is finely divided and free to arrange itself at will. "When the shell is thin$ and we regard it as formed of matter spread upon its inner surface, <r is called the "surface density " of the distribution, and its value at any point of the inner sur face of the shell may be regarded as a measure of the amount of matter which must be spread upon a unit of surface if it is to be uniformly covered with a layer of thickness equal to that of the shell at the point in question. 46. Surface Distributions. It often becomes necessary in the mathematical treatment of physical problems, on the assump tion of the existence of a kind of repelling matter or agent, to imagine a finite quantity of this agent condensed on a surface in a layer so thin that for practical purposes we may leave the thickness out of account. If a shell like that considered in the last section could be made thinner and thinner by compression 86 SURFACE DISTRIBUTIONS. while the quantity of matter in it remained unchanged, the volume density (p) of the shell would grow larger and larger without limit, and a would remain finite. A distribution like this, which is considered to have no thickness, is called a sur face distribution. The value at a point .* of the potential function due to a superficial distribution of surface density a- is the surface integral, taken over the distribution, of -, where r is the dis tance from P. It is evident that as long as P does not lie exactly in the distribution, the potential function and its derivatives are alwa} T s finite and continuous, and the force at any point in any direc tion may be found by differentiating the potential function partially with regard to that direction. If p were infinite, the reasoning of Article 22 would no longer apply to points actually in the active matter, and it is worth our while to prove that in the case of a surface distri bution where a- is everywhere finite, the value at a point P of the potential function due to the distribution remains finite, as FIG. 33. P is made to move normally through the surface at a point of finite curvature. To show this, take the point (Fig. 33), where P is to cut the surface, as origin, and the normal to the surface at as GKEEN S THEOREM. 87 the axis of , so that the other coordinate axes shall lie iu the tangent plane. If the curvature in the neighborhood of is finite, it will be possible to draw on the surface about a closed line such that for every point of the surface within this line the normal will make an acute angle with the axis of x. For convenience we will draw the closed line of such a shape that its projection on the tangent plane shall be a circle whose centre is at and whose radius is 7, and we will cut the area shut in by this line into elements of such shape that their pro jections upon the tangent plane shall divide the circle just mentioned into elements bounded by concentric circumferences drawn at radial intervals of A, and by radii drawn at angular distances of A<. If a;, 0, are the coordinates of the point P, a; , y\ z* those of a point of one of the elements of the area shut in by the closed line, and a the angle which the normal to the surface at this point makes with the axis of x, the size of the surface element is approximately - - , where u 2 = z 2 -\- ?/ 2 , and the COSa value at P of the potential function due to that part of the sur face distribution shut in by the closed line is The quantity all or sec a cos a V(a x ) 2 -}- u 2 is always finite, for, whatever the value of the quantity under the radical sign in the last expression may be when x. # , and " are all zero, it cannot be less than unity, and therefore Vi must 1 JL>i0 $ be finite even when P moves down the axis of x to the surface itself. If V and F 2 are the values at P of the potential functions due respectively to all the existing acting matter and to that 88 SURFACE DISTRIBUTIONS. part of this matter not lying on the portion of the surface shut in by our closed line, we have V=V l + V%, and, since P is a point outside the matter which gives rise to V^ the latter is finite ; so that Fis finite. The reader who wishes to study the properties of the deriva tives of the potential function, and their relations to the force components at points actually in a surface distribution, will find the whole subject treated in the first part of Riemann s Schwere, Electricitat und Magnetismus. Using the notation of this section, it is easy to write down definite integrals which represent the values of the potential function at two points on the same normal, one on one side of a superficial distribution, and at a distance a from it, and the other on the other side at a like distance, and to show that the difference between these integrals may be made as small as we like by choosing a small enough. This shows that the value of the potential function at a point P changes continuously, as P moyes normally through a surface distribution of finite super ficial density. If matter could be concentrated upon a geo metric line, so that there should be a finite- quantity of matter on the unit of length of the line, or if a finite quantity of matter could be really concentrated at a point, the resulting potential function would be infinite on the line itself, and at the point. 47. The Normal Force at Any Point of a Surface Distribu tion. In the case of a strictly superficial distribution on a closed surface where the repelling matter is free to arrange itself at will, the inner surface of the matter (and hence the outer surface, which is coincident with it) is equipotential, and the resultant force at a point B just outside the distribution is normal to the surface and numerically equal to 4?r times the surface density at 23. This shows that the derivative of the potential function in the direction of the normal to the surface has values on opposite sides of the surface differing by 4 TTO-, and at the surface itself cannot be said to have any definite value. GREEN S THEOREM. 89 It is easy, however, to find the force with which the repelling matter composing a superficial distribution is urged outwards. For, take a small element o> of the surface as the base of a tube of force, and apply Gauss s Theorem to a box shut in by the surface of distribution, the tube of force, and a portion o> of an equipoteutial surface drawn just outside the distribution. Let F and F be the average forces at the points of o> and <o respectively, then the surface integral of normal forces taken over the box is F u Fa). and this, since the only active matter is concentrated on the surface of the box (see Section 31), is equal to 27ro- (o, where o- is the average surface density at the points of the element o>. This gives us F=F 2 7TO-,,. O) Now let the equipotential surface of which co is a part be drawn nearer and nearer the distribution ; then lim = 1, lim F = 47ro- , and F= 27r<r . F is the average force which would tend to move a unit quan tity of repelling matter concentrated successively at the differ ent points of co in the direction of the exterior normal, but the actual distribution on co is COCT O , so that this matter presses on the medium which prevents it from escaping with the force 27Tcr 2 co; and, in general, the pressure exerted on the resisting medium which surrounds a surface distribution of repelling matter is at am* point 2 TTOT per unit of surface, where or is the surface density of the distribution at the point in question. We may imagine a super licial distribution of matter which is fixed, instead of being free to arrange itself at will. In this case the surface of the matter will not be in general equipoten tial, but, if we apply Gauss s Theorem to a box shut in bv a slender tube of force traversing the distribution, and by two surfaces drawn parallel to the distribution and close to it, one on one side and one on the other, we may prove that the 90 SURFACE DISTRIBUTIONS. normal component of the force at a point just outside the dis tribution differs by 4 TTO- from the normal component, in the same sense, of the force at a point just inside the distribution on the line of force which passes through the first point. It is sometimes convenient to denote the "charge" on a small area about a point P on a surface distribution by A , and the rest of the distribution by A", and to consider sepa rately the effects of A and A". If P l and P 2 are points on the normal to the surface drawn through P and near the surface on opposite sides of it ; if NI, N^ are the components in the direction PP l of the forces at P^ due to A and A" respectively, and if N z , N 2 " are the corresponding components at P z in the direction PP Z , then if P l and P z approach P, lim [-ZV/ + W + NJ + ^V 2 "] - 4 TTO-, where a- is the density of the distribution at P. The force due to A" changes continuously as P^ moves toward P 2 , however small A may be, so that lim N," = - lim N 9 " and lim (NJ + N 2 ) = 4 TTO-, and, by choosing A small enough, we may make N~i to differ in numerical value as little as we please from lim N 2 or from 27TO-. If the surface distribution is equipotential, and if it shuts in a region of no force, then if P l is in this region, JV~/ = JV/ , so that JVi" and N 2 " can be made to differ as little as one pleases in numerical value from 2 no- by making A small enough. Let the element of area covered by A be <o and the surface density of the charge on it o-, then the force with which A is urged in a direction normal to the surface by A" is coo- 2 TTO- within an infinitesimal of higher order than <o. That is, whatever the sign of o-, the surface distribution may be said to urge the surrounding medium outwards with a pressure in force units per unit of area which at P has the value 2 Tro- 2 , as we have already seen. It is easy to show that even if the surface distribution is not equipotential the components at P l and P 2 of the force GREEN S THEOREM. 91 in any fixed direction parallel to the surface approach the same limit as P l and P 2 approach P. At any point P of an equipotential surface covered with a superficial distribution of density o- the normal second deriv ative of V has a discontinuity* of 4 TTO- ( + ) where E l \J*l MZ/ and R% are the radii of curvature at P of two mutually per pendicular normal sections of the equipotential surface. 48. Green s Theorem, Before proving a very general theo rem due to Green, t of which what we have called Gauss s Theorem is a special case, we will show that if S is any closed surface and U a function of a*, y, and z, which for every point inside S is continuous, and single-valued, f C Cl) x U dx dy d~ = Cu- J) n x ds, [140] where the first integral is to include all the space shut in by S, and the second is to be taken over the whole surface, and where D n oc represents the deriva tive of x taken in the direction of the exte rior normal. To prove this, choose the coordinate axes so that S shall lie in the first octant, and divide the space inside the contour of the projection of S on the plane yz into elements of size dydz. On each of these elements erect a right prism cutting S twice or some other even number of times. Let us call the values of U at the successive points where the edge nearest the * C. Neumann, Math. Ann. 1880. Th. Horn, Zeitschr. f. Math. u. Phys. 1881. t George Green, An Essay on the Application of Mathematical Analy sis to the Theories of Electricity and Magnetism. Nottingham, 1828. 92 SURFACE DISTRIBUTIONS. axis of x of any one of these prisms cuts S ; U l} U 2 , 7 3 , U 2n respectively ; the angles which this edge makes with exterior normals drawn to S at these points, a 1? a 2 , a 3 , a 2w ; and the elements which the prism cuts from the surface S ; ds^ ds 2 , ds s , -ds, 2n . It is evident that wherever a line perpendicular to the plane yz cuts into S, the corresponding value of a is obtuse and its cosine negative, but wherever such a line cuts out of S t the corresponding value of a is acute and its cosine positive. Keeping this iu mind, we shall see that although the base of a prism is the common projection of all the elements which it cuts fro m $, and in absolute value is approximately equal to any one of these multiplied by the corresponding value of cos a, yet, since dxdy, ds^ c?s 2 , etc., are all positive areas and some of the cosines are negative, we must write, if we take account of signs, dydz = dsjcosci! = +^2cosa 2 = ds 3 cosa 3 = If the indicated integration with regard to x in the left-hand member of [140] be performed and the proper limits introduced, we shall have U,+ 7 4 -...], [141] where the double sign of integration directs us to form a quan tity corresponding to that in brackets for every prism which cuts $, to multiply this by the area of the base of the prism, and to find the limit of the sum of all the results as the bases of the prisms are made smaller and smaller. Since we may substitute for dydz any one of its approxi mate values given above, we may write the quantity within the brackets U\ COS ttj ds 1 + U 2 COS a 2 ds 2 + U 3 COS a ;i (7.S., -J- , and this shows that the double integral is equivalent to the sur face integral, taken over the whole of $, of 7 cos a, whence we may write C C Cl>,U-da;dyciz= Cuco&ada, [1-42] GREEN S THEOREM. 93 where the first integral is to be taken all through the space shut in by S, and the second over the whole surface. Let P or (x, y, z) be any point of S, a, /?, and y the angles which the exterior normal drawn at P to S makes with the coordinate axes, and P a point on this normal at a distance AH from P. The coordinates of P are # + A?i*cosa, ?/ + A>? -cos/2, z -f- An* cosy, and if W=f(x, y, z) be any continuous function of the space coordinates, W P > =f(x 4- AH cos a, y 4- AH cos/2, z + AH cosy) =/(a, y, z) 4- AH cos a DJ+ AH cos/? and -f A whence lim TF ~/ 7/> = A, TF P = cosa A/+cos/8Z>,/+cosy Z),/. [143] If, as a special case, TK= aj, we have .D n ;e = cosa; so that [142] may be written C CCD x U-dxtlydz= CuD n x-ds, [144] which we were to prove.* Green s Theorem, which follows very easily from this result, may be stated in the following form : If U and V are any two functions of the space coordinates which together with their first derivatives with respect to these coordinates are finite, continuous, and single-valued throughout the space shut in by an}* closed surface /S, then, if n refers to an exterior normal, * This theorem has been virtually proved already in Sections 29 and 36. 94 SURFACE DISTRIBUTIONS. = u- D n V- ds - C C Cu- V 2 V- dxdydz [145] = Cv-D n U-ds- C C Cv-V 2 U. dxdydz, [146] where the triple integrals include all the space within S and the single integrals include the whole surface. Since D X U> D X V= D X (U- D X V) - U- D X 2 V, we have C C (*D X U- D x V- dx dy dz = C C CD X (U- D x V)dxdydz- C C Cu- D X 2 V> dxdydz; but, from [144], C C CD X ( U-D x V)dxdydz = Cu-D x V- D n x . ds, whence f C C(L> X U> D X V) dxdydz = CU D x V D n x-ds- CC Cu.D x 2 V-dxdydz. [147] If we form the two corresponding equations for the deriva tives with regard to y and z, and add the three together, we shall obtain an expression which, by the use of [143], reduces im mediately to [145]. Considerations of symmetry give [146]. If we subtract [146] from [145], we get IT A U- V 2 F- V- V 2 U)dxdydz -D n V- V-D n U)ds. [148] In applying Green s Theorem to such spaces as those marked T (} in the adjoining diagrams, it is to be noticed that the walls of the cavities, marked /S and $", as well as the surfaces, GREEN S THEOREM. 95 marked S, form parts of the boundaries of the spaces, and that the surface integrals, which the theorem declares must be taken FIG. 35. over the complete boundaries of the spaces, are to be ex tended over S and S" as well as over S. We must remember, however, that an exterior normal to T at S points into the cavity C . If U and V both satisfy Laplace s Equation, the second member of [148] is equal to zero. If within the closed surface S the functions X, U, and V are continuous, and if the first derivatives of U and V are continuous (the first derivatives of A and the second deriva tives of U and V being finite), (A D y V) + D = (A -D Z V)-] dxdydz [149] + D V (X- D y U} + D z (A - D g U) ] dxdydz. Special Cases under Green s Theorem. Applications. I. If in [145] we put U= 1, we learn that if Fis any function which within and on the closed surface S is finite and contin- 96 SURFACE DISTRIBUTIONS. nous, together with its derivatives of the first order, the surface integral of D n V taken over S is equal to the volume integral of V 2 V taken through the space shut in by S. If V happens to satisfy Laplace s Equation within S, the surface integral is equal to zero. This result should be compared with Gauss s Theorem, treated in Section 31. II. If in [145] we make 7 equal to F, the potential function due to any distribution of matter, and assume that, in the general case, some of this matter is spread superficially on a surface S (or on a number of such surfaces), we may shut in S by two other surfaces, Si and S 2t parallel and very close to it. We may then apply Green s Theorem to so much of the space within a spherical surface, with centre at some con venient fixed point and radius r large enough to include the whole distribution, as does not lie between Si and S 2 . This gives C C Cv \*Vdxdydz, where the first surface integral is to be extended over the spherical surface, the second over S^ and the third over S 2 , it being understood that HI represents a normal to Si taken in the direction away from S, and n 2 a normal to S 2 taken in the direction away from S. Since V is continuous at S, while its normal derivatives are discontinuous in the manner indicated by the equation D V -f- D^ V = 4 TTO-, the limit of the sum of the two surface integrals taken over Si and S z as these surfaces approach S is 4?r J Va-ds. The value of the first surface integral is equal to 4 -n-r 2 times the average value of V D r V on the surface ; and, if this be written in the form 4?r GREEN S THEOKKM. 97 [average value of F(/- 2 Z> r F)], it is evident that the integral approaches zero as the radius r is made infinite, so that the field of the triple integrals may embrace all space. Since V-r= 47iy>, the whole second member of the equation represents 4?r li m FA& extended over all the distribution, and this is 8 TT times the intrinsic energy of the distribution. The first member of the equation represents the volume integral of the square of the resultant force extended over all space. We may write this result in the form III. If in [145] we make U = V= n, any function which within the closed surface S satisfies the equation Y -V = 0, we shall have D, t u dS. [151] IV. If in [148] V is the potential function due to two dis tributions of active matter, M v inside the closed surface S and J/ 2 outside it, and if U=-> where r is the distance of the point (x, y, z) from a fixed point 0, we must consider separately the two cases where is respectively without S and within S. A. If is without S, V 2 ( - J = for points within the sur face. Also, V 2 F = 4 ?rp, so that 98 SURFACE DISTRIBUTIONS. The triple integral is evidently equal to the value at the point of the potential function due to M v alone. If we call this Fi, and notice (see [143]) that D n r at any point of S is the cosine of the angle 8 between r and the exterior normal to $, we have ds + dS = - 4 ,F, [152] If $ is a surface equipotential with respect to the joint action of M l and Jf 2 > and if we denote by F s . the constant value of V on S, we have and it is easy to show, by the reasoning used in Section 31, /~cos 8 O >& that I dS = 0, whence - . <2 J r* HlM^? j B. If is a point inside $, whether or not it is within MI, and if S is equipotential with respect to the action of M l and FIG. 37. Jf 2 , we will surround by a small spherical surface of radius r and apply [148] to the space lying inside S and without the spherical surface. In doing so, it is to be noticed that S forms part of the boundary of the region we are deal ing with, and that an exterior normal to the region at S will be an interior normal of the sphere. GREEN S THEOREM. 99 Since for all points of the region we are considering v4 - )=0, we have or, since dS = r 2 dto } where dot is the area which the elemen tary cone the base of which is dS and the vertex O intercepts on the sphere of unit radius drawn about O, cls + r j?* ds _ ,, J A . r. It is easily proved, by the reasoning of Section 31, that f J : 4 and it is clear that if r be made smaller and smaller, the third integral of [155] approaches the limit zero. If F is the average value of F on the surface S , C Frf w = F 1 frfoi = F 4 TT : and as r is made smaller and smaller this approaches the value 4 ?r F , where F is the value of F at 0. The value, when r is zero, of the triple integral is evidently F 1? and we have If F 2 is the value at of the potential function due to M 2 alone, V = l\ + F a , so that 100 SURFACE DISTRIBUTIONS. If S is not equipotential with respect to the action of MI and M 2 , we have ";r* s ~/ r - Z) "(r)* s - [154 ] V. If in [148] we make U =-> where r is the distance of the point (aj, y, z) from a fixed point 0, and if F= v, any func tion harmonic everywhere within the closed surface S, we shall have - dS, [155 A ] I* if is within , and / V ^ =/ A (~Vw, [155,,] if is outside /S. VI. The closed surface S encloses a region TI and excludes the rest of space, T z . A function V is continuous and has finite first and second derivatives everywhere in the field of Green s Theorem. The first derivatives are everywhere con tinuous except at certain surfaces, Si in T l and S z in T Z1 where the tangential derivatives are continuous, and the nor mal derivatives discontinuous in the manner indicated by the equation At infinity V vanishes like the Newtonian Potential Func tion due to a finite distribution of matter. If U is the recip rocal of the distance from a fixed point 0, and if we apply Green s Theorem to U and F, using successively as fields, 5\ when is in T 2 , 7\ when is in T ly T z when is in T t , and GREEN S THEOREM. 101 T 2 when O is in jT 2 , and representing by n a normal to S pointing into T 2 in all cases, we learn that the expression is equal respectively to If there is no surface at which the normal derivative of F is discontinuous, and if F satisfies Laplace s Equation everywhere within S, the expression is equal to zero or to the value of V at according as is without or within S. If, now, S is a spherical surface of radius a, and if O x is distant Zj from the centre C, the distance from (7 of 2 , the inverse point of O l with respect to S may be denoted by / 2 , where ^ = a 2 . If r^ and r 2 represent the distances of any point P from O l and 2 respectively, then, if P lies on S, 102 SURFACE DISTRIBUTIONS. 4 2 = r 2 2 + a 2 2 r 2 a cos (r 2 , n) cos (/-!, n) a cos (r 2 , ?i) _ a 2 y*! 2 li r 2 2 a^ In this case, and so that it is easy to eliminate D n V by multiplying the second equation by a/h and subtracting the members from those of the first equation. The result is 1 rr _ F(a 2 - Z t 2 ) ^^ ~ 4 Tra J J [a 2 + /i 2 - 2 ^ cos (a, /i)] 8/2 .This integral determines F at every point within S when its value is given at every point on S. If O x is at the centre of S, I,. = 0, and ^ = a, so that V = - - 5 f |Fc?^, or the 4 7T(l J J average value on a spherical surface S, of a function F, har monic within and on S is the value of V at the centre of & It follows from this that a function which is harmonic about a point cannot have at either a maximum or a minimum value. GREEN S THEOREM. 103 If a function V is constant on any analytic surface S, is harmonic without S, and if it vanishes at infinity like a New tonian Potential Function, and V is the potential function in outer space due to a super ficial distribution on S of surface density D n V/ir. VII. A function V has the value zero everywhere on the closed surface Si, and the constant value C on the closed sur face S 2 , shut in by S^ In the space T, between Si and S 9 , V is harmonic. If we apply Green s Theorem, in T, to V and to the reciprocal of the distance from any point in T, we learn that where both normals point out of T. V is, therefore, the potential function due to surface distri butions on Si and S z numerically equal to D H V/v at every point. VIII. If the closed surface S shuts in a region T, and if the functions V and V\ which are equal at every point of S, are finite and continuous with their derivatives of the first order at every point of T, and if within T, V does and V does not satisfy Laplace s Equation, then the integral a + (*>* *T + (A *T] dxdydz, extended throughout T is less than the corresponding integral Qy ^ttWy + (WY + (W If we write V = V + u, u vanishes at every point of S, but is in general different from zero. 104 SURFACE DISTRIBUTIONS. = Q v+ Qu + 2 [ D * u D * V+E>J<< I> v V+D s u D 2 C Cn-D n VdS-2 C C C u-^Vdxdydz Now, since the integrands of Q u and $ r are made up of squares, and since neither u nor V are constants, both Q u and Q v are positive, so that Q v - > Q v . IX. There cannot be two different functions, W l and W^ which have equal values at every point of Si and S z (two closed surfaces the first of which shuts in the second), and between these surfaces are everywhere harmonic. If we suppose, for the sake of argument, that two such functions exist and call their difference u, it is clear that u is harmonic between the surfaces and that it vanishes at every point of both Si and $ 2 . If, therefore, in [145] we make U V = u, we learn that 2 ] dxd >jdz = 0, where the integral extends over all the space between Si and $ 2 . Since the integrand cannot be negative, it must be zero- at every point, so that D x u = D y u = D z u = and u is constant. But u = on Su therefore it is identically equal to zero and Wi = W It is easy to show that two functions which have equal normal derivatives at every point of Si and $ 2 , and are har monic everywhere between the surfaces, can differ only by a constant. X. We may now give an old proof of a theorem, originally discovered by Green from physical considerations, which is usually called Dirichlet s Principle by Continental writers, GREEN S THEOREM. 105 but in English books is generally attributed to Sir W. Thom son.* This theorem asserts that there always exists one, but no other than this one, function, v, of x } y, z, which (1) is continuous, and single-valued, together with its first space derivatives, throughout a given closed region T ; (2) at every point of the region satisfies the equation V 2 v = ; and (3) at every point on the boundary of the region has any arbitrarily assigned value, provided that this can be regarded as the value at that point of a single-valued function, continuous all over this boundary. There is evidently an infinite number of functions which satisfy the first and third conditions. If, for instance, the equa tion of the bounding surface S of the region is F(x, y, z) = 0, and if the value of v at the point (x, y, 2) upon this surface is to be f(Xj y, z), any function of the form 3>(a-, y, z)-F(x, y, z) +f(x, y, z) would satisfy the third condition, whatever continuous function < might be. If we assign to the function to be found a constant value C all over S, v = C will satisfy all three of the conditions given above. If the sought function is to have different values at different points of S, and if for u in the integral Q = which is to be extended over the whole of the region, we sub stitute any one of all the functions which satisfy conditions (1) and (3), the resulting value of Q will be positive. Some one at least of these functions (*;) must, however, yield a value of Q which, though positive, is so small that no other one can make Q smaller. f Let h be an arbitrary constant to * W. Thomson, Lioumlle s Journal, 1847. Dirichlet s Vorlesungen. Bacharach, Abriss der Geschichte der Potent ialtheorie. t A principle which will doubtless lead to a justification of this by no means self-evident assumption was pointed out by Hilbert in a remark able paper read before the Deutsche Matheinatiker-Vereinigung in 1899. 106 SURFACE DISTRIBUTIONS. which, some value has been assigned, and let w be any func tion which satisfies condition (1) and is equal to zero at all parts of S, then U = v + hw will satisfy conditions (1) and (3), and, conversely, there is no function which satisfies these two conditions which cannot be written in the form U= v + hw, where h is an arbitrary constant, and w some function which is zero at S and which satisfies condition (1). Call the minimum value of Q due to v, Q v) and the value of Q due to U, Q Ut then which, since w is zero at the boundary of the region, may be written, by the help of Green s Theorem, u -Q v = -2hCC CwV >2 vdxdyd Now, since Q v is the minimum value of Q, no one of the infinite number of values of Q v Q v formed by changing h and w under the conditions just named can be negative ; but if V 2 u were not everywhere equal to zero within T, it would be easy to choose w so that the coefficient of 2 A in the expres sion for Qu Q v should not be zero, and then to choose h so that Qu Q v should be negative. Hence V 2 t> is equal to zero throughout T, and there always exists at least one function which satisfies the three conditions stated above. Compare VIII. There is only one such function ; for if beside v there were another u = v 4- hw, we should have, since the coefficient of h is zero when V 2 w = 0, and that Q u may be as small as Q v , h& must be zero, whence either h = or Q = 0, and if O = 0, iv is zero. Therefore, GREEN S THEOREM. 107 u r, and there is only one function which in any given case satisfies all the three conditions given above. XI. The potential function F, due to a volume distribution of finite density p in the region T and a superficial distribu tion of finite surface density o- on the surface S, is everywhere continuous, and it so vanishes at infinity that, if r is the dis tance from any finite point, each of the quantities rV, -r 2 D r V, as r becomes infinite, approaches the limit M t where M is the amount of matter (algebraically considered) in the whole dis tribution. The first derivatives of V are everywhere finite, and they are continuous except on S, at every point of which tangential derivatives are continuous, while the normal deriva tive is discontinuous in the manner indicated by the equation where i and n 2 are the normals to the surface drawn away from it on each side. The second derivatives of V are every where finite, and they are continuous except at surfaces where p is discontinuous. At any point on such a surface the tan gential second derivatives are continuous, but the normal sec ond derivative is discontinuous by an amount equal to 4 TT times the discontinuity in p reckoned in the direction opposite to that in which the derivative is taken. Everywhere, except at surfaces of discontinuity in p, V satisfies Poisson s Equa tion, V 2 V = 4 -n-p, and without T, where there is no matter, this degenerates into Laplace s Equation. For a given value of p in the given region T, and a given value of a- on the given surface S, only one function has all these properties. Assuming that there are two such functions, V and F , let their difference be the function u. At every point of S, A, v + A,, v --= A, r f + A, v = - 4 TTO-, so that 2) n u + D n u = 0, 108 SURFACE DISTRIBUTIONS. and even the normal derivatives of u are continuous at every point of S. At surfaces of discontinuity in p, the derivatives of u are all continuous and u satisfies everywhere Laplace s Equation. The limits, as r becomes infinite, of ru and r*D r u are zero. Since u with its first and second derivatives is everywhere continuous, we may imagine a spherical surface of large radius r, drawn about any finite point 0, as centre, so as to enclose all the attracting mass and apply Green s Theorem in the form of [151] to u inside this surface. The numerical value of the surface integral uD.udS taken over the spherical surface is no greater than the area of the surface (4?rr 2 ) multiplied by the largest value which u D r u has on the surface, or 4?r [greatest value of (i(,r 1 D r u)~]. If, now, the radius of the surface be indefinitely increased, this expression approaches the limit zero so that the integral taken over all space has the value zero. Since the integrand is made up of squares which can never be negative, we must have at every point of space D x u = D y u = DM = 0. Therefore, u is constant in all space ; and since it is zero at infinity, it must be everywhere zero, so that V and V are identical. It is to be understood that T may be made up of several distinct regions, and that S may consist of several distinct surfaces. GREEN S THEOREM. 109 49. The Surface Distributions Equivalent to Certain Volume Distributions. Keeping the notation of IV. in the last article, let S be a closed surface equipotential with respect either to the joint action of two distributions of matter, M l inside S and J/2 outside it, or (when J/ 2 equals zero) to the action of a single distribution within the surface ; and let Fj, F" 2 , and V be the values of the potential functions due respectively to J/i alone, to J/ 2 alone, and to J/ t and M 2 existing together. If a quantity of matter were condensed on S so as to give at every 2) y point a surface density equal to , the whole quantity of 4?r matter on the surface would be and this, by 31, is equal in amount to J/i. Let us study the effect of removing 3/i from the inside of S and spreading it in a superficial distribution J/i over S, so that the surface density D V at every point shall be In what follows, it is assumed 4?r that we have two distributions of matter, one inside the closed surface and the other outside. It is to be carefully noted, how ever, that by putting J/ 2 equal to zero in our equations, all our results are applicable to the case where we have an equipotential surface surrounding all the matter, which may be all of one kind or not. The value, at any point 0, of the potential function due to the joint effect of J/ 2 and the surface distribution J/i , would be F v _ v 1 CD, > o y\ ; I 4 -. " / dS. If is an outside point, we have, by [153], so that the effect at any point outside an equipotential surface of a quantity J/j of matter anyhow distributed inside the sur face is the same as that of an equal quantity of matter dis tributed over the surface in such a way that the superficial 110 SURFACE DISTRIBUTIONS. _ 2) y density at every point is w , where V is the value of the 4 IT potential function due to the joint action of MI and any matter (M 2 ) that may be outside the surface. If is an inside point, we have, which shows that the joint effect of M 2 and J// is to give to all points within and upon the surface the same constant value of the potential function which points upon the surface had before J/i was displaced by M { . If, therefore, MJ and M 2 exist without J/i, there is no force at any point of the cavity shut in by $; or, in other words, the force due to MJ alone is at all points inside S equal and opposite to that due to M 2 . If J/i and M 2 exist without MJ, the cavity enclosed by S is, in general, a field of force. MI acts as a screen to shield the space within S from the action of M 2 . The surface of M^ is equipotential with respect to all the active matter, so that there is no tendency of the matter com posing the surface distribution to arrange itself in any different manner upon S. Since M^ exerts the same force on every particle outside S that M-i did, and since action and reaction are equal and opposite, every particle of M 2 exerts on MI forces the- result ant of which is equal to the resultant of the forces with which the same particle urged M^ The resultant effect, therefore, of the action, of M 2 on If/ is the same as the resultant effect of its action on M lt Now the whole system of forces applied to the surface distribution by M z and by the repulsions for one another of its own parts is equivalent to a tension from without of 2 no- 2 dynes per square centi meter applied all over S, and since the internal forces form a system in equilibrium, the resultant effect of M z on M l is equal to the resultant effect of the tension just mentioned on MJ. If two closed surfaces, ^ and S 2 , which mutually exclude each other, shut in, respectively, the two portions, M 1} M 2) of a GREEN S THEOREM. Ill distribution 3/, and are level surfaces of M s potential function, it is easy to see that a superficial distribution on Si of density a- = D n F/4 TT would act on a particle without Si just as MI does, and that a similar distribution on S.> would act on parti cles outside of S 2 as M z does. The action of J/ x on M z is the same as the resultant effect of the tension 2 -n-a- 2 or (D n F) 2 /8?r considered as acting all over S z . The surface integral of D n V/4:7r extended over any closed surface has been called by Maxwell the " electric displacement " through the surface. 50. Vectors. Stokes s Theorem. The Derivatives of Scalar Point Functions. It is frequently convenient to define a vector by giving the values (tensors) of its components paral lel to the coordinate axes ; and if for our present purposes we call these " the components of the vector," no confusion will arise. The expression (Q r , Q y , Q^ denotes a vector, Q, the components of which parallel to the axes of x, ?/, and z are respectively equal to Q x , Q y , and Q z . The direction cosines of the vector are the ratios of Q x , Q y , Q z to V Q x 2 + Q y 2 + Q*. The letter which represents a vector is often used in scalar equations to denote merely the tensor. Sometimes, however, the heavy face letter (Q) is used to denote the vector, while its tensor is represented by the same letter in ordinary type. Any three scalar point functions can be considered the com ponents of a vector point function. Scalar and vector point functions are sometimes called " distributed " scalars and vectors. Where there is no danger of any misunderstanding a vector point function may be called simply a vector. The scalar function D x Q r -\- D v Q y -\-D z Q s is called the diver gence of Q, and if this quantity vanishes identically, Q is said to be a solenoidal vector. The force due to any finite distri bution of matter attracting or repelling according to the " Law of Nature is solenoidal in empty space. The negative of the divergence of a vector is called its convergence. The vector, the components of which taken parallel to the coordinate axes are the three scalar point functions, Aft - Aft,, Aft - Aft, Aft - Aft. 112 SURFACE DISTRIBUTIONS. is called the curl of Q\ and if these components vanish at every point of a region, Q is said to be lamellar in that region. If the vector R is the curl of the vector Q, Q is said to be a vector potential function of R. The force due to a finite dis tribution of attracting or repelling matter is lamellar within and without the distribution. The curl of any vector is itself solenoidal. If two vectors have the same curl, their difference is a lamellar vector. The lines of a vector are a family of curves, one of which passes through every point of space, and each of which has at every one of its points the direction of the vector at the point. The differential equations of the lines of the vector Q are evidently dx / Q X = dy / ^ = dz / Q . after the values of Q x , Q v , and Q z have been substituted, we have two equations of the form dx dy = <f>(x, y, z), -^ = if/(x, y, ); whence we get, by differentiating, d 2 x dx = DA -^ + DA * + D z *, and, by eliminating y between the first and third equations, and x between the second and fourth equations, two equations of the second order between x and z and between y and z respectively. The integrals of these last equations are the equations of the lines of the vector. Sometimes the variables may be separated at the start, and then the work is much simplified. The lines of the vector ( x 2 , y, z) have the equa tions y = Az, a?log(2?y)=l, and those of the vector (3xz yz, xz + yz, ), the equations x = (B + A -f- Bz) e* z , y (A-\- Bz) e 2z , where A and B are arbitrary constants. If n represents the exterior normal of any closed surface S, the integral taken over S of the exterior normal component of the analytic vector Q is GREEN S THEOREM. 113 f#cos(n> Q)dS = CQ [cos (x, Q) cos (x, n) + cos (#, Q) cos (y, w) + cos (z, Q) cos (2, 71)] d = f[<?x cos (x, n) + Q y cos (?/, 7i)-h $ 2 cos (2, 7i)]cZS; and this is equal to the volume integral of the divergence of Q taken through the space within S. The integral of the exterior normal component of any analytic solenoidal vector, taken over any closed surface, is zero. An important theorem due to Sir George Gabriel Stokes may be stated as follows : The line integral taken around a closed curve s, of the tan gential component of an analytic vector point function Q, is equal to the surface integral taken over any surface S, bounded by the curre, of the normal component of the curl of the vector, the direction of integration around the curve forming a right- handed screw rotation about the normals, or J [ Q x cos (x, s) + Q y cos (y, s) + Q z cos (z, s)~] ds Aft - A^) cos (*, n) + (A<?*-^ S )COSO/, 71) -f (D x Q y - D y Q x ) cos (z, n)-] dS. [158] To prove this, we may evaluate first so much of the double integral as involves Q x , that is, \D Z Q X cos (y, 71) - D y Q x cos (z, n)-]dS. Let the area S be divided into quadrilateral elements by means of equally spaced planes parallel to the planes of zy and xy respectively, and consider especially one of these elements, \S, the projection of which on the xz plane is Ax Az, so that \S> cos (?/, n) = Ax Az approximately. 114 SURFACE DISTRIBUTIONS. That corner of the element A$ which has the least x and g coordinates shall be the point P } and that side of the element which passes through P and is parallel to the plane of yz shall be represented by As^ Since A^ is perpendicular both to the normal to S at P and to the axis of x, cos (x, Sj) = 0, and cos (n, Si) = cos (x, ri) cos (x, s^) + cos (y, ri) - cos (y, s^ + cos (z, ri) cos (g, Sj) = 0, cos (g, w) cos (z, COS (y, 7l) or = - cos (y, FIG. 38. Moreover, 2)^ = + (D y Q x )cos(y, 8 1 )-\-(D z Q x )cos(z, and dS = dx dz - - - - cos (y, n) dx dz ds l N -L cos (y, n) . = ds^dx cos (z. Si) - ; ( cos (y, n) GREEN S THEOREM. 115 Hence, J J \_D Z Q r cos (y, w) Z> v (^ r cos (2, ?i) ] dS [cos(y, ri) cos (2, s^)D z Q x cos (g, ??)cos(^,g 1 )D y ^ J .]^ 1 </ar cos(?/, ?;) -//i / / = I I ^> Q JJ * r ftt COS (*> S + DyQz COS If we perform the integration with respect to s l and intro duce the limits, it will appear that this integral may be found by proceeding around the contour s in the direction indicated in the theorem and determining the line integral of dx Q x ds = Q x cos (x t s) ds, where ds is an element of s. If we treat in a similar manner those portions of the double integral which involve Q y and Q z , the theorem will be evident. According to the definition used in the preceding sections, the numerical value of the directional derivative of any scalar point function u, at any point P, in any fixed direction PQ } is the limit, as PQ approaches zero, of the ratio of U Q u p to PQ, where Q is a point on the straight line PQ between P and Q . The gradient h u of the function u at P is the direc tional derivative of u at P taken in the direction in which u increases most rapidly. This direction is normal to the surface of constant u which passes through P. h u * = (D x uy + (D y u) z + (D z u)*. The directional derivative of any scalar point function at any point in any given direction is evidently equal to the product of the values of the gradient and the cosine of the angle between the given direction and that in which the function increases most rapidly. The vector, the components of which parallel to the coordi nate axes are numerically equal to D x u, D y u, D z u, has been 116 SURFACE DISTRIBUTIONS. called the vector differential parameter of u. The numerical value (tensor) of this vector at any point is the gradient of u at the point, according to some writers ; others use "gradient " to represent the vector itself. The lines of the vector are curves which cut orthogonally the surfaces of constant u, that is, the family of surfaces the equation of which is u = c, where c is a parameter constant for any one surface of the family. If f(x, y, z) is any scalar point function, any vector func tion the lines of which cut the surfaces of constant f normally must have components R D x f, R D y f, R D z f, where R is some function of x, y, and z. The curl of this vector has components D z f-D y R-D y f-D z R, D x f-D z R-DJ-D x R, D y f- DJR, D x f- D y R, and the cosine of the angle between the vector and its curl is zero, so that these two vectors are per pendicular to each other. If a vector has a curl which is not perpendicular to it at every point, no family of surfaces exists the members of which cut the lines of the vector orthogonally at every point of space. Every plane vector point function has for its curl a vector perpendicular to its plane. The vector (3 yz, xz, xy) is not lamellar, but it is perpendicular to its curl : its lines cut orthogonally the family of surfaces x z yz = c, as do the lines of the lamellar vector (3 x*i/z, x z z, x s t/), each component of which is x 2 times the corresponding component of the first. If the ratios of the corresponding components of two vector point functions are all equal to the same scalar point function, the vectors have the same lines. Two lamellar vectors may have the same lines, thus : the lines of every vector of the form [_f(x)j 0, 0,] are parallel to the axis of x, and every such vector is lamellar, whatever analytic function /may represent. We may define the numerical value of the normal deriva tive at any point P of a scalar point function u, taken with respect to another scalar point function v, to be the limit, as PQ approaches zero, of the ratio of U Q u,, to V Q v p , where Q is a point so chosen on the normal at P of the surface of constant v which passes through P that V Q v p is positive. GREEN S THEOREM. 117 If (u, v) denotes the angle between the directions in which u and v increase most rapidly, the normal derivatives of u with respect to r, and of v with respect to u, may be written h H cos (u, v) /h v and h v cos (11, v) / h u respectively. If h u = h v , these derivatives are equal. The derivative of xyz with respect to x + y + z has at the point (1, 2, 3) the value 11/3. The derivative at the same point of x + t/ + z with respect to xyz is 11 /49. 51. The Attraction of Ellipsoids. If we transform the equation *.**. * to parallel axes, using a point A , which lies on the surface and has the coordinates ( ar , y 0t z ) as origin, and then denote by the angle which any radius vector drawn through A makes with the x axis, the equation of the surface in polar coordinates takes the form /cos 2 sin 2 cos 2 <^ sin 2 sin- v d~ b~ c~ _ / " \ cos ?/ sin cos <f> Z Q sin sin 1- ^ 1 o If A Q were at that extremity, A, of the axis which has the coordinates ( o, 0, 0), the equation would be /cos 2 V a 2 2 sin 2 6 cos 2 <ft sin 2 sin 2 ~~ ~~ we will denote the coefficient of R in this equation by 12(0, 0). Let us compare the x components of the attraction at A and at A, due to a homogeneous ellipsoid of density p bounded by this surface. If, with each of these points as origin, a set of (conical) surfaces of constant 9 with the constant difference A0, and a set of (plane) surfaces of constant < with the con stant difference A<, be imagined drawn, the ellipsoid will be 118 SURFACE DISTRIBUTIONS. divided into elementary " cones " in two ways. The vertices of all the cones of one system will be A, and the vertices of all the cones of the other system will be A . To every cone of the first system corresponds a cone with parallel axis belonging to the second system, but whereas every cone of the first system yields a positive contribution to the x force compo nent at A, some of the corresponding cones of the second system yield negative components to the corresponding force component at A . We shall find it convenient to write in parentheses after R and r 1 the value of and < to which they belong, and to note that r\ n _ e> n + ^ = - r\ e ^ } . If the values of and <f> which correspond to a given cone of the first system are and < j the values of and < which belong to the corresponding cone of the second system may be either and < , or ?r an( i v + <o- The contribution of any cone of the first system to the x component of the force at A is f cos r 2 sin OdrdOd<j> = P R (9t ^ sin and the contribution of the corresponding cone of the second system to the x component of the force at A Q is either or pr\ v - 6t ^ + 7r) sin cos OdOdfa as the case may be. If, now, we group together two cones of the first system corresponding to (0 , < ) and (0 , TT + < ) respectively, we may write the positive contribution coming from this pair in the form 4 cos 2 The values of and < for the corresponding cones of the second system are one of the pairs (0o> $0 5 ^o? "" + <(>)> (^o? ^o 5 v ~~ &M ^o)? (TT , TT + </> ; , TT + <^> )? or (" ~ ^o? w + ^o j ^ ~~ ^o? <o)- GREEN S THEOREM. 119 The two values of 6 and < of either of these pairs give equal and opposite values to ( ?/ sin cos <f> ZQ sin sin <j>\ ~~~ ~* C( so that the positive contributions of this pair of cones of the second system is This contribution to the x component of the force at A is to the contribution of the corresponding cones of the first sys tem to the corresponding force component at A as x to a. Therefore, the x component at A of the attraction due to the whole ellipsoid is to the corresponding component at A as x to a. If, then, we know the values (X lt Y 1} Z^ of the attraction due to a homogeneous ellipsoid bounded by the surface at points on the surface at the negative extremities of the semiaxes a, b, c, we may find the numerical values of the com ponents parallel to the coordinate axes of the attraction at any point ( x w y , z ) on the surface from the equations The attraction X at A can be easily found* by adding together the contributions coming from all the elementary cones with vertices at A into which the ellipsoid is divided, that is, /ir/2 /<2Jr Xi = p I sin cos OdQ I R^ ^\d^ or, since */0 c/0 *See Routh s Analytical Statics, Vol. II, 182-221. Tarleton s Mathematical Theory of Attraction, 21-24 and 82-105. Schell s Theorie der Bewegung und der Krafte, pp. 690-716. 120 SURFACE DISTRIBUTIONS. 2 abW cos b 2 c 2 cos 2 6 4- 2 c 2 sin 2 (9 cos 2 < 4- a 2 6 2 sin 2 sin 2 2 a&V 2 cos u -- v u + v cos 2 </> 4- w sin 2 Now u 4- w cos 2 </> + w sin 2 < / d(f/ i ~ ~ ) 2 V(M + v) (u + w) Hence, X, = 4 aftc.p f 7 -- sinflco */o V (^ 2 cos 2 + a 2 sin 2 0) (c 2 cos 2 + a 2 sin 2 0) or, if s = a 2 tan 2 0, X 1 = 2a 2 ^ 7 rp f" - _ p Jo ( -f ^ /1 ( + *!> l/ *( + O l/ and [159] ?/ Z f , 7TO ^ I - Jo (s + a 2 ) 1/2 = 2 abc7rpz M = z Q M Q . At the positive ends of the axes of the ellipsoid the force components are - X v - Y lt - Z lf If the ellipsoid were made of matter of density p, repelling according to the " Law of Nature," the force components at the positive ends of the axes would be + X lt + lf + Z v GREEN S THEOREM. 121 f (#u 2/D ^i) is a point on the ellipsoid + 75 + "2 = 1> tt C !, A.f/1, X^i) is a corresponding point on the similar ellipsoid ?/ 2 3,2 2/2 + ~T~2 = 1> an( ^ tne straight line which joins these A o \ c two points passes through the origin. It is to be noticed that 7f , Z f , MJ have the same values for all similar ellipsoids, no matter what their actual dimen sions may be, and that the components of the attraction at corresponding points on two similar homogeneous ellipsoids of equal density p are to each other as the linear dimensions of the ellipsoid. Since the attraction of a homogeneous ellipsoidal homoeoid is zero (Section 12) at all inside points, we may draw through any point P within a homogeneous ellipsoid bounded by a surface SQJ a surface S, concentric with S and similar and similarly placed, and affirm that the attraction at P is equal to the attraction of so much of the whole ellipsoid as lies within S. If OP cuts S in jP , the attraction components at P are JT= 2 abcTrpxK 0) Y= 2 abc-KpyL^ Z 2 abc or - xK , - yL , - z3I ; therefore, the resultant attractions at internal points on any straight line drawn through the centre of a homogeneous ellip soid are parallel in direction. They are proportional in inten sity to the distances of the points from the centre. The potential function V within a homogeneous ellipsoid of density p bounded by the surface + 7-, + , , = 1 is such a 2 b 2 <- that its derivatives with respect to x, y, and z are respectively equal to 2 abcirpxK^ 2abcTTpyL Q1 2abcTrpz3I^ where A" , L w J/ have the same values at every point of the solid, so that V = alcirp(G Q - A> 2 - L v y 2 - J/^-), in which G is a constant to be determined by computing abcTrpG , the value of the potential function at the centre. 122 SURFACE DISTRIBUTIONS. The polar equation of an ellipsoidal surface of semiaxes a, b, c, when the origin is at the centre and 6 is the angle which any radius vector through the origin makes with the a axis, is , 2 = _ ?w _ b 2 c* cos 2 B + a 2 c 2 sin 2 $ cos 2 < + a 2 b 2 sin 2 6 sin 2 < u 4- v cos 2 </> H- w sin 2 < nZn s+r^ j rBi t/o /7r/2 /**/ I sin (9^ I /o */o Using the method of reduction already employed in finding the value of X^ we learn that G is an elliptic integral of the first kind, K M L w and J/ are elliptical integrals of the second kind. If a > b > c, (5 + a 2 ) > (5 + ft 2 ) > (5 + c 2 ) and ^r o < < M , and, unless 5 is zero, (5 + a 2 )/(s 4- # 2 ) < a 2 /6 2 and (5 + 6 2 )/(s 4 c 2 ) < 6 2 /c 2 . The equation for V may be written in the form if z* ~~ -rrp so that the equipotential surfaces within a homogeneous ellip soid are a set of ellipsoidal surfaces coaxial with the given ellipsoid and similar to each other. The axes are in the same order of length as are those of the ellipsoidal mass, but are more nearly equal. The outer surface of the attracting ellipsoid is not equipotential. The differential equations of the lines of force within a homogeneous ellipsoid are evidently GREEN S THEOREM. 123 dx/xK = so that if the reciprocals of 7f , L m M are represented by k, I, m, The two ellipsoidal surfaces are confocal if a 2 = a 2 + A, 6" - b + A, c 2 = c 2 + A. We will assume for convenience that A is positive. A point P on the second surface S is said to correspond to a point P on the first surface S, if # : x = a : a, y : y = b : b, z : z = c : c. If Pj and P 2 are any two points on S, and P/, P 2 the corresponding points on S , the distance PiP 2 is equal to the distance P/Pa [Ivory s Theorem], as may be seen by substi tuting for a , b , and c in the following equation their values in terms of a, b, and c. To the points on a chord EF of , drawn parallel to the x axis, correspond the points on a parallel chord E F of 5 . The lengths of these two chords are as a to a . To the points in a slender prism Q, of cross-section AyAz, within /S, one edge of which is the line EF, correspond the points in a slender prism Q , of cross-section Ay Az , or Ay &z-b c /bc, within S , and one edge of this is the line E F . If Q and Q are made of homogeneous matter of equal den sity, the x component of the attraction at any point P , on the larger ellipsoid S . due to Q, is [Section 6] equal to 124 SURFACE DISTRIBUTIONS. and the x component of the attraction due to Q at the point P on S corresponding to P is pky&zb c f 1 1 ~~bc \PE~ ~ P> The quantities in the parentheses are equal, by Ivory s Theo rem, and the two attraction components are to each other as bc:b c . If the whole space inside S 1 is filled with homoge neous matter of density p, the x component at any point P , on S , of the attraction of so much of the mass as lies within 8 be is equal to the product of and the x component of the b c attraction of the whole mass at the inside point P which lies on S and corresponds to P . We have already found an expression for the last-named force component. To find, then, the attraction at the outside point P (x , ?/ , 2 ), due to a homogeneous ellipsoid of density p bounded by the surface S, or -^ + y^H 2 = 1> we mus ^ nrs ^ find the positive a x 2 ?/ 2 z 1 2 value of A which satisfies the cubic - + (- ^ = 1. a 2 + A b 2 4- A c 2 4- A and thus determine the axes of the ellipsoidal surface S through P confocal with S. If we call this value of A, A , the point P on S which corresponds to P on S has the coordi- / ax by CK \ nates ( ? . . > _ )> and the x component of the attraction at P due to an ellipsoid of density p bounded by /S" would be ds (s + a 2 ) 3 / 2 ( S + ft 2 ) 1 / 2 ^ + c 2 ) 1 / 2 If we multiply this result by bc/b c , we shall get the result sought. If we substitute s + A for s in the integral and remember that x : x a : a , we may write the x component of the attraction of the ellipsoid at the point P in the form ds f A GREEN S THEOKEM. 125 where m is the mass of the ellipsoid. The components parallel to the axes of y and z of the attraction at P are, similarly, ds _ Z = - %mz j (s + 6 Lj ds (s 4- a 2 ) 1 / 2 (s 4 6 2 ) 1/2 (s 4 c) 3 / 2 = - 2 abcvpz M = - | w* J/. We know that, if we substitute in the equation the coordinates of any point in space, the largest root of the equation corresponds to an ellipsoid passing through the point, and is negative, zero, or positive according as the point lies within, on, or without S. Following Dirichlet, let us imagine a function u of the space coordinates, which shall have the value zero at every point within or on S, and, at every point outside of S, shall be equal to the positive root of the equa tion .F(A) which belongs to that point ; and let us con sider the integral jr i. fY-i x * l ? ^ > \ V= irabcp I 1 -- - -- * --- 1 P Ju \ s + a* s + b* 8 + f/ ds (s + a*) 1 * (s + 6 2 ) 1/2 (s + c 2 ) 1 / 2 which evidently vanishes at infinity. For inside points where u is zero, V is identical with the value just found for the potential function within a homogeneous ellipsoid of density p. Since V involves x explicitly and also implicitly through u, we have, in general, at any outside point, -r\ y __ _ Q 7 Cf)JC TrabcpD x u / x 2 y 2 z 2 *+* M-hu 126 SURFACE DISTRIBUTIONS. but, from the definition of u, the coefficient of D x u vanishes when u is positive, so that the integral alone remains and gives the value already found for the x component of the attraction at an outside point due to a homogeneous ellipsoid of density p bounded by S. At S, D x V is continuous : V gives every where, therefore, the value of the potential function due to a homogeneous ellipsoid of density p bounded by S. If we note that r(^ + _j_ + ^_\ J \s -f- a 2 s 4- b 2 s + c 2 / (s + a ds *) 1 * (s + 6 2 ) 1/2 (s + C 2 ) 1 2 -2 (s -f a 2 ) l/2 (s -f P) 1/f (i + c 2 ) 1 and that the equation F(u) = yields 2 x ( x 2 ?/ 2 D * U = (a 2 + u) / \(a* + w) 2 + ( //2 + w) 2 + (with similar values for D tf ? and D z ii), so that , for an outside point, and zero for a point within S, it is easy to see that V satisfies * Laplace s Equation without S and Poisson s Equation within /S, as it should. 52. Logarithmic Potential Functions. When a distribution of matter attracting or repelling according to the "Law of Nature " is such that by a proper choice of axes of reference for a set of orthogonal Cartesian coordinates the density can be made to depend on two of these coordinates only, the dis tribution evidently extends indefinitely far in both directions parallel to the third axis. Such a distribution is sometimes said to be "columnar." Any infinitely long cylinder the density of every filament of which is the same throughout the whole length of that filament, though different filaments * Picard, Traitt ft Analyse, Vol. I, p. 177. GREEN S THEOREM. 127 may have different densities, is a columnar distribution. If we choose for z axis a line parallel to these filaments, the components of the force taken parallel to the x and y axes at any point involve x and y only, and there is no force compo nent parallel to the axis of z. Since the z coordinate will not appear in any of our equations, we may represent a columnar distribution by its trace in the xy plane, if we keep in mind the fact that the distribution itself extends to infinity in both directions perpendicular to this plane. It is evident from the work of Section 6 that a fine, homo geneous filament of cross-section A^ 1? made of repelling matter FIG. 30. of density p lt urges a unit mass at a point at a distance r from the filament with a force of fy \ A :L ^ absolute kinetic force units. It follows that if the trace of a columnar distribution in the xy plane is an area A^ the force components at the point (x, ?/, z) parallel to the axes of x and y are 2p l (x-x l )dA l v CC 2 PiG/- Vi) dA i "ff^ y y \ x i where p^ is the density at any point the x and y coordinates of which are x l and y l respectively, and where the integrals are to be extended over the whole of A lf The integral v = +ffpi ] g [(*i - *Y + (^i - y) 2 ] dA i =ff 2 PI lo s ^^i 128 SURFACE DISTRIBUTIONS. extended over A is called from its form the "logarithmic potential function " belonging to the distribution, and X= + D X V, Y= + I) y V. In the general case the columnar distribution must be con sidered to be made up partly of filaments of positive matter and partly of filaments of negative matter, so that the density is positive for some values of x and y and negative for others. Under these circumstances X and Y represent the force com ponents which would act on a unit quantity of positive matter concentrated at the point (a?, y, z). It will be convenient to denote the amount of matter (reckoned algebraically) in the unit length of a columnar distribution by M. It is evident that at an infinite distance (in the xy plane) from the trace A l of a columnar distribution the logarithmic potential function becomes infinite, unless M is zero, while the force components vanish in any case. It is easy to prove that, if M is zero, V so vanishes at infinity in the xy plane that, if r is the distance from any finite point in the plane, r V and r 2 D r V have finite limits as r increases indefinitely. If M is not zero, V becomes infinite at infinity in such a way that the quantities ( V 2 M log r), (r.D r V-2M),(r. V-D r V- Jf logr), and (V-r>\ogr D r V) all approach the limit zero when r becomes infinite. That X, Y, and V are finite at every finite point in the xy plane outside of A l is evident ; that no one of them is infinite at any point within A l can be proved by transforming the integrals which define them to polar coordinates, using the suspected point as origin. If n is the exterior normal of any closed curve s in the xy plane, and r the distance from any fixed point in the plane, the line integral of cos (n, r)/r taken around s is equal to zero, TT, or 2 ?r, according as is without, on, or within s. From this it follows that the line integral around any closed curve in the xy plane, of the normal outward component of the force due to any columnar distribution of repelling matter the lines of which are perpendicular to that plane, is equal GREEN S THEOREM. 129 to 4?r times the mass of the unit length of so much of the columnar distribution as is surrounded by the curve. AVc may regard this as Gauss s Theorem applied to columnar distributions. If a function u involves x and y and does not involve z, no confusion need be caused by denoting D x -u + Du by \" 2 n. Using this notation, Green s Theorem for functions of the two variables x and y may be written in the form >J* D * u < + D,u D y iv) dA = Cu D H w -ds- C Cu V 2 w dA = Cw D n u -ds C Cw V-u dA, where the line integrals are to be extended around a closed curve s in the xy plane, within and on which u and w with their first derivatives are continuous, and the double integrals extended over the area shut in by s. If in this equation we make u = 1 and w the logarithmic potential function V, due to a columnar distribution, we get and this, according to the special form of Gauss s Theorem, just stated, is equal to Since the form of the curve * may be chosen at pleasure, it must be true that at every point V 2 V = + 4irp. It is desirable to notice that the plus sign here precedes 4 -n-p, whereas in Pois- son s Equation, as applied to the Newtonian Potential Func tion of a finite mass, the corresponding sign is minus. This and many other differences of sign that appear in our equa tions might have been removed if the opposite sign had been given to the integral which defines the logarithmic potential 130 SURFACE DISTRIBUTIONS. of a columnar distribution, but if this had been done a positive mass would have given rise to a negative potential function, and this might have caused confusion. If a portion of a columnar distribution consists of a surface charge on a cylindrical surface, we may conveniently construct a small quadrilateral in the xy plane by drawing two normals across the ends of an element of the trace of the cylindrical surface and two very near curves parallel to the trace element, one on one side and the other on the other. If, then, we apply Gauss s Theorem to this quadrilateral, we shall learn that at every point of the trace the sum of the normal derivatives of V taken away from the curve on each side is 47r<r. If a closed curve s be drawn in the xy plane so as to include the trace of a portion of a columnar distribution the lines of which are perpendicular to that plane and to exclude the trace of another portion, and if Fi and V z represent the parts of the potential function V belonging to these two portions of the distribution, we may apply Green s Theorem to V and the logarithm of the distance from a fixed point in the plane. If n represents a normal pointing outward from s, we shall find that r) /r COS (71, i) (* T\ Tr ~\ ^ * ds I D n V log r ds is equal to the value at of 2 TT F 2 , if is within s ; and to the value at of 2 ?r F 1? if is without 5. If s happens to be a curve on which F is constant, log r ds is equal to the value at of FI, if is without s, or of V, F 2 , if is within s. The reader may compare these results with those given in equations [153] and [157]. If a function w =f(x, y) has the value zero at every point of a closed curve s t in the xy plane and the constant value C all over another closed curve s a , shut in by s lf and if between s l and s 2 ? w is everywhere harmonic, we may apply Green s GREEN S THEOREM. 131 Theorem to w and the logarithm of the distance from a fixed point in the plane and prove that where the normals point outward on s t and inward on s 2 , is equal to 0, the value of w at 0, or (7, according as is without !, between s l and 2 , or within s 2 . Surface charges, of density . " ? applied to Sj and s 2 would, therefore, give rise to the potential function w between s l and s 2 . If a function ?r =f(x, y), harmonic at all finite points, has the constant value c on a closed curve s in the xy plane and becomes infinite at infinity in this plane in such a way that limit (w 2 fi log r) = 0, or limit (/ log r D r w w) = 0, where /A is a given constant, then at all points without s, if n is an interior normal, and w is the potential function due to a columnar distribution of superficial density D H w/4:v on the cylindrical surface of which s is the right section. The amount of matter in the unit length of this cylindrical distribution is /u. If within the closed curve s in the xy plane, w =f(x, y) is harmonic, we may apply Green s Theorem to w and the loga rithm of the distance r from a fixed point 1? within s, using as field the region within 5 and without a small circumference drawn around lP This yields 2 jrw ( = w at (\ = \ where n is the exterior normal to s. If r. 2 is the distance from a fixed point 0. 2 , without s, we may prove, in a similar way that = C[w- D n log r. 2 - log r 2 D n w~] ds, 132 SURFACE DISTRIBUTIONS. or If s is a circumference of radius a with centre at (7, and if O l and 2 are inverse points such, that <70! = l l} C0 2 = l>, IJ* = a?, then r 1 /r 2 is constant all over s, f D n wds = ) ) V*wdxdy = 0, and 2 TTW at Oj = J w [Z> n log ^ - Z> H log r 2 ] ds. Moreover ^ I> n log r^ = cos (T-J, ri), r. 2 J) n log ?- 2 = cos (r 2 , w), If = a 2 + rj 2 2 a? 1 ! cos (r a , ^), Z 2 2 = a 2 + r 2 2 2 ar 2 cos (r 2 , n), and the value on s of n/r 2 is h/a, so that taken around the circumference. If we introduce polar coordinates with origin at the centre of s and denote the coordinates of X by Z x and < 1? we shall have __ ato ^ 2 2 -- This is sometimes called "Poisson s Integral. 7? At the centre of the circumference where ^ = 0, w = f wds. ZiraJ EXAMPLES. 1. If the potential function due to a certain distribution of matter is given equal to zero for all space external to a given closed surface S and equal to <f> (x, y, z), where < is a continu ous single-valued function (zero at all points of $), in all space GREEN S THEOREM. 133 within S j there is no matter without S, there is a superficial distribution of surface density upon S, and the volume density of the matter within S is P = ~ ^ W* + D*<t> + Z>,*|. [Thomson and Tait.] 2. Show that, if w is constant on the closed surface S and is harmonic within S, it is constant in the space enclosed by S-, and that if W vanishes at infinity and is everywhere har monic, it is everywhere equal to zero. 3. If two functions, w t and w 2 , which without a closed sur face S are harmonic and vanish at infinity, have on S values which at every point are in the ratio of X to 1, \ being a con stant, then everywhere u\ = \u< 2 . 4. The functions u and v have the constant values u v and \\ on the closed surface S l and the constant values u. 2 and v 2 on the closed surface 3 within S r Between S l and S. 2 , u and v are harmonic. Show that (u - M!) (v, - vj = (u - Vl ) (u. 2 - MI ). 5. Outside a closed surface S, u\ and w. 2 are harmonic and have the same level surfaces. w l vanishes at infinity, while w. 2 has everywhere at infinity the constant value C. Assum ing that a scalar point function v is expressible in terms of another, u, if, and only if, D x v/D x u = D y v/D y u = D z r/D z u, show that u\ 2 is of the form Bu\ 4- C. 6. Show that there cannot be two different functions, W and JF , both of which within the space enclosed by a given surface S (1) satisfy Laplace s Equation, (2) are, together with their first space derivatives, continuous, and (3) are either equal at every point of S, or satisfy on S the equation D n W=D n W, and are equal at some one point. 134 SURFACE DISTRIBUTIONS. 7. Show that, given a set of closed mutually exclusive surfaces, there cannot be two different functions, W and W, which without these surfaces (1) satisfy Laplace s Equation, (2) are, with their first space derivatives, continuous, (3) so vanish at infinity that rW, rW, r*D r W, r*D r W, where r is the distance from any finite fixed point, have finite limits, and which satisfy one of the following relations : (1) at every point on the given surfaces W = W, (2) at every point of every surface D n W = D n W, 8. At every point of a portion (or the whole) of a closed surface S (or of a set of closed surfaces) the functions w l and w 2 have equal values, and at every point of the remainder of S these functions have equal normal derivatives. Outside and on S both functions are harmonic, and they both vanish at infinity in some manner not more closely defined. Each of the integrals J D^dS, \ D n iv 2 dS has evidently the same finite numerical value when taken over S or over any other surface which encloses S. Show that w l and w 2 are identical. If the values of w^ and iv. 2 at a point P, the coordinates of which referred to any fixed point as origin are (r, 0, <), instead of approaching zero as r is made to increase indefi nitely, both approach the limit f(6, <), / being a continuous function, when, with any values of and <, r is made infinite, w l and w 2 are identical. 9. The given closed surface S l shuts in the given closed surface S 2 . The given function w is harmonic between S v and S 2 . Show that no other function than w, harmonic be tween S l and &, has the same value that w has at every point of Si and the same value of the normal derivative at every point of S 2 . Show also that any such function which has the same value of the normal derivative at every point of Si and S 3 that the normal derivative of w has differs from w at most by a constant. No other function than w^ harmonic between Si and $ 2 , has the same value that w has at every point of $ 2 the same value of the normal derivative at every point of GREEN S THEOREM. 135 10. The harmonic function <r. which BO vanishes at infinity that, if r is the distance from any fixed finite point, the limits of r r and rD r V are not infinite, has an open zero level sur face Si as well as a series of closed level surfaces of which one is Sf Show that in the region T, between S l and .S r <r is the potential function due to surface distributions on Si and S. defined by the equation 4 w = /> FT. where n points out of T. The whole charge on the two surfaces is zero. 11. Outside the closed surface N. upon which its value is given at every point, the function <r is harmonic except at certain points, P^ P+, Py, etc., where it becomes infinite in such a way that, if r t represents the distance from P^ <r m l /r t is harmonic at J\. where m t is a constant belonging to the point P^ At infinity i.r vanishes like a Newtonian potential function. Prove that r is unique. If <r is a Newtonian potential function, what do you know about the distribution which gives rise to it ? 12. The functions V, JF, 9, 17 with their first space deriva tives are continuous, everywhere without a given closed sur face S, and they vanish at infinity like a Newtonian potential function due to a finite distribution of matter, r and JT have the same values at every point of S, but outside S\ , B, and O satisfy Laplace s Equation and W does not. The surface integrals of the normal derivatives of 9 and O taken over V are equal, but has the same value all over S, and Q a continuously variable value. Show that, if the integrations embrace all space outside r ~ (/ " <fff l> "r + ( J P r ir) > !) - / 136 SURFACE DISTRIBUTIONS. Hence, show that the energy of a given charge spread on a given surface S is least when the arrangement is equipo- tential. 13. Everywhere within the closed surface S the two scalar point functions V and V 1 are continuous with their first deriv atives. Over a given portion of S, V and V have equal values, while over the remainder of S both D n V and D n V are equal to zero. The vectors q and q have the components \D X V 9 \D V V, \D Z V and \D X V , \D y V , \D Z V respectively, where X is a positive analytic scalar point function. Show that, if q is solenoidal and q is not solenoidal, the integral extended over the whole space within S is less than the integral extended over the same region. 14. Gravitating matter of given uniform density is confined within a given closed surface, but its volume is less than that enclosed by the surface. Prove that its potential energy is a maximum, if the matter forms a shell of which the given sur face is the outer boundary, while the internal boundary is an equipotential surface. 15. Let = /! (x, y) and y = f 2 (x, y) be two analytical functions of x and y such that the two families of curves are orthogonal. Let V be any function of x and y which, with its first space derivatives, is continuous, within and on a closed curve s, drawn in the coordinate plane. Let h^ and h^ be the positive roots of the equations v - (*>.*) + (^vn v - (DM? + (z>^). fv\ Prove that s, the surface integral of h% k^- DA \) taken all GREEN S THEOREM. 137 over the area enclosed by s, is equal to the line integral taken around s of Fcos(, ri), where n is an exterior normal and (, n) represents the angle between n and the direction in which increased most rapidly. Show that the corresponding theorem in three dimensions may be expressed by the equation //JV A D, (jjjj) dr =ff rcos (fc n)d& 16. The operator [(Z> x ) 2 + (>)" + (A) 2 ] applied to any of the quantities xyiz v2, xiy ~v2 , etc., yields zero : is every analytic function of any one of these quantities harmonic ? 17. The product of two harmonic functions, u, v, is itself harmonic if, and only if, the level surfaces of u and v are orthogonal. The product of three harmonic functions, u, v, w, is itself harmonic if, and only if, the level surfaces of u, v, and ic are mutually orthogonal. 18. The function w of the two variables x and y is har monic in the xy plane everywhere outside of the mutually exclusive closed curves s l and s 2 . Upon these curves w has given constant values. At infinity, u- vanishes in such a manner that, if r is the distance from any finite point in the xy plane, r h j^ (r log r D r w - w) = 0. Show that w is the potential function without Sj and s 2 , due to superficial distributions defined by the equation 4 TTO- = Z^w, upon the cylindrical surfaces of which s x and s 2 are the traces. In the formula just given the normal points outward at 5 X and s 2 . 19. The function w of the two variables x and y is har monic everywhere in the xy plane except at certain points, P,, P 2 , P 3 , etc., where it becomes infinite in such a manner that, if r k is the distance from P kJ u- 2 ^ k log r k is harmonic at P k where ^ is a constant belonging to P k . Upon a certain open 138 SURFACE DISTRIBUTIONS. curve, s, w has the value zero, and everywhere at an infinite distance from the origin w so vanishes that show that on either side of s, w may be considered as the logarithmic potential function due to a distribution of elec tricity of density a- = -f on the infinite cylindrical surface 4 7T of which s is a right section, and to distributions upon lines normal to the xy plane which cut the plane at so many of the P points as lie on the chosen side of s. 20. If the normal component of a vector is zero at every point of a closed surface S, and if within and on $the vector is everywhere solenoidal and lamellar, its. components are equal to zero at every point within S. If the normal component of a vector is given at every point of S, and if everywhere within S the curl and the divergence have given values, the vector is determined. If q and q are vectors the normal components of which vanish at every point of S, and if within S, q is solenoidal with curl k, while q is lamellar with divergence Z>, wheref^)and D are given scalar point functions, q + q is *7 the unique vector, the normal component of which is zero at every point of S, and which within S has the curl k and the divergence D. 21. The normal derivative of u with respect to v is (D x u D x v -f- D y u D y v + D z u D z v)/h*. 22. If u = xyz, v = 2 x + y + z, the values at (1, 1, 1) of D v u and D u v are 2/3 and 4/3. 23. The gradients of u and v are numerically equal at every point, though not in general coincident in direction, if, and only if, u + v and u v are orthogonal functions. If the gradients of u and v agree everywhere in direction though not in magnitude, v is expressible as a function of u, so that GREEN S THEOREM. 139 24. If the components parallel to the axes of x and y of the solenoidal vector (if, v, 0), which has no component parallel to the z axis, are independent of z, a vector, directed parallel to the z axis, which has for its intensity any partial inte gral (Q z ) of u with respect to y which satisfies the condition D X Q Z = v, is a vector potential function of the original vector. Thus : (0, 0, x*y + if z 9 ) is a vector potential func tion of (z 2 + 3?/ 2 , 9x s -2xy, 0). The value at the point (x, y, z) of the derivative of Q z , taken in a direction perpen dicular to the z axis and making an angle a + 90 with the plane of xz, is D X Q S cos (a + 90) + D V Q Z sin (a + 90), or r>,,Q z cos a D X Q. sin a, or u cos a + v sin a, and this is the resolved part of the vector .(u, u, 0) at the same point in a direction parallel to the xy plane and making an angle a with the plane of xz. We learn, therefore, that the numer ical value at any point P of the derivative of Q^ taken in any direction s parallel to the xy plane, is equal to the component of the vector (M, v, 0) in a direction parallel to the xy plane and perpendicular to s. Show that the inter section of any plane parallel to the xy plane with a cylin der of the family Q z = constant is a line of the vector (u, v, 0). Show also that D X *Q Z + D*Q Z = - (D x v - D,u), the negative of the component parallel to the axis of the curl of (u, v, 0). 25. A vector parallel to the x axis of intensity independ ent of z and equal to the negative of a partial integral of w with respect to y, and a vector parallel to the y axis of inten sity independent of z and equal to a partial integral of w with respect to x, are vector potential functions of the vector (0, 0, w), provided w is independent of z. For example : the vectors [y 2 - 3x*y +/(), 0, 0] and [0, x* - 2xy + <<, 0] are vector potential functions of the vector (0, 0, 3x 2 2y). 26. If the lines of a vector are circles parallel to the xy plane with centres on the z axis, and if the intensity of the vector is a function f(r) of the distance r from that axis, a vector, everywhere parallel to the z axis, of intensity F(r), 140 SURFACE DISTRIBUTIONS. where f(r) = D r F(r) is a vector potential function of the original vector. Is this original vector solenoidal ? 27. If the lines of a vector are straight lines parallel to the xy plane and emanating from the z axis, and if the intensity of the vector is a function /(/) of the distance r from this axis,/(r) must be of the form c/r if the vector is solenoidal. A vector with such lines as these cannot be solenoidal if the intensity at every point is a given function of the angle which the line of the vector through that point makes with the xz plane. 28. The lines of the vector [jc -f(x, y), y -f(x, y), 0] are straight lines parallel to the xy plane and emanating from the z axis, and its curl is of the form (0, 0, y D x f x - D y f). If/ is expressible as a function of the angle tanr*.(y/a;), y D x f x Dyf is also expressible as a function of this angle, but if / is expressible as a function of r = Vx 2 -f- y 2 , y D x f x D y f vanishes and no vector of the form J can be a vector potential function of the vector [0, 0, <(>)] If the ratio of y to x be denoted by //-, and if /(/n) = the vector |> /(/*), ?//<, 0] is a vector -\- 1 potential function of the vector [0, 0, <(/K)]. 29. The lines of the vector [ y -f(x, y}, x -f(x, y), 0] are circles parallel to the xy plane with centres on the 2 axis, and its curl is of the form (0, 0, 2f+x-D x f+ V D y f). Show that if / is expressible as a function of r, the distance from the z axis, so is 2f+x- D x f + y D y f, and that, if \_y-F(r\ x-F(r), 0] is a vector potential function of the solenoidal vector [0, 0, <(/)]. Show also that if /is expressi ble as a function of the angle tan" 1 (y/x), that is, as a function of the ratio, ^ of y to x, 2f+ x D x f+ y - D y f is expressible GREEN S THEOREM. 141 as a function of //,, and that [ % y /(/")> ~k x /(/*)> 0] is a vector potential function of [0, 0, /(//.)]. 30. The difference between the values at any two points .4 and B of any analytic scalar point function V is equal to the line integral taken along any path from A to B of the tangential component of the vector (D X V, D y V<, D.V). 31. The only families of plane curves which are at once the right sections of possible systems of equipotential cylin drical surfaces in empty space due to columnar distributions of matter which attracts according to the " Law of Nature," and also the generating curves of possible systems of equipo tential surfaces of revolution due to distributions of such matter symmetrical about the common axis of these surfaces, are families of concentric conies. Must every such family of conies be confocal ? [Am. Jour. Math., 1896.] 32. If a vector is determined at every point by means of the components (R, 0, Z) in the directions in which the columnar coordinates of the point increase most rapidly, the divergence of the vector may be written D r R + R/ r + D /r + D.Z. 33. The equation represents, when a, b, and c are fixed, a family of confocal quad- ric surfaces of which X is the parameter. If a>b>c, and if x, y, and z are chosen at pleasure, the cubic in X has three real roots (it, v, w) ; one between a 2 and ft 2 , corresponding to a parted hyperboloid, one between b 2 and c 2 , correspond ing to an unparted hyperboloid, and one between c 2 and oo, corresponding to an ellipsoid, so that through every point of space three surfaces of the family can be drawn, and it is easy to see that these cut each other orthogonally. The direction cosines of a surface of constant X have the values D x \/h, D y \/h, D x \/h, where /V 2 = (T^A) 2 + (D^ + (AX) 2 . D x X = - 2 x/(a? + X) D^F, and ?i 2 = - 142 SURFACE DISTRIBUTIONS. Belonging to every point in space are three values of A (?/, v, w), and three values of h (k u , h v , /*,), and, if we sub stitute u, v, and w successively for A in the equation FQC) = 0, we shall get three linear equations in x 2 , y 2 , z 2 from which we may obtain expressions for x, y, z in terms of u, v, w. h 2 = - 4 [( 2 + ) (b 2 + u) o 2 4 M>] /[( M - v ) ( u - w )] and h v 2 and h w 2 have corresponding values which, substituted in gives Laplace s Equation in terms of the orthogonal curvi linear coordinates (M, v, w). Prove that if we assume that a solution of this equation exists which involves w only and vanishes when w is infinite, the equation which determines this solution takes the form D w \[(a 2 4 w) (b 2 4 iv) (c 2 4 w)] 1 2 D w V \ = 0, /rim 71 (d 4 tv) l/2 (b 2 4 iv) l/2 (c 2 4 w) 1/a = C f" ^ J w (a 2 4 *^) 1/2 (6 2 4 ^6) 1/2 (c 2 4 w) 1/a Hence, show that a set of confocal ellipsoids are possible external equipotential surfaces, and that if M is the mass of the corresponding distribution the potential function is given by the last equation, in which, since a very large value of w corresponds to an ellipsoid little different from a sphere of radius Vt0, C is to be determined by the equation Find the density of a superficial distribution on a surface of the w family, the potential function of which at all outside points shall be the function just defined. 143 34. The curl of the curl of a solenoidal vector such that the three functions which give the strengths of its components parallel to the coordinate axes satisfy Laplace s Equation vanishes. If the lines of a vector are all parallel to a plane, and the vector has the same value at all points in any line per pendicular to the plane, the vector is perpendicular to its curl. 35. A certain vector, the tensor of which is f(x, y, z), is at every point directed exactly in the direction of the straight line which joins the origin with the point in question ; show that the vector is not necessarily lamellar, but that it is per pendicular to its curl. If all the components of a vector are functions of x and y only, or if all are functions of x only, or if one component vanishes and the other components are functions of x, y, and 2, the vector may or may not be perpendicular to its curl. 36. If (Q x , Q y , Q z ) are the components of a vector Q, (\i, pi, v : ) the curl components, (\. 2 , /x 2 , v 2 ) the components of the curl of the curl of Q, and so on, At = D y Q z - D z Q y , \, = D x (Div Q) - V 2 &, X 3 = V 2 Xi, A. 4 V 2 A 2 , and so on. How are these equations changed if Q is a solenoidal vector ? 37. If the harmonic function f(x, y, z) represents the x component of a vector which is both solenoidal and lamellar, the y and z components must be of the form where \f/(y, z) is a solution of the equation 38. A certain vector (X, Y, Z) is not perpendicular to its curl (K x , K y , K g ). Show that the scalar function F, deter mined from the equation 7i x D X F + K y D y F 4- K z D Z F = - (K^ + K y Y 144 SURFACE DISTRIBUTIONS. is the scalar potential function of a lamellar vector (L, M, N), which added to the first vector gives a new vector perpendic ular to its curl. Is this equation always integrable ? 39. A vector Q, with components (Q x , Q y , Q^, is continuous except at a certain surface S. In each of the regions sepa- rated by S, D x Q tJ = D y Q x , D X Q Z = V Z Q X , D y Q z = V z Q y , so that at every point within these regions the curl of Q vanishes. Investigate the value of the curl of Q on S when the normal (or a tangential) component of Q is discontinuous there. 40. Unless V 2 f= 0, a vector the x component of which is f(x, y, z) cannot be both lamellar and solenoidal. 41. Matter spread uniformly in a superficial distribution on a circular portion of a plane forms a "circular surface distribution." Two such distributions, each of radius a, are placed parallel and opposite each other at a distance 8 apart. If the density of one of these be + a- and that of the other cr, and if 8 be made to approach zero and o- to increase in such a manner that the product of a- and 8 is always equal to the constant ^ the resulting value of the potential function is said to be due to a " circular double layer " of radius a, and density p. Show that the limiting value of the potential function at a point P on the axis of the double layer and at a distance x from its plane is 2^^ (I x/^/a 2 + x 2 ), where the positive sign is to be used if P is on one side of the double layer, and the negative sign if P is on the other side. Is the potential function discontinuous at the double layer ? Is the force discontinuous ? 42. Assuming the surface of the earth as defined by the sea- level to be a spheroid of ellipticity e, prove that the mass of the earth in astronomical units is a Vo (1 + e f m), where g Q is the force t>f gravity at the equator, a the equatorial radius, and in the ratio of " centrifugal force " to true gravity at the equator. ELECTROSTATICS. 145 CHAPTER V. THE ELEMENTS OF THE MATHEMATICAL THEOKY OP ELEOTKIOITY, I. ELECTROSTATICS. 53. Introductory. Having considered abstractly a few of the characteristic properties of what has been called "the New tonian potential function," we will devote this chapter to a very brief discussion of some general principles of Electrostatics and Electrokinetics. By so doing we shall incidentally learn how to apply to the treatment of certain practical problems many of the theorems that we have proved in the preceding chapters. In what follows, the reader is supposed to be familiar with such electrostatic phenomena as are described in the first few chapters of treatises on Statical Electricity, and with the hypoth eses that are given to explain these phenomena. Without expressing any opinion with regard to the physical nature of what is called electrification^ we shall here take for granted that whether it is due to the presence of some sub stance, or is only the consequence of a mode of motion or of a state of polarization, we may, without error in our results, use some of the language of the old "Two Fluid Theory of Elec tricity " as the basis of our mathematical work. The reader is reminded that, among other things, this theory teaches that : (1) Every particle of a body which is in its natural state con tains, combined together so as to cancel each other s effects at all outside points, equal large quantities of two kinds of elec tricity with properties like those of the positive and negative " matter" described in Section 44. (2) Electrification consists in destroying in some way the equality between the amounts of the two kinds of electricity which a body, or some part of a body, naturally contains, so that there shall be an excess or charge of one kind. If the 146 ELECTROSTATICS. charge is of positive electricity, the body is said to be posi tively electrified ; if the charge is negative, negatively electrified. Either kind of electricity existing uncombined with an equal quantity of the other kind, is called free electricity. (3) When a charged body A is brought into the neighborhood of another body B in its natural state, the two kinds of elec tricity in every particle of B tend to separate from each other, one being attracted and the other repelled by ^L s charge, and to move in opposite directions. In general, a tendency to separation occurs in all parts of the body, whether it is charged or not, where the resultant electric force (the force due to all the free electricity in existence) is not zero. This effect is said to be due to induction. In our work we shall assume all this to be true, and proceed to apply the principles stated in Section 44 to the treatment of problems involving distributions of electricity. We shall find it convenient to distinguish between conductors, which offer prac tically no resistance to the passage of electricity through their substance, and nonconductors, which we shall regard as prevent ing altogether such transfer of electricity from part to part. 54. The Charges on Conductors are Superficial. When elec tricity is communicated to a conductor, a state of equilibrium is soon established. After this has taken place, there can be no resultant force tending to move any portion of the charge through the substance of the conductor, for, by supposition, the conductor does not prevent the passage of electricity through itself. Moreover, the resultant electric force must be zero at all points in the substance of a conductor in electric equilibrium ; for if the force were not zero at an} point, electricity would be produced by induction at that point, and carried away through the body of the conductor under the action of the inducing force. From this it follows that the potential function V, due to all the free electricity in existence, must be constant throughout ELECTROSTATICS. 147 the substance of any single conductor in electric equilibrium, whether or not the conductor be charged, and whether or not there be other charged or uncharged conductors in the neigh borhood. Different conductors existing together will in general be at different potentials, but all the points of an} one of these conductors will be at the same potential. Wherever V is constant, V 2 T"=0, and hence, by Poisson s Equation, p = 0, so that there can be no free electricity within the substance of a conductor in equilibrium, and the whole charge must be distributed upon the surface. Experiment shows that we must regard the thickness of charges spread upon conductors as inappreciable, and that it is best to consider that in such cases we have to do with really superficial distributions of electricity, in which the conductor bears a rough analogy to the cavity enclosed by the thin shells of repelling matter de scribed in the preceding chapter. The surface density at any point of a superficial distribution of electricity shall be taken positive or negative, according as the electricity at that point is positive or negative, and the force which would act upon a unit of positive electricity if it were concentrated at a point P without disturbing existing distribu tions shall be called ;i the electric force" or the strength of the electric field at P." It is evident, from Sections 45 and 46, that the electric force at a point just outside a charged conductor, at a place where the surface density of the charge is o-, is 47r<r, and that this is directed outwards or inwards, according as a- is positive or nega tive. In other words, D n V, the derivative of the potential function in the direction of the exterior normal, is equal to 4 TTO-, and the value of V at a point P just outside the conductor is greater or less than its value within the conductor, according as the surface density of the conductor s charge in the neighborhood of P is negative or positive. It is to be carefully noted that, although the surface of a con ductor must always be equipoteutial. the superficial density of 148 ELECTKOSTATICS. the conductor s charge need not bo the same at all parts of the surface. We shall soon meet with cases where the electricity on a conductor s surface is at some points positive and at others negative, and with other cases where the sign of the potential function inside and on a conductor is of opposite sign to the charge . It is evident, from the work of Section 47, that the resistance per unit of area which the nonconducting medium about a con ductor has to exert upon the conductor s charge to prevent it from flying off, is, at a part where the density is o-, 2rra 2 . 55. General Principles which follow directly from the Theory of the Newtonian Potential Function. If two different distribu tions of electricity, which have the same system of equipoten- tial surfaces throughout a certain region, be superposed so as to exist together, the new distribution will have the same equipo- tential surfaces in that region as each of the components. For, if FI and F 2 , the potential functions due to the two components respectively, be both constant over any surface, their sum will l>e constant over the same surface. Two distributions of electricity, which have densities every where equal in magnitude but opposite in sign, have the same system of equipotential surfaces, and, if superposed, have no effect at any point in space. Two distributions of electricity, arranged successively on the same conductor so that at every point the density of the one is m times that of the other, have the same system of equipo tential surfaces, and the potential function due to the first is everywhere m times as great as that due to the second. If the whole charge of a conductor which is not exposed to the action of any electricity except its own is zero, the super ficial density must be zero at all points of the surface, and the conductor is in its natural state. For if o- is not everywhere zero, it must be in some places positive and in others negative ; and, according to the work of the last section, the potential function F, due to this charge, must have, somewhere outside ELECTKOSTATICS. 149 the conductor, values higher and lower than T",, its value in the conductor itself. But this would necessitate somewhere in empty space a value of the potential function not lying between V and 0, the value at infinity ; that is, a maximum in empty space if VQ is positive, and a minimum if T" is negative ; which is absurd. A system of conductors, on each of which the charge is null, must be in the natural state if exposed to the action of no out side electricity. For, by applying the reasoning just used to that conductor in which the potential function is supposed to have the value most widely different from zero, we may show that the surface density all over the conductor is zero, so that no influence is exercised on outside bodies ; and then, suppos ing this conductor removed, we may proceed in the same way with the system made up of the remaining conductors. If a charge M of electricity, when given to a conductor, ar ranges itself in equilibrium so as to give the surface density a- =/(#, ;y, z) and to make the potential function T\, = I - J T constant within the conductor, a charge J/, if arranged on the conductor so as to give at every point the density <r= f(x,y,z) would be in equilibrium, for it would give everywhere the poten tial function I I ^lf_f = _ y^ an( ] this is constant wherever V^ J r is constant. Only one distribution of the same quantity of electricity M on the same conductor, removed from the influence of all other electricity, is possible ; for, suppose two different values of sur face density possible, o-j =f l (x^ y, z) and o- 2 =/ 2 (a 1 , y, z), then o- 2 = / 2 (.T,#, z} is a possible distribution of the charge J/. Superpose the distribution a-. 2 upon the distribution o-j so that the total charge shall be equal to zero ; then the surface density at every point is a- l o- 2 , and this must be zero by what we have just proved, so that ^ = o- 2 . Since we may superpose on the same conductor a number of distributions, each one of which is by itself in equilibrium, it is 150 ELECTROSTATICS. easy to sec that if the whole quantity of electricity on any con ductor be changed in a given ratio, the density at each point will be changed in the same ratio. 56. Tubes of Force and their Properties. We have seen that a unit of positive electricity concentrated at a point P just out side a conductor would be urged away from the conductor or drawn towards it, according as that point on the conductor which is nearest P is positively or negatively electrified. If we regard lines of force drawn in an electric field as generated by points moving from places of higher potential to places of lower poten tial, we may say that a line of force proceeds from every point of a conductor where the surface density is positive, and that :i line of force ends at every point of a conductor where the sur face density is negative. No line of force either leaves or enters a conductor at a point where the surface densit} 7 is zero, and no line of force can start at one point of a conductor where the electrification is positive and return to the same conductor at a point where the electrification is negative. No line of force can proceed from one conductor at a point electrified in an} T way and enter another conductor at a point where the electrification has the same name as at the starting-point. A line of force never cuts through a conductor so as to come out at the other side, for the force at ever} point inside a conductor is zero. Lines and tubes of force are sometimes called in electrostatics lines and tubes of " induction." When a tube of force joins two conductors, the charges Q } , Q 2 of the portions $ 15 $ a which it cuts from the two surfaces are made up of equal quantities of opposite kinds of electricity. For if we suppose the tube of force to be arbitrarily prolonged ELECTROSTATICS. 151 and closed at the ends inside the two conductors, the surface integral of normal force taken over the box thus formed is zero, for the part outside the conductors } ields nothing, since the re sultant force is tangential to it, and there is no resultant force at any point inside a conductor. It follows, from Gauss s Theorem, that the whole quantity of electricity (Qi + Qi) inside the box must be zero, or Qi = Q 2 -> which proves the theorem. If (T l and o-o are the average values of the surface densities of the charges on S\ and S 2 respectively, we have <riS i = Q l and o- 2 So = Q 2 , whence - = --,| [162] The integral taken over any surface, closed or not, of the force normal to that surface is called by some writers the flow of force across the surface in question, and by others the induc tion through this surface. If we apply Gauss s Theorem to a box shut in by a tube of force and the portions S^ S 2 which it cuts from any two equipotential surfaces, we shall have, if the box contains no electricity, F.,X., F 1 S 1 = 0. [163] where F l and F. 2 are the average values, over S t and S 2 respec tively, of the normal force taken in the same direction (that in which V decreases) in both cases. In other words, the flow of force across all equipotential sections of a tube of force con taining no electricity is the same, or the average force over an oquipotential section of an empty tube of force is inversely pro portional to the area of the section. FIG. 41. When a tube of force encounters a quantity m of electricity (Fig. 41), the flow of force through the tube on passing this 152 ELECTROSTATICS. electricity is increased by irm. If, however, the tube encoun ters a conductor large enough to close its end completely, a charge m will be found on the conductor just sufficient to reduce to zero the flow__of force (7) through the tube. That is, 47T It is sometinrelf convenient to consider an electric field to be divided up by a system of tubes of force, so chosen that the now of force across any equipotential surface of each tube shall be equal to 4?r. Such tubes are called unit tubes ;* for wherever one of them abuts on a conductor, there is always the unit quan tity of electricity on that portion of the conductor s surface which the tube intercepts. In some treatises on electricity the term " line of force" is used to represent a unit tube of force, as when a conductor is said to cut a certain number of " lines of force." It is evident that m unit tubes abut on a surface just outside a conductor charged with m units* of either kind of electricity, if the superficial density of the charge has everywhere the same sign. These tubes must be regarded as beginning at the con ductor if m is positive, and as end&nfthpTe if m is negative. If a conductor is charged at some places with positive elec tricity and at others with negative electricity, tubes of force will begin where the electrification is positive, and others will end where the electrification is nega tive. It is evident that no tube of force can return into itself. 57. Hollow Conductors, When the nonconducting cavity, shut in by a hollow conductor K (Fig. 42), contains FIG. 42. * They are sometimes called "unit Faraday tubes," to distinguish them from the more slender tubes of unit induction, of which 4 irm start from a body which has a positive charge m. ELECTROSTATICS. 153 quantities of electricity (m lt ra 2 , m 3) etc., or ^_^ m ) distributed in any way, but insulated from K, there is induced on the walls of the cavity a charge of electricity algebraically equal in quantity, but opposite in sign, to the algebraic sum of the electricities within the cavity. Call the outside surface of the conductor S and its charge J/ , the boundary of the cavity S t and its charge 3/<, and sur round the cavity by a, closed surface S, every point of which lies within the substance of the conductor, where the resultant force is zero. Now the surface integral of normal force taken over S is zero, so that, according to Gauss s Theorem, the algebraic sum of the quantities of electricity within the cavity and upon S t is zero. That is, ) =0, [164] and this is our theorem, which is true whatever the charge on S is, and whatever distribution of free electricity there may be outside K. If the distribution of the electricity within the cavity be changed by moving ??ij, m 2 , etc., to different positions, the distribution of Jl/jon^,- will in general be changed, although its value remains unchanged. If A" has received no electricity from without, its total charge must be zero ; that is, If a charge algebraically equal to J/ be given to /i, M^M-Mt. The combined effect of f (??i),the electricity within the cavity, and J/i, the electricity on the walls of the cavity, is at all points without S t absolutely null. For, if we apply [153] to/$, any sur face drawn in the conductor so as to enclose S { , we shall have D,y everywhere zero, since the potential function is constant within the conductor ; this show r s that FI, the potential function due to 154 ELECTROSTATICS. all the electricity within $, must be zero at all points without S ; but S may be drawn as nearly coincident with S t as we please. Hence our theorem, which shows that, so far as the value of the potential function in the substance of the conductor or outside it, and so far as the arrangement of M and of M r , any free electricity there may be outside /f, are concerned, M t and N (m) might be removed together without changing anything. The potential function at all points outside S t is to be found by con sidering only M and M 1 . If Si happens to be one of the equipotential surfaces of N (m) considered by itself, M t will be arranged in the same way as a charge of the same magnitude would arrange itself on a con ductor whose outside surface was of the shape $, , if removed from the action of all other free electricity. The potential function ( F 2 ) due to M and M is constant everywhere within S ; for if we apply [154J to a surface , drawn within the substance of the conductor as near S as we like, we shall have F S -F O = O, which proves the theorem. The potential function within the cavity is equal to F 2 + Fj, where Fi is the potential function due to M t and ^ (m). Of these, F 2 is, as we have seen, constant throughout K and the cavity (Section 31) which it encloses, while FI has different values in different parts of the cavity, and is zero within the substance of the conductor. Suppose now that, by means of an electrical machine, some of the two kinds of electricity existing combined together in a conductor within the cavity be separated, and equal quantities (g) of each kind be set free and distributed in any manner within the cavity. The value of V } within the cavity will probably be different from what it was before, but F 2 will be unchanged ; for the ELECTKOSTATICS. 155 quantity of matter in the cavity is unchanged, being now, alge braically considered. so that J/, is unchanged, although it may have been differently arranged on $,, in order to keep the value of Fi zero within the substance of the conductor. If now a part of the free electricity in the cavity be conveyed to S t in some way, the sub stance of the conductor will still remain at the same potential as before. For, if I units of positive electricity and n units of negative electricity be thus transferred to /S ,-, the whole quantity of free electricity within the cavity will be N (m) ?-(-, and that on S f will be 3/ 4 -+- / n : but these are numerically equal, but opposite in sign, and the charge on $<, if properly arranged, suffices, without drawing on 3/ to reduce to zero the value of FI in K. Since 3/ and 3/ r remain as before, F 2 is unchanged, and the conductor is at the same potential as before. So long as no electricity is introduced into the cavity from without K* no electrical changes within the cavity can have any effect out side S^. Most experiments in electricity are carried on in rooms, which we can regard as hollows in a large conductor, the earth. F 2 , the value of the potential function in the earth and the walls of the room, is not changed by anything that goes on inside the room, where the potential function is F= F! + F 2 . Since we are generally concerned, not with the absolute value of the poten tial function, but only with its variations within the room, and since F 2 remains always constant, it is often convenient to dis regard Fo altogether, and to call FI the value of the potential function inside the room. When we do this we must remember that we are taking the value of the potential function in the earth as an arbitrary zero, and that the value of FI at a point in the room really measures only the difference between the values of the potential function in the earth and at the point in ques tion. When a conductor A in the room is connected with the 156 ELECTROSTATICS. walls of the room by a wire, the value of Vj in A is, of course, zero, and A is said to have been put to earth. 58. Induced Charge on a Conductor which is put to Earth. Suppose that there are in a room a number of conductors, viz. : AI charged with M l units of electricity, and A z , A-^ A 4 , etc., connected with the walls of the room, and therefore at the po tential of the earth, which we will take for our zero. If the potential function has the value p v inside A^ every point in the room outside the conductors must have a value of the potential function lying between ^ and 0, else the potential function must have a maximum or a minimum in empty space. If p t is posi tive, there can be no positive electricity on the other conductors ; for if there were, lines of force must start from these conductors and go to places of lower potential ; but there are no such places, since these conductors are at potential zero, and all other points of the room at positive potentials. In a similar way we may prove that if p r is negative, the electricity induced on the other conductors is wholly positive. Now let us apply [154 B ] to a spherical surface, drawn so as to include A } and at least one of the other conductors, but with radius a so small that some parts of the surface shall lie within the room. If we take the point at the centre of this surface, we shall have = ! CD r V-ds + - 9 Cvds. [165] aJ o?J If M is the whole quantity of electricity within the spherical surface, there must be a quantity M outside the surface, either on the walls of the room or on conductors within the room. The value at of the potential function, V<>, due to the elec tricity without the sphere, is less in absolute value than , for it could only be as great as this if all the electricity outside the sphere were brought up to its surface. By Gauss s Theorem, ELECTROSTATICS. 157 therefore, Cvds = 4 * \_M + a TV] . [166] Now, if M l is positive, the integral is positive, for all parts of the spherical surface within the room yield positive differentials, and all other parts zero, so that the second side of the equation is positive. But aF 2 is of opposite sign to J/, and is less in absolute value ; hence, J/ is positive, and the total amount of negative electricity induced on the other conductors within the spherical surface by the charge on A^ is numerically less than this charge, unless some one of these conductors surrounds A 2 ; in which case the induced chargo comes wholly on this conduc tor, while the other conductors, and the walls of the room, are free. Some of the tubes of force which begin at A l end on the walls of the room, provided these latter can be reached from A 1 without passing through the substance of any conductor. 59. Coefficients of Induction and Capacity. If a number of insulated conductors, A 2 , A~, A 4 , etc., are in a room in the pres ence of a conductor A^ charged with 3/j units of electricity, the whole charge on each is zero ; but equal amounts of positive and negative electricity are so arranged by induction on each, that the potential function is constant throughout the substance of every one of the conductors. Let the values of the potential functions in the system of con ductors be 2ht Pzi P^ Pn etc. Since each conductor except AI is electrified, if at all, in some places with positive electricity, and in others with negative electricity, some lines of force must start from, and others end at, every such electrified conductor, so that there must be points in the air about each conductor at lower and at higher potentials than the conductor itself. But the value of the potential function in the walls of the room is zero, and there can be no points of maximum or minimum poten tial in empty space ; so that />, must be that value of the poten tial function in the room most widely different from zero, and Pn Put P^ etc., must have the same sign as />,. The reader may show, if he likes, that both the negative part 158 ELECTKOSTATICS. and the positive part of the zero charge of any conductor, ex cept AH is less than M. Let p n be the value of the potential function in a conductor AI charged with a single unit of electricity, and standing in the presence of a number of other conductors all uncharged and insulated. Then if ^ 12 , _p 18 , Pm etc., are, under these cir cumstances, the values of the potential functions in the other conductors, A<>, A& A^ etc., the potential functions in these conductors will be M l p\^ M\P\zi ^f\P\\-> etc., if A l be charged with M l units of electricity instead of with one unit. This is evident, for we may superpose a number of distributions which are singly in equilibrium upon a set of conductors, and get a new distribution in equilibrium where the density is the sum of the densities of the component distributions, and the value of the resulting potential function the sum of the values of their potential functions. If AI be discharged and insulated, and a charge M 2 be given to A 2 , the values of the potential functions in the different con ductors ma be written If now we give to A 1 and A 2 at the same time the charges M\ and M 2 respectively, and keep the other conductors insulated, the result will be equivalent to superposing the second distribu tion, which we have just considered, upon the first, and the con ductors will be respectively at potentials, M l p n -\-M. 2 p 2l , Mipu + Mzpzi, Mtpu + ^>fe etc. [167] If all the conductors are simultaneously charged with quanti ties J/i, Jfefa, MM M, etc., of electricity respectively, the value of the potential function on A k will be V = Writing this in the form V k = a. k + M k p kk , we see that if the charges on all the conductors except A be unchanged, a, will be constant, and that every addition of units of electricity to ft* KLK<Ti:< STATICS. 159 the charge of A k raises the value of the potential function in it by unity. If we solve the n equations like [168] for the charges, we shall get n equations of the form M k = F! g lt + T; g,, 4- T>. <] Ak + - + F,</u + - + V K q nk , [169] where the q s are functions of the p &. If all the conductors except A k are connected with the earth, M k = V k q^, and ? u . is evidently the charge which, under these circumstances, must be given to A k in order to raise the value of the potential function in it by unity. It is to be noticed that q kk and are in general different. Pm The charge which must be given to a conductor when all the conductors which surround it are in communication with the earth, in order to raise the value of the potential function with in that conductor from zero to unity, shall be called the capacity of the conductor. It is evident that the capacity of a conductor thus defined depends upon its shape and upon the shape and position of the conductors in its neighborhood. 60. Distribution of Electricity on a Spherical Conductor. Considerations of symmetry show that if a charge J/ be given to a conducting sphere of radius r. removed from the influence of all electricity except its own, the charge will arrange itself uniformly over the surface, so that the superficial density shall be everywhere <r = 4*1* The value, at the centre of the sphere, of the potential function due to the charge 3/on the surface is , and, since the potential r function is constant inside a charged conductor, this must be the value of the potential function throughout the sphere. If M is equal to r, = 1 : hence the capacity of a spherical conductor r removed from the influence of all electricity except its own. is numerically equal to the radius of its surface. 160 ELECTROSTATICS. 61. Distribution of a Given Charge on an Ellipsoid. It is evident from the discussion of honueoids in Chapter J. that a charge of electricity arranged (on a conductor) m the form of a shell, bounded by ellipsoidal surfaces similar to each other (and to the surface of the conductor), and similarly placed, would be in equilibrium if the conductor were removed from the action of all electricity except its own. We may use this prin ciple to help us to find the distribution of a given charge on a conducting ellipsoid. Let us consider a shell of homogeneous matter bounded by two similar, similarly placed, and concentric ellipsoidal surfaces, whose semi-axes shall be respectively a, 6, c, and (l+a)a, (1 -j-a)6, (1 -f a) c. If any line be drawn from the centre of the shell so as to cut both surfaces, the tangent planes to these two surfaces at the points of intersection will be parallel, and the distance between the planes is pa, where p is the length of the perpendicular let fall from the centre upon the nearer of the planes. If p is the volume density of the matter of which the shell is composed, the mass of the shell is M ^irabc [(1-h a) 3 1] p, and the rate at which the matter is spread upon the unit of sur face is, at any point, a = pS, where 8 is the thickness of the shell measured on the line of force which passes through the point in question. Eliminating p from these equations, we have M8 r 2a If, now, in accordance with the hypothesis that the thickness of the electric charge on a conductor is inappreciable, we make a smaller and smaller, noticing that 8 differs from pa by an infini tesimal of an order higher than the first, we shall have for a strictly surface distribution, If the equation of the surface of the ellipsoidal conductor is ELECTROSTATICS. 161 we have 1 _ , , + and * This last expression shows that, as c is made smaller and smaller, o- approaches more and more nearly the value M rilb /i-g-lT [172] \ a- lr and this gives some idea of the distribution on a thin elliptical plate whose semi-axes are a and b. For a circular plate, we may put a = b in the last expression, which gives M [173] for the surface density at a point r units distant from the centre of the plate. The charge M distributed according to this law on both sides of a circular plate of radius a raises the plate to potential so that the capacity of the plate is 7T 62. Spherical Condensers. If a conducting sphere A of radius / (Fig. 43) be surrounded by a concentric spherical conducting shell B of radii r, and r and charged with m units of electricity while B is uncharged and insulated, we shall have (1) the charge m uniformly distributed upon S, the surface of the sphere ; (2) an induced charge m (Section 57) uniformly distributed upon S t , the inner surface of B ; 162 ELECTROSTATICS. (3) a charge +ra (since the total charge of B is zero) uni formly distributed on 8 , the outer surface of B. FIG. 43. The value at the centre of the sphere of the potential function due to all these distributions is V A = + , and this is r r t r the value of V throughout the conducting sphere. The value of m the potential function in B is V B = o If now a charge M be communicated to U, this will add itself to the charge m already existing on S , and the charge on S t will be undisturbed. The values of the potential functions in the conductors are now and r = If now B be connected with the earth so as to make V B = 0, the charges on S and S t will be undisturbed, but the charge on 77? * S will disappear. V A is now equal to . If A were uncharged, and B had the charge M, this charge would be uniformly distributed upon S , for, since the whole charge on S is zero, the whole charge on # t must be zero also. It is easy to see that$ and { must both be in a state of nature, for if not, lines of force must start from S and end at S { , and others start at S { and end at /S, which is absurd. ELECTROSTATICS. 163 If A were put to earth by means of a fine insulated wire passing through a tiny hole in .5, and if B were insulated and charged with M units of electricity, we should have a charge x on $, a charge x on S/. and a charge 3/-J- x on S . To find .v, we need only remember that F 4 = - -+~-| = 0, whence .r may be obtained. ?i ?0 /0 If B be put to earth, and A be connected by means of the fine wire just mentioned, with an electrical machine which keeps its prime conductor constantly at potential Fi, A will receive a charge // and will be put at potential FI. To find y, it is to be noticed that there is a charge y on ,, and no charge on fl , which is y y put to earth. V A = = Fi, whence y mav be obtained. r r { If r= 99 millimeters and r t = 100 millimeters, y = 9900 F x . If a sphere, equal in size to A but having no shell about it, were connected with the same prime conductor, it too would receive a charge z sufficient to raise it to potential FI, and z would be determined by the equation V l =-- If r = 99, we have z = 99 F! ; hence we see that A, when surrounded by B at potential zero, is able to take one hundred times as great a charge from a given machine as it could take if B were removed. In other words, B increases ^4 s capacity one hundred fold. A and B together constitute what is called a condenser. FIG. 44. If A of the condenser AB, both parts of which are supposed uncharged, be connected by a fine wire (Fig. 44) with a sphere 164 ELECTROSTATICS. A which has the same radius as A, and is charged to potential F!, A and A will now be at the same potential [F 2 ], and A will have the charge a;, and ^4 the charge ?/. The total quantity of electricity on A at first was rFi, so that a; + ?/ = ? Fi, and _ ?/ _ a; a; a; i o whence a? and y may be found. The reader may study for himself the electrical condition of the different parts of two equal spherical condensers (Fig. 45) , FIG. 45. of which the outer surface ! of one is connected with an elec tric machine at potential F 15 and the inside of the other, $ , is connected with the earth. The two condensers, which are sup posed to be so far apart as to be removed from each other s influence, illustrate the case of two Leyden jars arranged in cascade. 63. Condensers made of Two Parallel Conducting Plates. Suppose two infinite conducting planes A and B to be parallel to each other at a distance a apart ; choose a point of the plane A for origin, and take the axis of x perpendicular to the planes, so that their equations shall be a; = and x = a. Let the planes be charged and kept at potentials V A and V B respectively. It is evident from considerations of symmetry that the potential function at the point P between the two planes depends only upon P s x coordinate, so that DF=0, /),F=0, A, 2 F=0, ZVF=0. ELECTROSTATICS. 165 Laplace s Equation gives, then, D;-V=Q, whence D X V=C, and V=Cx + D. If x = 0, V= V A ; and if x = rt, V= V B ; so that F=F-F, + F and />,!" The lines of force are parallel between the planes, and the surface densities of the charges on A and B are V V V V - and -- - respectively. 4 TTCl 4 TTCt If we take a portion of area S out of the middle of each plate, there will be a quantity of electricity on *S 4 equal to * A ~ * , 4:ira and an equal quantity of the other kind of electricity on S B . The force of attraction between S A and S B will be 2 ir<r S. or s (V B -V A Y STT a- If S B be put to earth, the charge that must be given to S A in order to raise it to potential unity is S In other words, the capacity of S A is inversely proportional to the distance between the plates. In the case of two thin conducting plates placed parallel to and opposite each other, at a distance small compared with their areas, the lines of force are practically parallel except in the immediate vicinity of the edges of the plates ;* and we may infer * See Maxwell s Treatise on Electricity and Maynetism, Vol. I. Fig. XII. 166 ELECTROSTATICS. from the results of this section that the capacity of a condenser consisting of two parallel conducting plates of area , separated by a layer of air of thickness a, when one of its plates is put to S V earth is very approximately - for large values of - 4?ra a 64. Capacity of a Long Cylinder surrounded by a Concentric Cylindrical Shell. In the case of an infinite, conducting cylinder of radius ?* ( , kept at potential V t and surrounded by a concentric conducting cylindrical shell of radii r and r r , kept at potential V n , we have symmetry about the axis of the cylinder, so that D^ V 0, and Laplace s Equation reduces to the form whence, for all points of empty space between the cylinder and its she11 But V=V t I when r = r,, and F= F_ whenr = ?- , p;iog+F.u>g hence F= - ^ - =A [175] and FIG. 47. The surface densities of the electricity on the outer surface of the cylinder and the inner surface of the shell are respectively ELECTROSTATICS. 167 V * and -*. so that the charge on the unit of length of the cylinder is V V 1 - i and the charge on the corresponding portion of the inner surface of the shell is the negative of this. We may find the capacity of the unit length of the cylinder by putting V = and F t = 1, whence capacity = - If r in this expression is made very large, the capacity of the cylinder will be very small. In the case of a fine wire connecting two conductors, r { will be very small, and there will be no conducting shell nearer than the walls of the room, so that the capacity of such a wire is plainly negligible. 65. Charge induced on a Sphere by a Charge at an Outside Point, The value at any point P of the potential function due to i m l units of positive electricity concentrated at a point AU and m 2 units of negative electricity concentrated at a point A z , is V = - 2 , where r, = A,P and r z = A,P. r i r z It is easy to see that if m l is greater than m^ so that where A > 1, V will be equal to zero all over a certain sphere which surrounds A 2 . If (Fig. 48) we let A,A 2 = a, A 1 = 8 lf A Z = 8,, OD = r, it is easy to see that X-a a g*X g 6l ~ r " : - 168 ELECTROSTATICS. and a = = = r. [176] Oj 02 If PR represents the force f t due to the electricity at A^ and PQ the force f a due to the electricity at A 2 , the line of action of the resultant force F (represented by PL) must pass through the centre of the sphere, since the surface of the sphere is equi- potential. FIG. 48. The triangles A 1 PO and A 2 PO are mutually equiangular, for they have a common angle AOP, and the sides including that angle are proportional (?* 2 = 8i8 2 )- Hence, from the triangles QPL and A L PA 2 , by the Theorem of Sines, - [177] sin a 2 sin c^ sin (a 2 c^) whence F== & == h oArn^ p 179 -. * 4* * 3 Now, according to Section 49, we may distribute upon the spherical surface just considered a quantity w 2 of negative elec tricity in such a way that the effect of this distribution at all points outside the sphere shall be equal to the effect of the charge ra 2 concentrated at A 2 <, and the effect at points within the sphere shall be equal and opposite to the effect of the charge raj concentrated at A t . Since F is the force at P in the direc- ELECTROSTATICS. 169 tion of the interior normal to the sphere, we shall accomplish this if we make the surface density at every point equal to cr, where 4TO . = -F= -"* i = -(8i -OM! ; [180 ] r a rr a and if we now take away the charge at ./L, the value of the po tential function throughout the space enclosed by our spherical surface, and upon the surface itself, will be zero. If the spheri cal surface were made conducting, and were connected with the earth by a fine wire, there would be no change in the charge of the sphere, and we have discovered the amount and the distri bution of the electricity induced upon a sphere of radius r, con nected with the earth by a fine wire and exposed to the action of a charge of MJ units of positive electricity concentrated at a point at a distance ^ from the centre of the sphere. If now we break the connection with the earth, and distribute a charge m uniformly over the sphere in addition to the present distribution, the potential function will be constant (although no longer zero) within the sphere, and we have a case of equi librium, for we have superposed one case of equilibrium (where there is a uniform charge on the sphere and none at A^ upon another. The whole charge on the sphere is now " 1 and the value of the potential function within it and upon the surface, 17 ^ i m i m y |- r 6j r If the conducting sphere were at the beginning insulated and uncharged, we should have M= 0, and therefore ._Viif!\ and r= ^i. ^i] ri a ) *>i If we have given that the conducting sphere, under the influ ence of the electricity concentrated at A is at potential FI, we 170 ELECTROSTATICS. know that its total charge must be V\r -- ~> and its surface It is easy to see that the sphere and its charge will be attracted toward A l with the force and the student should notice that, under certain circumstances, this expression will be negative and the force repulsive. If m-L = m 2 , the surface of zero potential is an infinite plane, and our equations give us the charge induced on a conducting plane by a charge at a point outside the plane. The method of this section enables us to find also the capacity of a condenser composed of two conducting cylindrical surfaces, parallel to each other, but eccentric ; for a whole set of the equipotential surfaces due to two parallel, infinite straight lines, charged uniformly with equal quantities per unit of length of opposite kinds of electricity, are eccentric cylindrical surfaces surrounding one of the lines, A^ and leav ing the other line, AU outside. We may therefore choose two of these surfaces, distribute the charge of A l on the outer of these, and the charge of A 2 on the inner, by the aid of the principles laid down in Section 49, so as to leave the values of the potential function on these surfaces the same as before. These distributions thus found will remain unchanged if the equipotential surfaces are made conducting. The reader who wishes to study this method more at length should consult, under the head of Electric Images, the treatises of Gumming, Maxwell, Mascart, Tarleton, and Watson and Burbury, as well as original papers on the subject by Murphy in the Philosophical Magazine, 1833, p. 350, and by Sir W. Thomson in the Cambridge and Dublin Mathematical Journal for 1848. ELECTROSTATICS. 171 66. The Energy of Charged Conductors. If a conductor of capacity C, removed from the action of all electricity except its own, be charged with 3/j units of electricity, so that it is at potential V l ^, the amount of work required to bring up to C the conductor, little by little, from the walls of the room, the additional charge Am, is A IF, which is greater than V l A3f or 1L . A3/, and less than ( V l + Aj, V) AJ/ or 3/1 + AJ/ . AJ/. If the charge be increased from M l to J/ 2 by a constant flow, the amount of work required is evidently The work required to bring up the charge M to the conductor at first uncharged is then M 2 CV- MV so- T- [18o] This is evidently equal to the potential energy of the charged conductor, and this is independent of the method by which the conductor has been charged. If, now, we have a series of conductors A^ A& A^ etc., in the presence of each other at potentials I 7 ",, F 2 , V 3 , etc., and having respectively the charges M, M 21 M& etc., and if we change all the charges in the ratio of x to 1, we shall have a new state of equilibrium in which the charges are x J/ 1? .i-J/ 2 , xM s , etc. ; and the values of the potential functions within the conductors are xV^ xV 2 . a-T> etc. The work (ATF) required to increase the charges in the ratio x + Aa; instead of in the ratio x is greater than Aa-) (xV 2 ) + (M 2 Aa?) (xV s ) + etc., or Aaj[ Jtfi F! -f 3/o F 2 -f 3/s F ; -f etc.] , and less than (.>: -h AJ;) Aar [ M l V, + J/ 2 F 2 + J/ 8 F 3 -h etc.] ; 172 ELECTROSTATICS. hence the whole amount of work required to change the ratio x 1 , x 2 . irom to is W 2 - W l = * ~ IM 1 F! + M, F 2 + M s V z + etc.]. [186] If in this equation we put x l and x 2 = 1, we get for the work required to charge the conductors from the neutral state to potentials V 1} F 2 , F" 3 , V), [187] a particular case of the general formula stated in Section 27. The work required to make any combination of changes of charge on any system of fixed conductors is evidently equal to the difference between the intrinsic energies of the system in its original and hnal states. If V k , V k represent the initial and final potentials on the kth conductor, and e k and e k the original and final charges, V k e k . Since the final energy is independent of the manner in which the changes are produced, we may suppose that the changes take place gradually and at the same relative rate for all the conductors, so that at any instant the charge of each conductor has received the same fraction of its whole increment or decrement that every other conductor has received, it being understood that in the general case some charges will be in creased and others decreased. At the instant when the change accomplished is to the whole change as x : 1, the charge of the /tth conductor is e k + x(e k e k ), and the value of the poten tial function in this conductor is V k + x(VJ V k ). In order to increase x by Ax, the charge must be increased by the amount Ax (e k e k ), and to bring this up from infinity an amount of work equal approximately to ELECTROSTATICS. 173 must be done, and for the whole system the corresponding work is y\e* f - e k )[ V k + x ( V k - F<.)] \x. To find the work required to bring about the whole change, this expression must be integrated with respect to x between the limits and 1. This process yields V t + FiO (e t - *,), and by comparing this with the result stated above we learn that we ma also write E -E = ( V k - T,) (e, + e t ). We learn incidentally that /^ ^ = /^ c t -^V> an d we see that if all but two (A^ A 2 ) of any system of conductors are either put to earth or are insulated and without charge, i J r i + e 2 V z = <?! JY + e z F 2 . If ei = 1, e z = 0, ei = 0, e z = 1, 7^ 2 = r/, and if T x - 1, JT,, = 0, IV = 0, r 2 = 1, ei = e z , so that a unit charge given to AI, while A z is uncharged and insulated, raises A ^ to the same potential that A l would have if it were uncharged and insulated while A 2 had a unit charge ; and the same quantity of electricity is induced on A 2 when it is put to earth, while A l is charged to potential unity as would be induced on A if it were put to earth and A 2 charged to potential unity. Using the notation of Section 59, this shows that p rk = p kr and that q rk = q kr we may write, therefore, E = - i (Pit? + p&* + p*p* + + ^..O 174 ELECTROSTATICS. If the conductors are fixed so that the p s and ^ s are con stant, we may learn from differentiating this last equation that, if all the charges but e k are kept constant, U Ck E = V k , and if the values of the potential function in all but one of the conductors (the &th) are unchanged D V] E= e k . If the system changes its configuration, the p s and <? s are in general changed, and we learn that if the charges are kept constant during the change, A- / but that if by suitable changes in the charges the potentials are unchanged, In the latter case, A V k , or \A(e r _p rjfc ) = 0, so that ,or or 2 A _^ + 2 A"^ 4- If, therefore, <j> is any coordinate, which defines the con figuration, limit /A ^ 7 \ li " mit A system of conductors with constant charges when left to itself tends to obey the urgings of the reciprocal forces between its parts, and therefore to diminish its intrinsic energy. If, in this case, the single coordinate <f> is free to change and is increased by A<, the energy after the change is E -f- A ^, where A J is really negative. The mechanical work done by the forces is D^E- A<. If, now, <f> had been changed as before by the same small increment, A<, while the potentials were kept constant by bringing up to each con ductor from without the necessary quantity of electricity, the energy after the change would have been E -f A"^, where ELECTROSTATICS. 175 \"E is really positive. The energy has therefore increased by an amount practically equal to the former loss. Practi cally the same amount of mechanical work has been done as before, and enough energy has been introduced from without to do this work and to add an equivalent amount besides this to the potential energy of the system. The contribution, therefore, from outside sources is about 2 A"^. These state ments applied to a small change in <f> are based on the exact equation D^ E = D^ E, proved above. 67. If a series of conductors A l} A 2 . A 3 , etc., are far enough apart not to be exposed to inductive action from one another, and have capacities Ci, C 2 , (7 3 , etc., and charges 3/i ? 3/ 2 , 3/ 3 , etc., so as to be at potentials FI, F 2 , F 3 , etc., where 3/ 1 =C 1 F 1 , 3/2 = CoF., 3/3= (7 3 F 3 , etc., we may connect them together by means of fine wires whose capacities we may neglect, and thus obtain a single conductor of capacity The charge on this composite conductor is evidently l+J+J + ...= and if we call the value of the potential function within it F, we shall have whence F= * **+ - " [188] a formula obtained, it is to be noticed, on the assumption that the conductors do not influence each other. The energy of the separate charged conductors before being connected together was V <-A U 2 U s [189] 176 ELECTROSTATICS. and the energy of the composite conductor is M* + M s + . ) (6\ Fj + C 2 V 4- 2 + C s +. . a * W J L J which is always less than E, unless the separate conductors were all at the same potential in the beginning. 68. Specific Inductive Capacity. In all our work up to this time we have supposed conductors to be separated from each other by electrically indifferent media, which simply prevent the passage of electricity from one conductor to another. We have no reason to believe, however, that such media exist in nature. Experiment shows, for instance, that the capacity of a given spherical condenser depends essentially upon the kind of insulating material used to separate the sphere from its shell, so that this material, without conduct ing electricity, modifies the action of the charges on the con ductors. Insulators, when considered as transmitting electric action, are sometimes called dielectrics. Given two condensers of any shape, geometrically alike in all respects, with plates separated in the one case by a homogeneous dielectric, A, and in the other case by another homogeneous dielectric, 7>, the ratio of the capacities is found to be the same whatever the shape or dimensions of the condensers when these same two dielectrics are used. If this ratio is unity, the dielectrics are said to have the same electrical inductivity or the same specific, inductive capacity. If the ratio of the capacities of the first and second con densers is n, A is said to have an inductivity n times as great as that of B. The electrical inductivity of dry air at the standard pressure and temperature being chosen as a standard, the electrical inductivities of all other known substances are ELECTROSTATICS. 177 positive quantities which in the case of any one specimen, though somewhat dependent upon conditions of temperature and pressure, may be considered independent of the electrical stress to which the substance may be exposed. The letter //. is often used to represent the inductivity of a medium. It is generally assumed, for the sake of definiteness, that outside all the material media upon which we can experiment, the ether extends indefinitely in all directions and the inductivity of the ether is assumed to be sensibly the same as that of air under standard conditions. We cannot expect that a non- homogeneous dielectric will have the same inductivity through out, so that in the general case we must assume that /u. is a function of the space coordinates. The vector formed by multiplying the force by the scalar quantity -p- is sometimes 4 7T called the displacement. The force is occasionally called the electrical intensity, or the electromotive intensity. We may best sum up the results of experiments upon the behavior of dielectrics in electric fields by stating some gen eral equations which may be used in solving any problem. We shall find it convenient to write down first, for the sake of comparison, the simplified forms of these equations which we have shown to be characteristic of the electric field about any distribution of electricity when air is the only dielectric. If X, Tj Z are the force components parallel to the axes, and if V is the potential function, so that X = -D X V. Y = -V y V, Z = -D Z V, we know that when /* = 1, (1) D X X + D, t Y + D Z Z = + 4 Trp, except at surfaces where p is discontinuous. (2) The surface integral of the normal (outward) com ponent of the force taken over any closed surface is equal to 4?r times the amount of matter (algebraically reckoned) within the surface ; or 178 ELECTROSTATICS. (3) At a charged surface all the tangential components of the force are continuous, but the normal components are dis continuous in the manner indicated by the equation -ZVi + JVa = + 4 TTO-, where JV^ and N z represent the normal force components taken away from the surface on both sides. If the charged surface is not equipotential, the lines of force which cross it are in general refracted ; for, if fa is the angle which a line of force in reaching the surface makes with its normal, fa the angle which the same line makes with the normal on leaving the surface on the other side, and, if 2\ and 1\ are the tangential components of the force, 2\ N-^ tan fa, T 2 N 2 tan fa, and since T = T 2 , N z tan fa + Nj_ tan fa 0, or, since the normal component is discontinuous, (4 TTO- Ni) tan fa -f- j^ tan fa = 0. (4) V so vanishes at infinity that rV and r 2 D t .V have finite limits. If we now introduce a new vector (called the induction) equal to the product of the scalar point function ^ and the force, we may write down a set of equations, very like those which we have just enumerated and equivalent to them when /* = 1, which will give the force components and the potential function in terms of the charges when /x is different from unity and (in the general case) determined by different analytic functions of the space coordinates in different por tions of space. In general, (1) D x (p.X} + D y 0* Y) + D z (pZ) = + 4 ^ [191] at every point in space, except at surfaces where either p or /x, is discontinuous. Since in all cases X= D X V, Y = D y F, Z = D z F, this equation may be written D x (pD x V) + D, (pD y V) + D z (v.D 2 V) = - 4 * p . [192] ELECTROSTATICS. 179 In a dielectric of uniform inductivity it becomes (2) The integral, taken over any closed surface, of the out ward normal component of the induction is equal to 4 IT times the amount of matter within the surface or C [193] (3) If the surface of separation between two different dielec trics which are in contact with each other has a charge of superficial density, o-, all the force components tangent to the surface are continuous. If /^ and /x 2 are the inductivities of the two media, the normal component of the induction is dis continuous in the manner indicated by the equation /A^Y! -f- /x 2 .V 2 = + 4 TTO-, or nt D ni V + n* I\ V = - 4 TTO-. [194] If this surface has no charge, o- = 0, and the normal component of the induction is continuous, though the normal force com ponent is discontinuous: evidently, the law of refraction of the lines of force is, in this case, tan fa : ^ = tan fa : /x 2 . Whether or not o- is zero, J\\ tan fa -f JN T 2 tan fa = 0. At a charged surface where the dielectric is continuous, (4) F is everywhere continuous, and it so vanishes at infin ity that r Fand r 2 D r V have finite limits. The first derivatives of V are everywhere continuous, except at charged surfaces and surfaces where the inductivity is discontinuous : here the tangential derivatives of V are continuous and the normal derivatives have the properties just discussed. The lines of force and the lines of induction are coincident. It is well to notice that what we have here called the induction and what is usually called induction in perfectly hard magnets are different special cases, as will be shown later on, of a 180 ELECTROSTATICS. much more complex vector which appears in some general problems. It is easy to prove with the help of [149] a series of theo rems concerning the potential function analogous to those already found for the case where /t* = 1. For instance : if the closed surface S l shuts in the closed surface S z , there cannot be two different functions, V and V, which (1) between S 1 and $2 satisfy the equation D x (pD x w) + D y (pD v w) + D z (f,D x w) = 0, where /A is a given, everywhere positive, analytic function of the space coordinates, (2) are continuous in that region with their first derivatives, and (3) are equal at every point of Si and S z . Assuming, for the sake of argument, that two such functions exist, we may call their difference u and note that u and its first derivatives are continuous between Si and S ZJ and that u vanishes at every point of these surfaces. Since u satisfies the equation + D Du + D (*.Du = between Si and $ 2 , we may conveniently make A = p, U=V=u in [149], for both integrals in the second member of the equa tion vanish, and we learn that when extended over the region in question. Since /t is posi tive, and the integrand can never be negative, D x u = D y u = D z u = 0, and u is a constant. But u = on Si and S 2) hence V and V are identical. If, while satisfying conditions (1) and (2), V and V are required to have equal normal derivatives at every point of S l and $ 2 > it is easy to prove in a similar manner that one can differ from the other only by a constant. ELECTROSTATICS. 181 With given values of the volume density in given regions of space, and with given values of the superficial density on given surfaces, the force components and the potential func tion are, in general, different when /x = 1 and when /*, is dif ferent from 1, and, if the dielectric is heterogeneous with surfaces of discontinuity in tt, not equipotential surfaces, the forms of the lines of force are very different in the two cases. If the dielectric of a given condenser, the plates of which are the surfaces Si and S 2 , is air, and if these plates have given charges, V must satisfy Laplace s Equation between Si and . while at every part of the condenser plate D n V = 4 TTO-. If, now, a homogeneous dielectric of inductivity /u. be substituted for the air, the new potential function V satisfies Laplace s Equation between Si and S a (since /x is constant andp is zero), and at every point of Si or S 2 , D n V = Now V / p. satisfies all these last conditions, and since two functions which do so can at most differ by a constant, we may write V = V/IL + C. The force in any direction at any point in the dielectric is 1 /tt as great in the second case as in the first. If /Si and S 2 , instead of having given charges, had been kept at the given potentials f\ and F 2 , the density of the charge at any point of either plate would have been tt times as great in the second case as in the first, while the potential function (and the force) would have had the same value at every point, which ever dielectric was used. The capacity of the condenser is, in this case, equal to The generalized form [192] of Poisson s Equation, when expressed in terms of the orthogonal curvilinear coordinates u, v, w as independent variables, becomes 182 ELECTROSTATICS. /* [ V A, 2 F + V D? V 4- A,,, 2 D* V+D U V- + (/> F I> x w + D t , V D x v + D,, V D x w) (I)./* /> X N + />/* D x v + A,/* - D x w) + (D u F. D y + D v V- D v v + D w F- >//;) (D M/ u, JD^w + />/* J9 y t; + D w /x D^w) + (D u F- Z> 2 % + D v F- Z> z v 4- D M ., F- D z w) (Z) Mf i D s u + D vt L D z v + D w fi D z w) = --rrp. If /x is a function of one of the coordinates, u, only, the family of surfaces on which u is constant are possible equi- potential surfaces due to a distribution of electricity in this dielectric, provided the special form of the equation just stated, obtained by putting D vfJL = D wfJL = D, F = D w V = P = 0, that is, provided the equation involves only w. Now D M /x//u, is, by hypothesis, a function of w only, so that the condition is that the ratio of V 2 u to h u * shall be independent of v and w, and this is the condition (Section 35) that must be satisfied when the dielectric is air, in order that the surfaces upon which u is constant may be possible equipotential surfaces. It is easy to see that if the space between two equipoten tial surfaces in air about a distribution of electricity be filled with a dielectric the inductivity of which is either constant or else a function only of the parameter of the original equi potential surfaces, the new equipotential surfaces will coincide with the old ones, though the value of the potential function on any particular surface will generally be changed. If in [149] we make u = F, the potential function due to any ELECTROSTATICS. 183 distribution of electricity, and if we make A. = /A, we may apply the equation to all space after we have enclosed by pairs of new surfaces all surfaces of discontinuity of /a, p, or D n V, and learn that the intrinsic energy of the distribution is equal to extended over all space. When the potential function, F, due to a given distribution (p, <r) of electricity with any given set of dielectrics has been found, we may ask what distribution (p , a- ) of electricity would have given this same potential function if all the dielectrics had been displaced by homogeneous air. The dis tribution (p f , a- ) is called the apparent charge to distinguish it from the distribution (p, a-) which is sometimes called the real or the intrinsic charge. From the apparent charge when found, V might be calculated by means of the familiar integrals When V is given, the quantity p is determined at all points where the equation has a definite meaning by V 2 F= 47rp and the quantity a- at all surfaces where the normal derivative of V is discontinuous by the equation N + -ZV 2 = 4 TTO- . Now D x ^D x ^ + D y (t,D y V) + D z ^D z r)=-^p, or jtV 2 V + (D X V- D x n+D y V- D y ^+D z V- D a/ *) = or -47r/zp +(.D x F. D xt L+D v V- D yt L+D z V- D^) = and this defines p . In every region where /u. is constant p = p//x. In the most general case of a surface where the normal derivative of V is discontinuous, there is a discontinuity in /x at the surface and a charge, <r, on the surface so that ^\ + frNt = 4 7TO-, A\ + N 2 = 4 7TO- , whence < = - + ^ ~ ^ = ^ + ^ 2(/Al ~ ^>. [197] 184 ELECTROSTATICS. In a particular instance there may be a surface charge with no discontinuity in the dielectric, in which case a- = o-///, ; or there may be discontinuity in the dielectric with no real surface charge, in which case The difference (p /a, o- o-) between the apparent charge and the intrinsic charge is sometimes called the induced charge. The solving of one or two simple problems will suffice to illustrate the use of the general equations which determine the potential function when the dielectric is not homogeneous. I. "A condenser consists of two concentric conducting spherical surfaces of radii a and b separated by a dielectric the inductivity of which at a distance, r, from the common c ~\~ r centre, 0, of the spherical surfaces is The inner plate, of radius a, has a charge E. The outer plate is at potential zero. The potential function in the dielectric is evidently a function of r only ; what is its value ? " Since the induction through any closed surface is equal to 4?r times the intrinsic charge within, we may imagine a spherical surface drawn in the dielectric with centre at and radius equal to r and then assert that, if F is the force, c 4- r v 4 Ti-r 2 Z - F = 4 TrE so that F = Z> V = r (c -+- ?) and V log The capacity of the condenser is c r (b + c) c -f- log ~- 7 The apparent surface density on the inner a (b + c) plate is a- = E/\_kira(a + c)], the intrinsic surface density is JZ T /4?ra 2 , and the density of the charge induced at the inner surface of the dielectric is EC / [ IT a* (a + c)]. II. "A condenser consists of two large, plane, conducting plates parallel to each other and separated by three slabs, s lt ELECTROSTATICS. 185 5 2 , s 3 , of dielectric of thickness a, b, and c respectively, and of inductivity 1, /x, and 1. What is the capacity of the con denser per unit area of one of its plates ? " Take the axis of x perpendicular to the faces of the plates with the origin in the first plate, which shall be kept at potential zero. It is evident that the potential function is a function of x only, so that D x 2 V = in each slab of dielectric and F must be of the form Lx + J/". Denote the functions which give the potential in the three slabs by When j- = 0, J\ = 0. When x = a, - DJ\ + ^D Z V Z = 0. and 1\ = F 2 . When x = (a + b), - pD, F 2 + D z F 3 = 0, and F 2 = r s . We have, therefore, J\ = L^, F 2 = A (a: + a/t - a) //i, F, = A(/xx + & - fyt)// 4 - When a: = 0, D^ V = 4 TTO- = L lt and, if F 8 = 1 when x = a +b + c, we get <r = 47r(ft + [J.C + b) and this is the capacity per unit area of the first plate. 69. Polarized Distributions. Imagine two homogeneous bodies, P and ^V, of equal but opposite densities, p and p, of the same dimensions, and occu pying at the same time the same space, in which, of course, the resultant density is zero. If P be moved without rotation through a small distance h, in some direc tion, there will be a space of no density common to P and X, a space of density p where P extends F IG . 49. beyond ^V, and a space of density p where A" extends behind P. The thickness of the shell of matter, measured on the exterior normal to the space of no density, is A. If, now, h be made to approach zero, and p 186 ELECTROSTATICS. be imagined to increase without limit so as to keep the product pk always equal to a given constant /, and we have in the limit merely a superficial distribution, of density o- = / cos (A, n), on the boundary of the space originally occupied in common by P and N. Since the direction of h is fixed in space, and n is an exterior normal, the distribution consists partly of negative matter and partly of positive matter in equal amounts. The surface density is equal to zero at points of contact of the distribution with tangents parallel to the direction of h. If this distribution be divided up into filaments parallel to h, it is clear that the charges on the ends of every filament are equal and opposite, and that each is equal in amount to ql, where q is the cross-section of the filament in question. It is easy to see from this that if the distribution were placed in a uniform field of force of intensity F, this field would exert upon any such filament of length I a couple of moment J^-sin(A, F) qll, and upon the whole distribution a couple of moment .F-sin(7i, F) I times the volume of the space enclosed by the distribution. / is, therefore, numeri cally equal to the moment of the couple, per unit of volume, per unit field perpendicular to the direction of h. The dis tribution just described is said to be a uniformly polarized distribution. / is called the intensity of the polarization. If, for instance, P is a sphere of radius a with centre at 0, and if r* = x 2 4- y 2 + # 2 , the potential function, V(x, y, 2), due to its own mass, has, as we know, the value 2 TT/O (a 2 1 r 2 ) at inside points, and the value 4:7rpa 3 /3r at outside points. After P has been displaced through a distance h parallel to the x axis, the potential function at any point (x, y, ), either in the space common to P and N or outside both, has the value V (x h, ?/, z) V (x, y, ). If the point is within both P and N, the value of this quantity is 2 irph (2 x k) 1 3, but if the point is without both P and ELECTROSTATICS. 187 X, the value is 2 7ra s p h(2x - h)/3 ,*. The limits of these expressions (4?r/.r/3 and 4=Tra 3 Ix/3 r 3 ) give the values of the potential function within and without a sphere uniformly polarized to intensity I parallel to the x axis. Within the sphere the equipotential surfaces are planes perpendicular to the x axis, the field is uniform, and since .Y= D X V, the lines of force are parallel to the negative direction of the x axis. Consider ations of symmetry show that the lines of force without the sphere are curves lying in planes through the axis of x. From the expression for V at outside points we learn that if is the angle which the radius vector drawn from the origin to any point makes with the x axis, the equipotential sur faces of revolution with out the sphere may be considered as generated by plane curves which belong to the family cos 6/i* = c. Curves of this family lying in a FIG. 50. plane are cut orthogo nally by curves in the same plane which have the equa tion r= &-sin 2 0, and this evidently gives the lines of force. Fig. 50 shows the forms of these lines and the direction of the force. It is to be noted that this direction changes abruptly at the surface ; on the jc axis without the sphere the force is directed from left to right, whereas within the sphere it is 188 ELECTROSTATICS. directed from right to left. This discontinuity is wholly explained, as a little simple computation will show, by the fact that at any superficial distribution of density o- every tangential component of the force is continuous, but the normal component is discontinuous by 4 TTO-. The potential function belonging to a uniform field of force of intensity X , the lines of which are parallel to the x axis, is X Q x, and if into such a field a sphere of radius a, uniformly polarized to intensity / parallel to the x axis, is brought, and if we define the constant x by the equation X Q =. 4 7r/x/3j the FIG. 51. potential function, referred to the centre of the sphere as origin, will have the value 47r/x(l \)/3 at points within the sphere, and the value 4 TT/Z [> 3 / 3 (z 2 + y 2 + z 2 ) 3 / 2 - x /3] at outside points. The field within the sphere is now a uni form field of intensity 4 TT/(X 1) /3 directed parallel to the x axis : if x = 1? this force vanishes. The equipotential sur faces of revolution without the sphere could be generated by the revolution about the x axis of a family of curves the equation of which in the xy plane is 4 -rrlx [- s /3 r 3 x/3] c > where r 2 = x 2 -\- if. The equation of the family of curves which cut these orthogonally may be written, 27r/?/ 2 (2 a 3 /3 > 3 + x/3) = m, ELECTROSTATICS. 189 and this represents the lines of force. These lines may be easily plotted for any value of ^, by assuming in succession a series of values of r and computing the corresponding values of ?/. Figs. 51 and 52 show two characteristic forms which the lines may have. In the first ^ = + 1, in the second X = 3. Some slight theoretical interest attaches to the case for which ^ = 2, and the reader may care to plot for himself the corresponding curves. He should indicate the direction of the force at various points by arrows. The value, at inside points, of the potential function due to a homogeneous ellipsoid of density p, with axes coincident FIG. 52. with the coordinate axes, is given on page 121. If we call this p O (z, y, z), we may write O (x, y, z) = abdr(G<> - A> 2 - L^f - H&*), where G , K , L , M Q have the same values at all points of the mass. If, now, we consider an ellipsoidal distribution, uni formly polarized to intensity /, in a direction s, it is easy to see that the value of the potential function within the distribution is I[DlQ, cos (x, s) +D y l - cos (y, s) + D z to cos (z, s)] or 2 abcirI\K^ cos (z, s) + L Q y cos (y, s) -f M<p cos (z, s)], and that, if we regard the polarization as a vector and denote its components by A, B, and (7, the force components are - 2 TrabcAKfr - 2 TrabcBL Q , - 2 irabcCM Q . The field within the distribution is, therefore, uniform, and it has a direction defined by cosines which are to each other as A" -cos (x, s) : L -cos (y, s) : Jf -cos (z, s). It is to be noticed that this direction does not coincide with that of the 190 ELECTROSTATICS. polarization unless K = L = M (so that the ellipsoid is really a sphere), or unless the direction of polarization coin cides with that of one of the principal axes of the ellipsoid. In certain cases the elliptic integral = f */0 ds (s + a 2 ) 3 / 2 (s 4- 6 2 ) 1/2 (s + c 2 ) 1 / 2 and the corresponding integrals L and M can be easily evaluated. If, for instance, a = b = c, these quantities evi dently have the common value 2/3 a 3 . If the ellipsoid is a figure of revolution, we may find the values of K , L , M Q with the help of the integrals /ds (s-M^O + m 2 ) 1 / 2 1 t /(.s + m 2 ) 1 / 2 - (m 2 - Z 2 ) 1 ~ (m: 2 - I 2 ) 1 / 2 ?\(j + m 2 ) 1 / 2 + (m 2 - P) 1 + m 21 / 2 / ^ - 1 1 r * J ( /o?s (s + ^ 2 )(.9 + m 2 ) 3 / 2 1 (fa f ds \ J (s + Z 2 ) (s + m 2 ) 1 / 2 / - + m 2 ) 1 / 2 In the case of a prolate ellipsoid where > b, b = c, and ELECTROSTATICS, and the force components within the ellipsoid are 1 191 ( -, 2,c(* -* l-A TT~e} l- If, while & is constant, a be increased without limit, e approaches the limit unity, (1 e 2 ) -log[(l + e)/(l e)] the limit zero, and the ellipsoid becomes an infinitely long cylinder of revolution, for which the force components are 0, -2-rrB, -2-rrC. In the case of an oblate ellipsoid where a < b, b = c, and e = ^/b- -a 2 /b, and the force components within the ellipsoid are -4,^ -.- Vl - e 2 -2wC If, while b and e are constant, a is made to approach zero, e approaches the limit unity, the limiting values of the force components are 4 -rrA, 0, 0, and we have the case of a circu lar disc, in which, if the direction of polarization lies in the plane of the disc, the resultant force is zero. 192 ELECTROSTATICS. If the imaginary body P, instead of being of the same density throughout, had consisted of two homogeneous por tions of densities p and p 2 , to the left and to the right of their surface of separation, $; if the density of N had been at every point equal and opposite to that of P, and if the limits of pji and pji had been the constants ^ and / 2 , the resulting surface distribution, on the boundary of the space occupied originally by N and P conjointly, would have had the density o- = I t cos (h, n) to the left of the original posi tion of S, and the density o- = Z> cos (A, n) over the rest of the surface. There would have been on S a surface density <r = / l cos(A, n^) + J 2 cos (A, ?i 2 ), where n^ and n 2 represent exterior normals to the regions in which P had the densi ties p l and p 2 respectively. This distribution is therefore equivalent to two distributions uniformly polarized in the direction of h, and laid together so as to have the common surface S. If, again, the density of P had been given by the expression P = PO /(*, y, ), where p is a constant and / an analytic function of the space coordinates, then, if P had been displaced parallel to the x axis, there would have been, (1) a region common to P and N in which the density would have been po[f( x - h > y> z ) -f( x > y> )] or - p h D*f+ e where e is an infinitesimal of the same order as h, (2) a region of density p f(x h, y, z) where P extended beyond N, and (3) a region of density p f(x, y, z) where N extended behind P. If the limit of p h had been the constant A , and if A-/(z, y, z) had been denoted by A, the resulting distri bution would have had a surface density a- = A cos (x, n) over the boundary of the space originally occupied by N and P and a volume density p = D X A inside this boundary. This kind of distribution is called a non-uniform polarization of intensity A, the direction of the polarization being that of the x axis. We know from Green s Theorem that the ELECTROSTATICS. 193 surface integral of A cos (x, n) taken over any closed surface is equal to the volume integral of + D^A taken through the space bounded by the surface, so that the whole amount of matter, algebraically considered, in the distribution just discussed is zero. If such a distribution as this were placed in a uniform field of force of intensity F, perpendicular to the x axis, it would encounter a couple of moment M= F CCx <r-dS+ F C C Cx.p.dr = F C Cx A . cos(ar, n)dS - F C C Cx D^A dr - x Here, again, the volume integral of the intensity of the polar ization is a measure of the moment of the couple which would be exerted upon the distribution, if it were placed in a uniform field of unit strength perpendicular to the x axis. The intensity of the polarization at any point in a polarized distribution has been called the moment per unit volume of the distribution at the point. If a distribution polarized in the manner just described parallel to the x axis were placed in a uniform field (X , Y Q9 Z ), not perpendicular to the x axis, it would experience a couple the components of which would be If p V(x, y, z) is the potential function at (x, y, z) due to P in its original position, the potential function at (or, y, z) due to N and P, after P has been displaced parallel to the axis of x through the distance /*, is po [ V(x - h, y, z) - V(x, y, )], or - p Q h*D x V+e, 194 ELECTROSTATICS. where e is an infinitesimal of the same order as h. As li is decreased and p so increased that pji is always equal to A M the potential function at (x, y, z) due to the resulting distri bution becomes A D x V. Thus, if P is a sphere of radius , the density of which is proportional to the distance from its centre, we have p = p r, V = 7rp (4# 3 r 3 ) /3 if r < a, and V=Trp Q a*/r if r > a. The polarization in the resulting distribution is A r, where A is a constant to be chosen at pleasure ; the potential function has the value irA^xr within the polarized sphere, and -n-A^x /r* without it ; the moment of the sphere is 7rA a 4 . Imagine six coincident bodies, P 1? N^ P 2 , N 2 , P 3 , N 3 , of densities p<>fi(x,y,z), p fi(x, y,z), p f 2 (x, ;>/, z), -/o / 2 (*,y,), Po/sO, y> )> - Po/s(# 2A ) respectively. Imagine P 1? P,, P 3 displaced through distances A parallel, respectively, to the axes of Xy y, and z, then imagine h to decrease and p to increase in such a way that p () A is always equal to a constant M. If J//i (a;, y, ), J/A (a, y, s), Jfjf a (, y, ) be denoted by A 9 B, and C respectively, the resulting distribution has a surface density a A cos (#, ri)-\- B cos (y, 71) + (7 cos (, ?i) on the boundary of the space originally occupied by the six bodies, and a volume density p = (D X A + ^ ?/ 5 4- -A-^) in the region enclosed by this boundary. ^4, B, C are usually considered to be the components taken parallel to the coordinate axes of a vec tor, /, so that <r / cos (n, I) and p = (Divergence of /). The whole amount of matter in the distribution is zero. / is called the polarization, and the direction of I at any point is the direction at that point of the polarization. The lines of the vector / are defined by the equations and are called the lines of polarization. If through every point of a curve, s, in a polarized distribution, we draw a line of polarization, we shall get (unless s is itself a line of polarization) a polarization surface; if s is closed and the polarization surface tubular, the latter is called a tube of ELECTROSTATICS 195 polarization. The product of the cross-section of a very slender tube of polarization at any point, and the value at that point of /, is sometimes called the strength of the tube. The matter in a slender tube of polarization con stitutes a polarized filament. If the vector / is solenoidal. the distribution is wholly superficial, and the strength of every tube of polarization is constant throughout its length. Uniform polarization is a special case of solenoidal polarization. It is evident that the generally polarized distribution just mentioned may be regarded as formed by the superposition of three distributions polarized parallel to the axes of x, //. and z respectively, and it is easy to see that a uniformly polarized distribution in a uniform field (X m Y w Z Q ) will be acted on by a couple the components of which are the prod ucts of the volume of the distribution and the quantities BZ - CY W CX, - AZ W A Y, - BX. A short, extremely slender, right prism, uniformly polar ized in the direction of its length, forms a simple kind of polarized element. If 21 is the length of such an element, q its cross-section, and / the intensity of its polarization, 2qII may be called the moment of the element, for it represents the moment of the couple which would act upon the element if it were placed perpendicularly across the lines of a unit field. This product of the volume of the element and / we may denote by H. We know that the field of force due to the element is mathematically accounted for by a superficial negative charge, ql, on one end of the prism and an equal positive charge on the other end. Let Q be any point distant r from the negative end and r 2 from the positive end of the axis of the prism, and r from its centre. Let (>, /) be the angle between the direction of polarization and the line drawn from the centre of the axis to Q ; then, since = /~ + Z 2 + 2 rl cos (r, 7) and r.r = >~ + I 2 - 2 // . cos (r, /), 196 ELECTROSTATICS. the value at Q of the potential function due to the element is /lr-lr or or cos (r, The limit, Jf-cos(r, I)/iP, of the expression just found is called the potential function due to a uniformly polarized element, or to a space doublet. It will appear from the work which fol lows that a similar result might have been obtained from the use of a gener ally polarized element of any form. The lines of force due to a polarized element are shown in Fig. 53 ; they are the same as the external lines of force in Fig. 50. Before we attempt to find an expression for the potential function due to a generally polarized finite distribution, it is well to notice that if the vector / is discon tinuous at any surfaces, the distribution may be FIG. 53. considered as made up of a number of contin uously polarized portions abutting at these surfaces : we may confine our attention, therefore, to continuously polar ized distributions. If a given distribution of this kind has the volume density p in a region T and a surface density a- 1 on the closed surface S which bounds T , the potential function at the point (x, y, z) due to the distribution is ELECTROSTATICS. 197 where r is the distance from the point (x , y\ z ) in the volume or surface element to the point (x, y, z). If / is the value of the polarization at (x , y , z ), and if we substitute the values of p and <r in terms of / and its components A 1 , , C , we have r r[A cos (x , n) + B cos (y , n) + C cos (z f , n) ] ^ z dr . si- we may write -/// *,(). Sj*- * - D X >--^ C C A * -j J y with similar expressions for the other terms of the triple inte gral, all the double integrals are cancelled, and we have r = c D - r . r P cos (r, I ) J B cosfy , r) + C cosfz r} -J- ii L dr The quantity under the integral signs in this last expression 198 ELECTROSTATICS. is, as we have just seen, the potential function due to an element at (x 1 , y , z ), in which the polarization is / . If the polarization is solenoidal, the volume integral of p is equal to zero and a surface integral alone remains. We have seen that a polarized distribution is completely defined when the form of its boundary, S, and the values of the components of the vector / within it are known, and that its potential function has the same value (at least at outside points) as that due to an ordinary distribution of matter made up of a certain volume distribution within S and a certain superficial distribution on S. What we usually call a polar ized distribution is supposed to be quite different, however, in its physical nature from this ordinary distribution, which may be said to be mathematically equivalent to it. A simple illustration will make the character of this difference clear. If a number of small cubes, all uniformly polarized parallel to one edge, with common intensity /, were placed together, with their directions of polarization parallel, to form a larger cube, P, superficial distributions of equal and opposite den sities would come in contact, and the resulting distribution would appear to consist only of a positive charge uniformly spread on one face of the larger cube and an equal negative charge spread uniformly on the opposite face. That is, the potential function, at outside points, due to P, would be the same as that due to an indifferent body, P 1 , of the same dimensions as P, charged with a superficial distribution of density -f- / on one face and a superficial distribution of den sity / on the opposite face. If, however, we define the force at a point within a distribution to be the force which would urge a unit mass concentrated at the point, if an infinitesimal cavity were excavated at the point to allow of its introduc tion, the intensity of the force at a given point within P might be very different from that of the force at the corre sponding point in P. The first would be due merely to the surface charges already mentioned, whatever the shape of ELECTROSTATICS. 199 the cavity, while if an excavation were made in P by remov ing one of the very small uniformly polarized cubes of which it is made up, the surface charges on the adjacent cubes would appear, and, however small the cavity might be, these would be found to modify the force very appreciably. We must regard such polarized distributions as occur in nature as made up of polarized molecules, so that if any portion be broken off, across the lines of polarization, from a body in which the polarization is defined by the vector /, each portion is a polarized distribution defined by the same vector as before at every point, so that a surface distribution appears on each of the new faces formed by the fracture. Every magnet appears to be a polarized distribution of magnetic matter, and prob lems in magnetism, as the reader who has some knowledge of magnetic phenomena will see, can be conveniently attacked by the analysis of this section. If into a field of electric force a conductor or a mass of dielectric different in nature from that which it displaces be introduced, the field becomes changed in a manner completely explainable on the assumption that the conductor, or the dielectric, has become electrically polarized, and that the surrounding dielectrics are now and were polarized. Indeed, results of experiment compel us to assume that space which seems to be empty of ponderable matter is still occupied by a medium, the ether, capable of transmitting electrical forces. We must assume, also, that every medium with which we are acquainted, whether it be solid, liquid, gaseous, or ethereal, is susceptible to electrical and to magnetic forces, so that if a mass of any isotropic medium be placed in a field of electric (or magnetic) force, it becomes polarized by induction in such a manner that the direction of the electric (or magnetic) polarization coincides at every point with the direction of the resultant electric (or magnetic) force due to all the apparent electrical (or magnetic) matter in the universe, 200 ELECTROSTATICS. including that which belongs to the polarization of the medium itself. The ratio of the intensity of the polariza tion induced at any point of a medium by the resultant force at the point is called the susceptibility of the medium at the point under the given circumstances. Every medium has both an electrical susceptibility and a magnetic susceptibility, and these may be represented by very different numbers. The susceptibility of a medium to magnetic influences often depends upon the intensity of the inducing force; we may consider, however, that, if a medium is homogeneous, its elec trical susceptibility (k) has the same value throughout. A medium in a field of force may have an intrinsic polarization as well as the polarization induced in it by the field. A steel magnet in the earth s field illustrates this possibility. A given region may be at once a field of magnetic force and a field of electric force, so that any medium, when placed in this region, becomes both magnetically and electrically polar ized. Since the two polarizations are similar, we need speak in what follows only of one, if we keep in mind the fact that two quite independent polarizations may coexist. We shall represent susceptibility by k and inductivity by /n, with the understanding that different numerical values must be assigned in general to these quantities according as we are dealing with electrical or magnetic phenomena. In the most general case of either electrical or magnetic polarization we may imagine that an isotropic medium has (1) an intrinsic volume charge of density p w where p is a scalar point function, (2) superficial intrinsic charges over certain surfaces, and (3) an intrinsic polarization 7 , with components A w B w C , which may or may not be everywhere continuous. In addition to this it has (4) an induced polar ization which, as we have seen, has the direction of the resultant force coming from all the apparent charges in existence, including those which come from the intrinsic and induced polarization of the medium itself. We may assume for our present purposes that k depends upon the character ELECTROSTATICS. 201 of the medium, but not upon the intensity of the resultant force (Xj Y, Z). The whole apparent volume density, accord ing to the statements just made, is, in the general case, P = Po ~ [JMo + AA 4- A CO and this is equal, according to Poisson s Equation, to -V 2 r/4ir, or to (D^+D^Y+DtZ)/!*. If we denote 1 + 4 irk by /A, this equation may be written in several interesting forms, and may be regarded as a general ized Poisson s Equation. = 4 *[> - (D X A, 4- D,B + AQ 4 TT^o) + A (pZ + 4 7TQ = 4 + A [Z 4- 4 7r(A-Z 4-Q] = 4 7r The vector, the components of which are (/n-Y 4- 4- 4 7T^ ), (/xZ 4- 4 7rC ), or, what is the same thing, (X+4u4), (r + 47r5), (Z + 47rC), where .4, ^, (7 are the components, (&X4- 4,), (A:F4- A)> (^^ + Q, of the resultant polarization arising from the superposition of the intrinsic and the induced polarizations, is called the generalized induc tion. At a charged surface, the sum of the normal compo nents of the generalized induction, pointing away from the surface on both sides, is evidently equal to 4 TTO- O . The sum of the normal components of the force, pointing away from the surface on both sides, is equal to 47r<r, while every tan gential component of the force is continuous at the surface. If in a homogeneous medium incapable of being polarized inductively, where there is an intrinsic polarization 1^ (with components A , B w <7 ), but no intrinsic body or surface charges not accounted for by the polarization, the polarization within the medium coincides everywhere in direction with the 202 ELECTROSTATICS. force due to all the active matter in existence, including the polarization masses, and if the intensities of the polarization and force have everywhere the constant ratio X, the divergence of the polarization is evidently equal to A times the diver gence of the force, and Poisson s Equation becomes D X X + D y Y + D e Z = - 4 TrX (D X X + D v Y +D Z Z), so that the force and the polarization must be solenoidal. The converse of this theorem is evidently not true. Inductive bodies which are incapable of being intrinsically polarized are sometimes said to be electrically or magneti cally soft. Most isotropic substances seem to be electri cally soft. Bodies which can be intrinsically polarized, but which are assumed to be incapable of being polarized by induction, are sometimes said to be electrically or magneti cally hard. No absolutely hard media are known to exist, but the magnetic susceptibilities of some permanent magnets are comparatively small. The generalized Poisson s Equation becomes D x 0* X) + D y (p Y) + D g ^) = 4 7T Po in the case of a body which has no intrinsic polarization, and the generalized induction becomes the simple vector (/*X, /A Y, fjiZ) discussed in the last section. This vector coin cides in direction with the resultant force at every point. At a charged surface which also separates two media of different inductivities, the tangential components of the force are continuous, but the product of the tangential component of the force and the inductivity is clearly not continuous. The normal component of the force is discon tinuous by 4 TT times the apparent density of the charge on the surface, while the normal components of the induction are discontinuous by 4 TT times the density of the intrinsic charge on the surface. In general, therefore, in soft media, neither the tangential nor the normal components of the induction are continuous, but the directions of the force and ELECTROSTATICS. 203 induction coincide with each other close to the surface on both sides of it. If k is independent of the intensity of the resultant force, the volume density, - \_D x (kX) + D y (kY) + D z (kZ)~], due to the induced polarization in a homogeneous medium, may be written -& (D.^ + D V Y + D Z Z), or k-\ 2 V, and this van ishes at all points where there are no intrinsic body charges. We must, therefore, consider homogeneous dielectrics about charged bodies to be solenoidally polarized. If a mass, M, of a soft homogeneous medium of induc- tivity fi lt be introduced into a given field of force in an indefinitely extended homogeneous medium of inductivity /u 2 which contains no "real" charges at a finite distance from J/, the two media become solenoidally polarized, there is an apparent" charge, e , at the surface S, of J/, and the poten tial function V is now the sum of the given potential func tion F , which defined the field when the place occupied by M was filled with medium of inductivity ^ and the potential function F 7 , which might be computed from the expression Limit 2(rf /r), since it is equal to the potential function in a medium of unit inductivity due to a real charge, e . If n^ and n? are normals to S drawn respectively into and out of J/, we have at every point of S, Mi !>, V + H, - D nt V = 0, D, n V a + A, Fo = 0, and, if A. = ^ /^ A D ni V + D n2 V + (\- 1) D ni r () = 0, in which A is positive, and, since F is given, the last term of the first member is a given function, f(x, y, z). V is con tinuous at S, it is harmonic within and without S, and it van ishes canonically at infinity ; it is easy to show that all these conditions determine V (and, therefore, e ) uniquely. For if we assume that two different functions may satisfy the condi tions, and if we denote their difference by u, u must vanish canonically at infinity, be continuous at S. and satisfy Laplace s 204 ELECTROSTATICS. Equation within and without S. On S, ^ D ni u + /x 2 D n ^u = 0. If, now, we apply [149] to u, choosing for X of that equation the value ^ in the space within S and the value //, 2 in the space without S) it is evident that u must have everywhere the value zero. It is to be noted that changes in /x, and ^ which did not affect their ratio, would not affect V. If the strength of the given field had been greater in the constant ratio m than it was, the potential function V", due to the apparent charge on S } would have been larger in the same ratio, for [149] shows that the difference between the two functions V" and mV (both of which vanish canonically at infinity, are continuous at S, are harmonic within and without S, and at /S satisfy an equation of the form X D,,, V" + D n2 V" + m. /(x, y, *) = 0), is identically zero. If, when a hard body, M, solenoidally polarized intrinsi cally, is placed in a field of force in a homogeneous soft medium of inductivity unity, the directions of the polariza tion and of the resultant force are found to coincide within M, and, if the ratio of the intensities of these vectors is equal at every point of M to the constant k, the potential function within and without M is the same as if M were a homogene ous, perfectly soft medium of susceptibility k and induc tivity 1 + 4 irk, polarized inductively by the original field. To prove this we have only to compare the properties of the potential functions in the two cases. Let S be the bounding surface of M } let ;/, and n 2 represent respectively interior and exterior normals to S, and let F be the potential function due to the original field ; then F is harmonic within S, and on S, where F is continuous, -D ni F -4- Z> M2 F = 0. Let / be the polarization in the hard body M, and I" the polarization which would be induced if S were filled by a soft medium of inductivity 1 -f birk, and let F f and F" be the correspond ing potential functions. t The components of F and 7" in any ELECTROSTATICS. 205 direction at any point within S are respectively equal to k times the derivatives, at the point, in the given direc tion, of F + V and F + F". The density o- of the real charge of S in the case of the hard body is / cos ( / ) or k D ni ( F + F ) and the density a" of the charge which would be induced on S, if M were displaced by the soft medium, is - /" - cos (n l} I") or D ni ( F + F"). On S, An ( fo +v)+ A, ( v* + n = - 4 IT* . AJ ( * o + n, and (1 + 4 *k) A, ( F n + F") + A, ( FO + H = 0, or (1 + 4 TT*) - A! r + A 2 F = - 4 TT A t *w and (1 + 4 TrA ) - A t F" + A 2 r" = - 4 v k D ni F . If, now, u = V F", u is harmonic within and without S, it vanishes canonically at infinity, and it is continuous on S, where (1+4 TT&) A, M + Aa M = - Jt is eas 3 r to P rove J witn the help of [149], that under these circumstances u must be identically equal to zero, so that F and F" are identically equal. We know from work done earlier in this section that when a uniformly polarized sphere is placed in a uniform field of force of intensity X Q , so that the direction of the polarization and this field coincide, the resultant field within the sphere is a uniform field of intensity X$ 4^/3 in the direction of the polarization, and that the ratio of the polarization and the force is the constant k = 3I/ (3 X (l 4 TT/), or 3 /4 TT ( x 1), so that /=3JT [(1 +47rA-)-l]/47r[(l +4^ + 2]. We infer from this that if a sphere of soft medium of inductivity /x were placed in a uniform field of force of intensity X n in a soft medium of unit inductivity, the sphere would become uniformly polarized to intensity 3 X Q (^ l)/47r(/x + 2) and that the uniform field inside the sphere would have the direction of the outside field and the intensity 3JT /(/x + 2). Since in such a case as this the ratio of the inductivities of the inner and outer media is alone important, we may say 206 ELECTKOSTATICS. that, if a soft sphere of inductivity ^ were placed in a uniform field of intensity JT in a soft medium of inductivity /u, 2 , the sphere would become uniformly polarized to intensity 3 X 0x^-1) 74^/^4- 2), or and that the intensity of the uniform resultant field within the sphere would be 3X /(/u, ly //x 2 + 2) or 3/x 2 JF /(/z 1 -f 2/x 2 ). That part, JT (/A, /u, 2 ) /(^ H- 2/x 2 ), of the field within the sphere which is due to the polarization alone, is negative, if /A! > fj. 2 , and is then called the self -depolarizing force. If we note that in the analysis accompanying Figs. 51 and 52, x was defined by the equation -3T = 47r/x/3 and that consequently x = OiM + 2 )/(/*iM - !)> we ma J a Pp!y to our present subject all the work there done. These figures represent the lines of force for the cases /x 1 //x 2 = <x> and /ji, l /fjL 2 = % respectively; the first corresponds to a perfect conductor in a uniform electric field, or (approximately) to a sphere of very soft iron in a uniform magnetic field in air. The theory of the polarization by induction of a soft sphere in a uniform field was first given by Lord Kelvin, and this theory, with diagrams for /^//AO 2.8 and 1*1/1*2 = 0.48, may be found in his Reprint of Papers on Electrostatics and Mag netism. Very interesting figures, drawn for equal intervals of the function m and corresponding to /Xi//x 2 = 3, /u,i//x 2 = oo, are given on pages 373 and 374 in Professor Webster s Theory of Electricity and Magnetism. If an ellipsoid, made of inductively hard material and uni formly polarized [/ (A, B, (7)], be placed in a uniform field of force (X w Y w Z ), the resultant field within the ellipsoid will evidently be uniform and its components will be X Q - 2 7rabc.AK w Y -2 irabcBL M Z Q -2 vabcCM n ; the directions of the polarization and the field will not agree, however, unless these components are as A : B : C, which will be the case if ELECTROSTATICS. 207 A = kX / (1 + 2 irabcklQ, B = kY /(l+2 7rabckL ), C = kZ /(l + 2 irdbckM^, where k is any constant. If the intrinsic polarization of the ellipsoid satisfies these conditions, the ratio of the intensities of the polarization and the field is k. We infer from this that if a homogeneous ellipsoid of inductively soft material of susceptibility k be placed in a uniform field of force in a medium of unit inductivity, the ellipsoid will become uni formly polarized and that the components of the polarization will have the values just given. The resultant field within the ellipsoid will have the components X /(l+2irabckK ), Y /(l + 2irabckL ), Z /(l + 2 irabckM ) and the self-depolarizing force the components - 2 irabcKfiA, - 2 irabcL { ,B, - 2 TrabcM C. These results may be expressed in terms of the inductivity p. of the ellipsoid by writing (/w, l)/47r for k, and if, then, the ratio ^ / ^ be substituted for /x, the formulas will corre spond to the case of a soft homogeneous ellipsoid of induc tivity /M! in a field of force in a homogeneous medium of inductivity /u,,. If we remember the expression already found for the moment of the couple which a uniform field exerts upon a uniformly polarized distribution in it, we shall see that in the present case the components of this couple are 4 TTCibc (BZ - C F ) / 3, 4 Trabc ( CX Q - AZ ) / 3, and 4 nabc (A Y - BX^ / 3, or 8 irWbWk* YZ () (M - L Q ) /3 (1 + 2 TrabckM Q ) (1 + 2 7rabckL ), 8 7r- 2 ft 2 c 2 /t%X (JT - M Q ) /3 (1 + 2 vabckKJ) (1+ 8 ir 2 a 2 b 2 c*k*X F (L - JC ) / 3 (1 + 2 TrabckL^ (1 + 2 208 ELECTROSTATICS. and it is to be noted that, according to the reasoning of page 122, if a > b > c, K Q < L < M Q . If the lines of the field in the air make an acute angle with the plane of xz and are perpendicular to the z axis, Z = and X Q and Y are positive, the axis of the couple is the axis of z, the moment is positive (even for such negative values of k as occur in nature), and the ellipsoid tends to turn so that its long axis shall have the direction of the field. For perfectly hard, intrinsically polarized bodies the gen eralized Poisson s Equation becomes and if p = 0, as in the case of a hard magnet, the induction is solenoidal. Unless the intrinsic polarization happens to have the same direction as the resultant force, or vanishes, the lines of force and the lines of induction do not coincide. In some cases, the directions of the force and of the polariza tion are exactly opposed, and the lines of force and of induc tion are opposite in direction. Outside a hard magnet, where the intrinsic polarization is nothing, the lines of force and of induction are identical. At the surface of the magnet, where there is no intrinsic charge except that which belongs to the polarization, the normal component of the force is discontin uous, while the normal component of the induction is con tinuous. It is convenient, therefore, to regard the lines of induction as closed curves. The lines in Fig. 50 represent both lines of force and lines of induction, but it is to be noticed that inside the sphere, where X= 4 7T//3, the direc tion of the lines of force is from right to left, while the direc tion of the lines of induction (since /= 4- 8 ir//3) is from left to right. The straight line of force through the centre of the sphere is discontinuous in direction at the surface, while the corresponding line of induction is continuous. The ELECTROSTATICS. 209 tangential components of the force are continuous at the sur face of a magnet, those of the induction discontinuous. The normal component of the induction just within the surface is (X + 4 TT AO) cos (x, n) + ( Y + 4 TT S ) cos (y, ) + (Z + 4 TT C ) cos (2, w), or the normal component of the force plus 4ir times the density on the surface belonging to the intrinsic polarization. If we make a small cavity inside a generally polarized hard body, the force at any point of the cavity is the original value of F (that is, the negative of the gradient of the potential func tion at the point), minus the contribution due to the volume charge removed, plus the force due to the new surface charges which appear on the walls of the cavity. If the volume of the cavity be made smaller and smaller, the contribution due to the volume charge removed can be made as small as we like, while the effect of the surface charges may remain finite. Let the cavity be of the form of a piece of a slender tube of polarization lying between two near orthogonal surfaces. In this case there will be surface charges on the ends of the cavity, but none on the side walls. These charges, of density /, will be such as to drive a particle of positive matter in the centre of the cavity to that end of the cavity towards which the polarization is directed. If we reflect that a surface distribution of finite sii, which has at the point P the density r, repels a unit point charge infinitely near P with a force of 2 TTO-, but that an element of the surface at P, infinitely small with respect to the distance of a point charge from P, has no perceptible effect upon this point charge, it will be easy to see that, if the cross-section of the cavity is infinitely small compared with its length, the force due to the surface charges on the ends approaches zero as the whole cavity is made smaller, and the force at the centre of the cavity is F. If, however, the length of the cavity is infinitely small compared with its cross-section, the force due to the charges on the ends is 4 TTO-. or 4 TT/, so that the whole force is F + 4 TT/, or 210 ELECTROSTATICS. the induction at the point before the cavity was cut. If the infinitely small cavity were spherical, the force at its centre would be F + irl. It is to be carefully noticed that we have no means of determining an absolute inductivity for any medium, but only the ratio of the inductivity of the medium to the inductivity of some other medium taken as a standard. The unit quantity of electricity is defined to be that quantity which concentrated at a point at a distance of one centimetre from an equal quantity would repel it with a force of one dyne, when the dielectric is the ether. In any other homogeneous dielectric of inductivity /x times that of the ether, e of these units of electricity concentrated at each of two points distant r centi metres from each other would repel each other with a force of e^/pr* dynes, so that, if this medium had been used as a standard, the unit of electricity would have been larger in the ratio of v/x to 1 than it now is. If a charged conductor is enveloped by an infinite, homogeneous dielectric, we may assume the apparent charge on it to be its real charge and neglect the polarization of the dielectric (and we do this when the dielec tric is the standard substance which we assume to have unit inductivity, and hence no susceptibility) ; or we may suppose the dielectric to be polarized, and consider the apparent charge to be the algebraic sum of the real charge on the conductor and the charge belonging to the polarization induced on the dielectric. This we do when the dielectric is not the standard substance, assigning to it a susceptibility based on that of the standard substance which is the zero of our scale. A simple illustration will tend to make the rather complex relations which would attend a change in the choice of a standard substance more intelligible. A spherical conductor of radius a has a charge, E, and is surrounded by three spherical shells of homogeneous dielectric, concentric with it, the last reaching to infinity. The radii of the spherical surfaces which separate the first (inner) dielectric ELECTROSTATICS. 211 from the second, and the second from the third, are b and c respective!}-, and the inductivities of the dielectrics, referred to some standard substance, are /n l5 /x. 2 , and /* 3 , or 1+4 irk ly 1+4 TJ-A-O, and 1 + 4 7rA- 3 . If we apply Gauss s Theorem suc cessively to spherical surfaces concentric with the conductor and lying in the first, second, and third media, we learn that the force ( D r V) at a distance T from the centre has, in the three media, the values E / /^r 2 , E / ^ 2 r 2 , E / > 3 r 2 . The con ductor acts like a medium of infinite susceptibility. The induced polarizations in the three media are directed radially outward, and their intensities are ^E/^r 2 , kc.E/p.c.r*, k z E/w\ The densities of the apparent charges at the surface S l of the conductor and at the surfaces of separation S z , S s of the dielectrics, regarded as manifestations of the polarizations, are on S o- 2 = if* 2 (j.. 2 = fi l /x 2 /47r^ 2 /x 1 /A 2 on S& and o- = kEL* - kE = E - i*^ 2 on s These same densities might also be found by the aid of the ordinary characteristic equation of the potential function at an apparently charged surface, D n .V + D n ,V 47ro- . If instead of using the old standard we make the outer dielectric of this problem the standard, the unit of electrical quantity will be larger in the ratio of V^ to 1 than it was before, and the old charge on the condenser will be E/^JL- expressed in the new units. The strength of a field at any point being the force in dynes which would be experienced by a unit of positive matter placed at the point, the number which expresses the strength of a given field in the new units is Vj^ times the number which expresses the strength of the same field in the old units. The inductivities of the three dielectrics are now /M//^, /x 2 //x 3 , 1, and their susceptibilities are 3 or (A-! - A- 3 )/(l + 47rA- 3 ), x 3 or (A-o A- 3 ) (1 + 47rA- 3 ), and 0. 212 ELECTROSTATICS. The strengths of the fields in the three dielectrics are the intensities of the polarizations are E(^ - Ms)/4 T V^ nt* 9 ^(/* 2 - /x 3 )/4 TT V^ /x^ 2 , ; and the apparent charges on S l9 S 2 , S s have the densities E V/T 3 /4 Tra 2 ,*!, and ^ (/x 2 The sum of the apparent charges on Si, $ 2 , $ 3 is now .Z?/ V/x 3 , the real charge on the conductor expressed in the new units, and the sum of the induced charges is zero. In the case first treated, where the outer medium was supposed to be polarizable, the sum of the apparent charges on Si, S z , S 3 was E / ^ and this, being expressed in the old units, is equivalent to i?/ V/x 3 in the new. The sum of the induced charges was the difference between E / /x 3 and E or E(l ju, 3 )//>t 3 ; in this case, however, we must imagine the outer surface "at infinity" of the outer medium to have an induced charge in total amount equal to the integral of the normal component of the polarization (k 3 E/p. 3 r 2 ) over the surface, or 4 irk^E / ^ and this is equal to E(fjL 3 l)//n 3 , so that here, again, the whole amount of the induced charge is, of course, zero. It is to be noted that this finite charge at infinity does not affect the electrical field in any way. We have seen that when the outer medium is taken as a standard the inner medium has a susceptibility (^! /x 3 )/47r/x 3 , and this is sometimes called the susceptibility of a medium of inductivity ^ with respect to a medium of iiiductivity /x 3 . No medium has yet been found to be less electrically susceptible than the ether. Some bodies are less magnetically susceptible than the ether, so that their suscepti bilities are negative on the usual scale. These bodies are said to be diamagnetic. If a body of inductivity /*!, bounded by the surface S, is placed in a large mass of a medium of inductivity /x 2 , the outer ELECTROSTATICS. 213 surface of which is so far removed from the place of observa tion that the apparent charge on it contributes little to the field of force, the fact that the outer medium is really polarized may be lost sight of ; and if we attribute the apparent charge on S wholly to the polarization of the inner medium, instead of regarding it as the difference between the charge of one sign due to the polarization of the inner medium, and the charge of the opposite sign due to the polarization of the outer medium, the apparent susceptibility of this medium will be (/A! p^)/7rfjL<,. If fj. 2 is greater than /x x , this will be negative and the inner medium will seem to be polarized in a direction opposite to that of the force. If in any given case the direction of the vector / is every where perpendicular to the direction of its curl, it is possible to cut a polarized distribution by a set of surfaces, u = c, everywhere normal to the line of polarization. If surfaces of this family be drawn for small constant differences, Aw, of the scalar point function u, the distribution will be divided into shells, each of which is polarized normally to its surface. If AM is the thickness of one of these shells at a given point and / the average intensity of polarization on a line of polariza tion drawn through the shell at the point, J A;i is called the strength of the shell at the point. Since D H u = h u , the value of the gradient of ?/, the strength of a shell of infinitesimal thickness can be written I-du/h u . A shell is said to be simple if I/k u has the same numerical value all over it; otherwise the shell is said to be complex. If A, B, C are the intensities of the components of the vector /, the fact that the lines of / coincide with the nor mals to the surface u = c gives the scalar equations A/I= D x it/h u , B/I= D y u/k u , C/I= I)M/h^ and with the help of these the vector \D,C - D S B, D Z A - D X C, D X B - D y A^ 214 ELECTROSTATICS. which is the curl of /, may be written in the form \_D z u D !t ( T I //) - D yU D v u D x (I/h H ) - D x n - - If, now, the scalar quantity I/h u has the same numerical value over every surface of constant u, it must be, if not everywhere constant, a function of n only, so that D x (I/h H ) : A," = D v (I/h u ) : D y u = D z (///,.) : D g u, and if these relations are satisfied, the components of the curl of / vanish, and the polarization is lamellar. Every lamel- larly polarized distribution may be divided up into simple FIG. 54. polarized shells ; if the polarization is not lamellar, but if the directions of this vector and its curl are everywhere perpen dicular to each other, the distribution, as we have seen, may be divided up into shells, but these will not be simple. The potential function due to a polarized element of moment M has at a point, P, distant r from the element, the value M cosa/r 2 , where a is the angle which a line drawn from the element to P makes with the direction of polarization in the element. If a very thin simple shell be divided up into ele ments of length equal to the thickness (A??) of the shell and of cross-section equal to an element (A$) of one surface of the shell, the moment of each element is A$-/A?i, and if , the constant strength of the shell, be denoted by <, the ELECTROSTATICS. 215 potential function due to the shell has at any point, P, the value limit <^ cos(r, n)S/r* = <<o. where to is the solid angle subtended at P by the boundary of the shell. This value is positive if, in looking out from the vertex P within the conical surface which passes through the boundary of the shell, one sees the positive side of the shell. If, while the strength of the shell is unchanged and the boundary fixed, the shell itself be imagined deformed in any way, the value at P of the poten tial function due to the shell will be unchanged so long as P is on the same side of the shell. The potential function due to a closed simple shell of any form is zero at every outside point and 4 TT< at every inside point, where the positive sign is to be used if the positive side of the shell is turned inwards. If P and P are two points close to each other on opposite sides of a simple, very thin shell, S, of strength <l>, and if V P and V P . are the values of the potential function at P and P , due to S, we may imagine the shell closed by an additional shell also of strength 3> which shall add to the potential func tions at each of the near points P and P the quantity x. If P is within the closed shell, P 1 will be outside, so that V + x = 0, V 4 x = 4 irfc, or V - V = 4 ^. The potential function due to an infinitely thin, open or closed, simple polarized shell is, therefore, discontinuous at the shell by 4 TT times the strength of the shell. The potential energy of a magnetic north pole of strength m at a point, P, near a simple, finite magnetic shell is M<U>, and if P is on the positive side of the shell, ?>t<<o ergs will be done by the field on the pole if it be carried to infinity by any path. If the pole be carried around the edge of the shell from a point very near the shell on the positive side to a point very near the first but on the negative side, the work done on the pole by the field will l>e 4:inn& ergs. In general, c os(/>, ) . 216 ELECTROSTATICS. where r is the distance from dS to P, and n is the normal to the shell on the positive side. If the directions of both r and n were considered reversed, the value of the integral would be unchanged, but it would then more clearly represent the sur face integral, taken over the shell, of the normal component towards the negative side of the shell of the force due to mag netic pole at P. If, instead of a single pole at P, there is any collection of poles at different points or, indeed, any magnetic distribution, M, the mutual potential energy of the shell and this distribution is equal to <f> times the flux of magnetic force due to M in the negative direction through the shell. A simple magnetic shell in a magnetic field, H w due to matter outside the shell tends to move so as to decrease the FIG. 55. mutual potential energy of the shell and the field, and this quantity, as we have just seen, is equal to the negative of the product of the strength of the shell and the number N of lines (unit tubes) of force due to the field which cross the shell in the positive direction. The shell, therefore, tends to move so as to make N as great as possible. If the shell be displaced parallel to itself through a very short distance, du, in any direction, the limit of the ratio of the loss of energy (+-dN) caused by the displacement to du (i.e., 4> D U N) measures the force U, which tends to move the shell in this direction. If we suppose that the shell in being displaced does not encounter any of the magnetic matter which gives rise to the field, Zf will be a solenoidal vector within the cylinder ELECTROSTATICS. 217 generated by the shell, so that the integral of the normal out ward component of H Q taken over the surface of the cylinder will be zero. The shell in its initial and final positions forms the ends of the cylinder, and these together contribute dN to the surface integral, so that the convex surface must con tribute dN. If ds is an element, measured in the positive direction about the shell, of the curve which bounds the shell in its original position, and if dS is the element of the convex surface of the cylinder generated by ds, dS = ds du sin (du, ds) ; the magnetic induction through this element due to the magnetic matter outside the shell is HQ cos (n, 7/o) sin (du, ds) du ds, and this integrated with respect to s is equal to dN, or to Udu/&. Therefore, V = & ( H Q - cos (n, H Q ) sin (du, ds) ds, and the component in any direction (u) of the whole force on the shell may be expressed as a line integral taken around the curve which bounds the shell. The integrand vanishes at any point where u is parallel to H Q or to ds, but if at any point u happens to be perpendicular to the plane of H Q and ds, the integrand becomes H Q sin (H w ds), the component of the field perpendicular to ds. If, with this fact in mind, we choose at every point on the curve a direction, p, perpendicular to the plane of If^ and ds, so that cos (p, ds) = and cos (p, If ) = 0, and remember that cos (n, ds) = 0, cos (u, n) = 0, we may easily prove that H Q cos (n, H ) sin (du, ds) = H Q sin (H Q , ds) - cos (p, u). This shows that U may be mathematically accounted for by assuming that every element of the curvilinear boundary of the shell is urged in a direction perpendicular to the field and 218 ELECTROSTATICS. to the element, by a force numerically equal to the product of the length of the element, the strength of the shell, and the component perpendicular to the element of the field, JT . If the field is due to a single magnetic pole of strength m at a point, P, distant r from ds, the force on the element would be w<3>- sin(r, ds) -ds/r 2 , and the force exerted by the shell on the pole would be accounted for by assuming that every element, ds, of the boundary of the shell contributed an elementary component, w<J>-sin(>, ds)ds/r 2 , in a direction perpendicular to the plane of P and ds. VECTOR POTENTIAL FUNCTIONS OF THE INDUCTION. Every vector, K, which, except in a given finite region, T, is everywhere continuous, solenoidal, and lamellar, has in simply connected space outside T an easily found scalar poten tial function, W, which satisfies Laplace s Equation. We may assign to W at pleasure a numerical value at any given point, 0, and define the value of W at any other point, , to be the line integral of the tangential component of K taken along any path from to which does not cut T. The partial derivatives with respect to x, y, and & of W thus defined outside T are evidently equal at every point to the compo nents of .A" parallel to the coordinate axes, and, since K is solenoidal, V 2 W= 0. If K so vanishes at infinity that the limit of the product of its intensity and the square of the distance (>) from any finite point is finite, the limit of r 2 D r W is finite, and if we assign to W the value zero at any point at infinity, its value everywhere at infinity will be zero. If K is continuous and if it vanishes at infinity in the manner just described, and is known to be everywhere solenoidal and lamellar, it must vanish everywhere ; for, if we apply [151] to the harmonic function W within an infinite sphere, it will appear that IF, which vanishes at infinity, is identically equal to zero. The vector which represents the force in the case of a charged spherical conductor is solenoidal ELECTROSTATICS. 219 and lamellar within and without the conductor, and it vanishes properly at infinity, but it is discontinuous at the surface of the sphere. It is usually convenient to assume that the integral of the normal component of a vector, taken over any closed surface at which the vector and its first derivatives are continuous, is equal to the integral of the divergence taken through the space within the surface, even though at some inner surface the vector is discontinuous. On this assumption the vector just mentioned is not solenoidal on the surface of the conductor, for it has there divergence equal in total amount to 4?r times the charge. The line integral of the tangential component of a vector, taken around a closed curve on which this component is con tinuous, is generally used as a measure of the integral of the normal component of the curl of the vector taken over a cap, *S , bounded by the curve, even though at some curve on S the vector ceases to be continuous. A vector cannot be considered lamellar at a surface where, though its normal component is continuous, some of its tangen tial components are discontinuous. If two continuous vectors, U and U , which so vanish at infinity that r 2 U and r*U have finite limits, have at every point in space equal curls and divergences, and are lamellar and solenoidal outside certain given finite regions, they are identical ; for the difference between these vectors is every where lamellar and solenoidal, and it vanishes at infinity in such a manner that the product of its intensity and the square of the distance from any finite point is finite. This theorem may be extended to the case where U and U , though not everywhere continuous, have identical discontinuities. If Nj, &, T/!, t represent the numerical values at the point ( X D y\i %i) of the divergence and the curl components of the vector U, which outside a given region is everywhere continu ous, lamellar, and solenoidal. and which so vanishes at infinity that r 2 U has a finite limit ; if 220 ELECTROSTATICS. and if in which the integrations are to he extended over all space, or at least over all space where U is not lamellar and solenoidal ; we know from the theory of the Newtonian potential function, where similar integrals have been studied, that, if N, , 77, are the divergence and the curl components of U at (x, y, ), V*E = N, V*F X = - fe ^F y = - I,, V*F Z = - f. The divergence of the vector F, which has the components F x , F y , F s) is equal to and, since Z>.,(l/r) = - D Xl (l/r), and - ft D, (1 /r) = J)^, /r-D Xi (^/r), we may write this by the help of Green s transformation in the form * ti + D -^ + D ^ } /r dTl [f , cos (x, w) + 17, cos (y, w) + {i cos (,)] /r rf/S,, where the second integral is to be taken over the outer boundary of space. The integrand of the triple integral vanishes every where, because the vector (, rj, ), being the curl of another vector, is itself solenoidal. The field of the double integral is in a region where U is lamellar, so that the integral itself vanishes and F is seen to be solenoidal for all values of x, y, and 3. From these results it appears that the vector which has for its components (D X E plus the x component of the curl of F), (D y E plus the y component of the curl of F), (D Z E plus the ELECTROSTATICS. 221 z component of the curl of F) has everywhere the same curl and the same divergence as U and vanishes like it at infinity, so that it is identically equal to U. D X E, D y E, D Z E are the components of a lamellar vector, and the curl of F is solenoidal, so that the vector U, which is not everywhere either solenoidal or lamellar, is everywhere expressible, as was first shown by Helmholtz,* as the sum of a solenoidal and a lamellar vector. The equations U x = D X E + D y F s - D z F y , U y = D y E + D Z F X - D X F Z , give any vector, U 9 which is known to vanish properly at infinity, when its curl components and its divergence are known. If U is solenoidal, E vanishes and F is a vector potential function of U. Every lamellar vector has a scalar potential function the component of the gradient of which, at any point, in any direction, gives the intensity of the compo nent of the vector at that point, in that direction. The com ponent at any point, in any direction, of the curl of a vector potential function of a solenoidal vector gives the intensity of the component of the vector at the given point, in the given direction. Heaviside gives the name "circuital" to a vector which is solenoidal but not lamellar, and the name "diver gent " to a vector which is lamellar but not solenoidal. If p l is a function of x lf y^ 1? and if r 2 stands for the expression (x - a^) 2 + (y ytf -f (z z^ 2 , the familiar inte gral ill dx^du^dz^ extended over all space, is a function of X, //, Zj which Prof. J. Willard Gibbs in a remarkable paper t has denoted by the symbol Pot p. Using this notation, we may write 4 7 r =-PotN, 47r^ = Pot, 4 v F y = Pot 77, 4 ir^ = Pot ; * Crelle s Journal. Bd. LV, 1858. t Elements of Vector Analysis, 92. See also Heaviside s Electrical Papers, XXIV. 222 ELECTKOK1NEMATICS. and if we represent by Pot curl U the vector which has for its components Pot , Pot 77, Pot , we have the vector equation 4 irF= Pot curl V, and if U is solenoidal, 4 TT U= curl Pot curl U. If U is solenoidal, 4 TT 7 curl curl Pot CT = Pot curl curl U, and curl Pot U is a vector potential function of kirU, or Pot 7 is a vector potential function of a vector potential function of irU. In the case of any polarized distribution whatever, provided there is no intrinsic volume density p , the induction is solenoidal and has a vector potential function. II. ELECTROKINEMATICS. 70. Steady Currents of Electricity. When a charged body A is brought up into the neighborhood of a previously uncharged, insulated conductor J5, the two kinds of elec tricity which, according to our provisional theory, exist in equal quantities in every particle of B tend to separate from each other and, as a consequence, free electricity appears on B s surface, some parts of this surface becoming charged positively and other parts negatively. If A is brought into a given position and fixed there, the distribution on the surface of B quickly attains and keeps a value determined by the fact that the whole interior of B must be a region at con stant potential, or, in other words, that the resultant force at any point within B due to the free electricities on its surface must be equal and opposite to the force at that point due to all the free electricity outside B. If, now, A with its charge is moved to a new position, the old distribution on 2? s surface will not in general screen the interior of B from the action of ^4 s charge, and a new separation of electricity within B and a new arrangement or combination of the charge on the surface is necessary before a new state of equilibrium can be established. If A be moved continuously in any manner, there will be a con stant attempt on the part of the separated electricities to set ELECTROKIXEM AT LCS. - - 8 up a state of equilibrium, and hence at every point of B there will be, in general, some electrical change going on continually. If two conductors ^1 and B at different potentials be con nected by a fine Avire, the whole will form a single conductor, which can only be in a state of equilibrium when the value of the potential function due to all the free electricity in existence is constant throughout its interior, and there will be such a transfer of electricity through the wire as will establish this state of equilibrium in a very short time. If, however, by any device we can furnish unlimited quantities of electricity to A and B in such a way as to keep them at the same potentials as at the beginning, there will be a continual attempt to establish electric equilibrium within the compound conductor consisting of A, B, and the wire, and. as a result, there will be a continual transfer of electricity through the wire. The transfer of electricity from one place to another through a conductor is a very common phenomenon. Sometimes, as we have seen, electricity traverses the conductor for a short time only ; sometimes, however, the transfer goes on indefinitely, and, so far as we can judge from its attendant phenomena, at a constant rate, so that just as much of a given kind of elec tricity crosses any surface within the conductor in any one second as in any other : such a continuous steady transfer as this is called a " steady current." The existence of a steady current in a conductor implies a force tending to drive electricity through the conductor ; that is, it implies, at least in the absence of moving magnetic masses and of electric currents in the neighborhood of the conductor, free electricity somewhere in existence which gives rise to a potential function not constant throughout the conductor. No pail of a conductor through which a steady current is flowing- can accumulate free electricity as the time goes on, for such an accumulation increasing with the time would be accompanied by changes which must show themselves outside the conductor. We are led to assume, then, that if any closed surface be drawn inside a conductor which carries a steady current, just as much 224 ELECTROKINEMATICS. electricity of a given kind enters the region enclosed by the surface in any interval of time as leaves it during that interval. We have seen that at every point inside a conductor where there is a resultant electric force there will be an electric sepa ration which will go on as long as the force exists. Experi ment seems to show that the rate of separation of quantities of electricity is proportional to the magnitude of the force. Let P be a point of a small plane area u> inside a conductor, and let F be the average value during the interval from t to t -f- A of the component of the electric force normal to this area ; then in what follows we shall assume that the amount of positive electricity which crosses this surface, in the sense in which the force points, during the interval is k w F* A, where A; is a con stant depending only upon the material of which the conductor is composed and upon its physical condition. The average value of this flux per unit of time per unit of surface is, there fore, k F. If, now, <o and A are made to grow smaller and smaller in such a manner that P is always a point of o>, F ap proaches as a limit the negative of the value at P of the deriva tive, taken in the direction in which F acts, of F, the potential function due to all the free electricity in existence ; so that at any instant the value at a point, P, in any direction, n, of the rate of flow of positive electricity across a surface normal to ??, per unit of this surface per unit of time, is the value at P of -k-D n V. It follows from this that if any tube of force be drawn in a conductor which carries a steady current, there is no flow through the sides of the tube. Consider a region shut in by a tube of force and by two equipotential surfaces inside a con ductor through which a steady current is flowing. Let o^ and w 2 be the areas of the equipotential ends of the region, and let F l and F 2 be the average values of the normal force, taken in the same sense in both cases, over these ends. Applying Gauss s theorem to this region we have F 2 w 2 F^ = 47rQ, where Q is the amount of free electricity, algebraically considered, within the region. If the conductor is homogeneous, the amount of ELECTROKINEMATICS. 225 positive electricity which enters or the amount of negative electricity which leaves the region by one end per unit of time is kF l o^, and the amount which leaves it at the other end is kF 2 tu 2 . These amounts are equal, so that jF 2 w 2 F l w 1 = ; hence, Q = 0, and there is no free electricity at any point within a homogeneous conductor which carries a steady current. The free electricity which gives rise to the potential function the rate of change of which is proportional to the flow of electricity within the conductor, must then lie either outside the conduc tor, or on its surface, or both. It would not be difficult to prove that there must be a distribution of electricity on parts of the surface of even- conductor which carries a steady current and is in contact in some places with an insulating medium ; but the fact that a wire through which such a current is passing may be moved about so as to change its position with respect to outside bodies without changing the amount of the current will suffice to make it probable that a part, at least, of the free electricity that we have been considering moves with the wire. Since the density of the free electricity within a conductor which carries a steady current is zero, the potential function F, inside the conductor, must satisfy Laplace s Equation ; that is, V 2 F=0. It is easy to see, since there can be no accumulation of free electricity in any conductor which bears a steady current, that the amount of electricity which comes up on one side to the common surface of two such conductors which are in contact must be equal to that which goes away from this surface on the other ; that is, at every point of the surface, l\ D n Vi = A* 2 Z).,F 2 , where k\ and A\> are the spe cific conductivities of the two conductors, and D n V l and D n F 2 the values at the point, taken in the same sense in both cases, of the derivatives of F in the direction of the normal to the sur face, one on one side of the surface, and the other on the other. It is to be noticed that the boundary between two such con ductors may or may not be an equipotential surface. At every point of the common surface of a conductor and an insulating medium k-D n V=Q or D,,F=0; hence the equipotential sur- 226 ELECTROKINEMAT1CK. faces within the conductor cut the surface where the conductor abuts on the insulating medium at right angles. 70. Linear Conductors. Resistance. Law of Tensions. Let us consider the case of a linear conductor, that is, one in which all the lines of force are parallel to each other and to the sides of the conductor, so that every tube of force has a constant cross-section throughout that part of its length which lies in the given conductor. It will appear later on that any right cylin drical conductor, whatever the form of its cross-section, will be a linear conductor, if every point of one of its ends be kept at one constant potential, and every point of the other end at another. It will also be evident that such wires as are ordinarily used for making electrical connections are, to all intents and purposes, except perhaps at the very ends, linear conductors, whether these wires are straight or curved. Let the ends of a homogeneous long uniform straight wire of constant cross- section q, and of length /, be kept respectively at potentials V and F". Take the axis of the wire for the axis of aj, and the origin at that end of the wire at which the potential func tion due to all the free electricity in existence is F ; then every line of force inside the wire is parallel to the axis of x ; and since there is no force in any direction perpendicular to the axis of a?, Z> 3/ F=0 1 , Z) 2 F=0, and Laplace s Equation, which must be satisfied by F inside the wire becomes Z)/F=0, whence V Ax -f- B ; or, since V V when x = 0, and F= F" when x = l, r= (V-V )X | y, The steady current c which traverses the wire carries across every right section in the unit of time kg D X V units of posi tive electricity in the positive direction of the axis of x. That is, where k is the specific conductivity of the material out of which ELECTROKINEMATICS. I l i the wire is made. The quantity l/kq is called the resistance of the wire, kq/l its conductivity. The quantity & is a function of the temperature. In the case of a pure solid metal at any ordinary temperature a rise of 1 C. will increase 1/k by about 0.004 times its own value. This fractional increase is much smaller in the case of some alloys : for " manganin " at room temperatures it is not more than 0.00001. The analysis of this section assumes that the homogeneous linear conductor is at the same temperature throughout and that it is not surrounded by a changing magnetic field. It is an important physical principle, first enunciated in a slightly different form by Ohm, that if a fixed portion of the surface of a given homogeneous conductor be kept constantly at potential V^ and another fixed portion at potential V Z) while the rest of the surface of the conductor is in contact with an insulating medium, the ratio of V V to the steady current which traverses the conductor, as measured by the quantity of positive electricity per unit of time which either enters the conductor through the surface V = J\ or leaves it through the surface V = F 2 , is a quantity independent of V l and F 2 . This ratio is called the resistance of the conductor under the given circumstances. The resistance of a conductor depends not only upon its shape, the material of which it is composed, and the temperature and other physical conditions of this material, but also upon the shape, size, and position of those portions of the surface which are kept at the potentials V\ and F 2 . The resistance of so much of a tube of force drawn in a conductor which bears a steady current as lies between the equipotential surfaces V J\ and V = V z is the ratio of V\ V z to the amount of positive electricity per unit of time which enters the portion of the tube which we have been considering through the surface V = Fi, or leaves it through the surface V= V 2 , or crosses any section of the tube in the direction indi cated. Any electric change which, under the same conditions of temperature and pressure, will leave this tube of force still 228 ELECTROKINEMATICS. a tube of force and its equipotential ends still equipotential, however the value of the potential function may be changed, will, according to this law of Ohm, leave the resistance the same. Other things being equal, the resistance of a tube of force increases with the length of the tube and diminishes as the section of the tube is made greater. Suppose that we have a series of linear conductors joined end to end in a closed ring, so that the end of the nth conductor is in contact with the beginning of the first. Let F m and V m " be the values of the potential function at the beginning and end of the rath conductor, and r m the resistance of this conductor. Since the same current c must traverse every conductor of the series, we have 77 77 it 77 77 f 77 77" 77 77" y. . 1 1 -~ cr l) r 2 2 67 2> 3 3 ~ Lr 3? n V n ~~ 6/ M> and, if we add them together, we shall get r l + r 2 + r s -\ -f r n where F 2 FI" is the difference between the values of the potential function on opposite sides of the surface common to the second and first conductors, F 3 F 2 " the corresponding difference for the third and second conductors, and so on around the ring. If these differences are not all zero, the circuit is said to be the seat of an electromotive force. We may here assume that when any two conductors, at the same temperature throughout, but made of different mate rials, are placed in contact with each other, a discontinuity* of the potential function suddenly appears at their common * Although the language of the old "Two Fluid Theory" is used in this chapter, the reader is strongly urged to make himself acquainted with the physical theories now commonly used in accounting for electrical phenomena. See Dr. 0. J. Lodge s papers " On the Seat of the Electro motive Force in the Voltaic Cell," printed in the Philosophical Magazine for March, April, May, and October of 1885, and his " Modern Views of Electricity," a series of contributions to Nature, begun in 1886. ELECTROKINEMATICS. 229 surface. The amount of this discontinuity, which remains con stant after it has once been established, is the same for all points of the common boundary of the two conductors, and is independent of their size and shape, of the extent of surface in contact, and of the absolute values of the potential function on either side of the boundary. We shall represent the sudden fall in the value of the potential function encountered by pass ing from a conductor made of material A to a conductor made of material B across any point of their common surface b}~ the symbol A \ B. A certain class of substances, to which all metals belong, has the property that if L, M, and y are any three of these substances, all at the same temperature, L | J/+J/| X=L | JV. This class is said to obey " Volta s Law of Tensions." If a number of conductors made of different kinds of metals all at the same temperature be placed in line, the first in contact with the second, the second with the third, and so on, the algebraic sum of the jumps of the potential function encountered in going from the first conductor to the last through all the others is exactly the same in amount as the single jump which would occur at the common surface of the first and last conductors if they were put directly in contact with each other. Some other substances besides metals obey the Law of Tensions, but most liquids and solutions, whether in contact with each other or with metals, do not obey this law. The sum of the jumps in the potential function encountered in passing from copper to zinc by way of an iron conductor is the same, if the whole be at one temperature, as the jump encountered in passing directly from copper to zinc. But this is not equal to the sum of the jumps met with in passing from copper to zinc through sulphuric acid. Cu | Fe + Fe | Zn=Cu | Zn, but Cu | (H 2 SO 4 ) + (H 2 SO 4 ) | Zn ^ Cu | Zn. The numerator of the expression just found for the intensity of the current which traverses a closed chain of linear conduc- 230 ELECTKOKLNKALATICS. tors is evidently the algebraic sum of the "jumps" in the potential function encountered by travelling in the direction in which the current is supposed to move, from the first conductor to the last through all the others, and reckoning the jump at any boundary positive if the value of the poten tial function is increased as one crosses the boundary. If all the conductors which form the circuit are metallic and all at the same temperature, whether or not they are all made of the same kind of metal, this numerator is zero, and it follows that in order that a steady current may traverse a circuit of con ductors, one at least of the conductors must disobey the Law of Tensions. The same formulas apply to a circuit composed of conduc tors of any form if each of the common surfaces of contigu ous conductors is equipotential. Every slender tube of force in a homogeneous conductor which carries a steady current is also a tube of flow and constitutes a current filament. We shall hereafter apply the term linear only to conductors which have very small cross-sections. 72. Electromotive Force. We have seen that if a number of homogeneous conductors made of different materials be connected in series to form a heterogeneous conductor K, there will be discontinuities in the electrostatic potential function within K at the common surfaces of adjacent con ductors. If an equipotential surface A near one end of K be kept at potential V A , and an equipotential surface B near the other end of K y at potential V B , and if the algebraic sum of the discontinuities of potential between A and B, counting a step up as positive, is E, the current in K from A to B will be ( V A V B -f- E) /r, where r is the resistance between A and />. In such a case as this, V A V R is called the electrostatic or external electromotive force, and E the internal or intrinsic elec tromotive force. If K forms a closed circuit, all the electro motive force may be regarded as internal. In this connection ELECTROKINEMATICS. 231 it should be said that, although physicists are not all in agreement as to the magnitude of the discontinuity of poten tial at the surface of contact of any two given dissimilar conductors, there is no difference of opinion as to the alge braic sum of these discontinuities in the case of any closed circuit. If one end of a hetero geneous cylindrical con ductor K, of given resistance r, formed of homogeneous cylindrical conductors in series, be kept at a given poten tial F! and the other end at the given potential F 2 , the value of the potential function will depend very much upon the constitu tion of K. Three different cases are illustrated in Fig. 56, in which abscis sas represent resistances and ordinates the corre sponding values of V. In these figures A is sup posed to be an electro lyte, while Z, M, N are metals : F x = 2, F 2 = 0.5, A\N=O.S, A\M=1.S, N\M=Q. The current strength (indicated by the slope of the line which gives the value of F) is evidently different in the different diagrams. Fig. 57 represents V in a long chain made of two metals P, Q, and an electrolyte 7?, such that R \ P is small, R \ Q still smaller, and P \ Q zero. Here the ends are at the same FIG. 56. 232 ELECTKOKINEMATICS. potential, and there are no great potential differences any where in the chain, but the current (as indicated by the slope of the F line) is large, as is the sum of the small discontinuities which go to make up the electromotive force in the chain. A galvanic battery may be regarded as a chain of three or more generally non-linear conductors, at least one of which disobeys the Law of Tensions. The algebraic sum of the jumps in the potential function encountered by starting at that pole of a galvanic battery at which the potential is less, and passing to the other pole through the battery, is the electromotive force of the battery. The difference of poten tial between the poles of the battery, when they are not connected, measures this electromotive force. Chemical action goes on inside every FIG. 57. battery when its poles are closed ; some of its solutions are decomposed, and the products of this decomposition often appear at the boundaries of the liquid conductors inside the battery and decrease the electromotive force by changing the amount of jump in the potential func tion at each of these boundaries. For this reason the electro motive force of a battery in action may be much less than when the poles are open. If two points, P and Q, in a network of conductors which carry a steady current, be connected by an additional wire conductor, K, containing a battery of such electromotive force, e, and so directed as to prevent any current from passing through K, e measures the difference of potential between P and Q. It is easy to show that when the poles of a battery are closed by a conductor of resistance E, the difference between the values of the potential function at the ends of this conductor is RE / (B + R), where E is the electromotive force of the battery under the given circumstances, and B the resistance of the conductors which make up the battery itself. The steady current which flows through the circuit ELECTROKINEMATICS. 233 carries E / (B -f- 7?) units of positive electricity across every cross-section per unit of time. With a given battery the intensity of the current can be changed very much, of course, by increasing or decreasing the resistance of that part of the circuit which lies outside the battery. In the centimetre-gramme-second system of electrostatic [E.S.] absolute units, the unit of electric quantity is that quantity of electricity which, if it could be concentrated at a point in air, would repel a like quantity concentrated at a point 1 centimetre from the first with a force of 1 dyne. This unit is found inconveniently small, however, when one has to deal with such steady currents as are usually met with in practice, and the coulomb, which is equal to about 3 x 10 of these absolute units, is the practical unit of quantity most frequently used. The absolute E.S. unit of current carries the absolute unit of electricity past any point in its course each second. A current of a coulomb per second (equivalent to 3 X 10 9 of these absolute current units) is called an ampere. The absolute E.S. unit of resistance is 9 X 10 11 times as large as the practical unit called the ohm. The latter is the resistance of a column of pure mercury 1 square millimetre in section and 106.3 centimetres long, at C. The resist ance at C. of a wire of pure copper 1 millimetre in diameter and 1 metre long is about 0.01642 ohm. The absolute E.S. unit of difference of potential is equivalent to 300 practical units. The practical unit, called the volt, is such that if the two ends of a wire of 1 ohm resistance were kept at 1 volt difference of poten tial, the steady current which traversed the wire would carry past any cross-section 1 coulomb of electricity per second. A condenser which requires 1 coulomb of electricity to charge it, so that the difference of potential between its poles is 1 volt, is said to have a capacity of 1 farad. A con venient unit of capacity is the microfarad or the millionth 234 ELECTROKINEMATICS. of a farad. It is equivalent to 900,000 absolute E.S. units of capacity. The capacity of a conducting sphere 9 kilo metres in radius would be 1 microfarad, that of the earth something over 700 microfarads. The capacity of a nautical mile of such ocean telegraph cable as is usually laid may be taken to be about 4, microfarad. 73. KirchhofJ s Laws. The Law of Divided Circuits. From what has been proved in the preceding sections about conduc tors which carry steady currents, follow two theorems of much practical importance, called KirchhofPs Laws. I. If several wires which form part of a network of conductors carrying a steady current meet at a point, the sum of the inten sities of all the currents which flow towards the point through these wires is equal to the sum of all those which recede from it ; or, in other words, the algebraic sum of all the currents which approach the point through the wires which meet there is zero. II. If, out of any network of wires which form a complex conductor and carry a steady current, a number of wires which form a closed figure be chosen, and if, starting at any point, we follow the figure around in either direction, calling all cur rents which move with us positive, and all discontinuities of the potential function which lift us from places of lower potential to places of higher potential positive, the algebraic sum of the products formed by multiplying the resistance of each conductor by the current running through it, is equal to the algebraic sum of the jumps in the potential function which we encounter in going completely around the figure. The first of these laws is an immediate consequence of the fact that there can be no growing accumulation of free elec tricity anywhere in a circuit which bears a steady current. To prove the second law, let a 1? a 2 , a s , a n be n linear con ductors, which, taken in order, form a closed figure, itself a part of a complex conductor which carries a steady current. In passing from a^ to a n through all the other conductors, let ELECTROKINEMATICS. 235 Vj and V" be the values of the potential function at the beginning and end of the jih conductor, and let r,. and c j be respectively the resistance of this conductor and the value of the current running through it. Then, from the definition of the term " resistance," we have the following equations : or, adding them all together, -T c H r n -rj-rf.+vj-vj + which is the statement of this law. If electricity is free to pass from a point P to another point P by two wires of resistance t\ and n, respectively, and if a steady current be flowing from P to P , the current will be divided between the two wires in the inverse ratio of their resistances or in the direct ratio FIG. 58. of their conductivities. For, if V and V be the values of the potential function at P we have V V = c^ and V V = c. 2 r. 2 , whence ^ : c n and P , = r, : /v Moreover, or Cl + < v- r - ( V - V) - 1 - + - ) ; >-i rj The expression in the second number of the last equation is, by the definition of the term, the resistance of the com pound conductor formed of the two which join P and P . It is evident that the conductivity of this conductor is the 236 BLECTROKINEMATICS. sum of the conductivities of the two wires of which it is composed. If n conductors be joined up in parallel to form a compound conductor, the conductivity of the latter is the sum of the conductivities of the constituents, and its resistance is the reciprocal of the sum of the reciprocals of their resistance. If four conductors the resistances of which are p, q, r, and s form a quadrilateral (Fig. 58) one pair of vertices of which are connected by a wire of resistance g and the other pair by a conductor of resistance b containing a battery of electromotive force E, we have an arrangement of much practical importance, which is often called Wheatstone s Net. If we denote the strength of the current through the cell, in the direction indi cated by the arrow in the figure, by (7, and the currents in the other conductors by C p , C Q , C r , C s , and C g respectively, Kirchhoff s Laws yield the equations c = c p + c q = c r +c,, c p = c g + c r , C q =C s -C g , p.C p -q.C q + ff.C g = 0, If we substitute the values of C, C p , C q obtained from the first three equations in the last three, we shall get a system of three linear equations involving the three unknown quantities C g , C r , C a , which can be easily solved. These equations are and if we denote the determinant of the coefficients, - \ pr(q + *) + qs (p + r) + b (p + q) (r + s) + g[b(p + q + r + s) + (q + s)(p + or -gr(b + q + 8) + qb(g + r + s) ps 2 + qrs ELECTROKIXEMATICS. 237 it is easy to see that C g = E(qr-ps)/\ C r = E(gq-\- $2) + sg 4- sq) / A, C q = E (g r + gp + rp + ps) / A . C7 = E (gq -\- sp + sg -f sq + pg + rp + rg + rq) / A. The resistance (It) of the net pqrsg, computed from the equa tion CE/(b + R), is [g (g + s) (p + r) + j?r (y + s) + qs (p + r)] [<7 O + ? + > + s} + (p + q) (r + *)] If no current passes through the resistance g, we have qr =ps, C p = C r , C q = C a and, as we may see by multiplying out and cancelling, and C, / C = (p + r) / (p + q + r + 5). It is evident, from an inspection of the Kirchhoff equations belonging to the three cases, that if the resistances of the linear conductors which go to make up a given network are fixed, and if C lt C 2 , C 3) are the currents in the different members when these members contain the electromotive forces EU E 2 , E s , - and C/, C 2 f , C B , , the corresponding currents when the electromotive forces are E, E 2 , E^ -, C l -\- C/, C 2 + CJ, C 3 -\- C s , - would be the currents if the electromo tive forces were E l -j-^ , E 2 + E 2 , E z + E z , . . .. Let P and Q, any two points in a network of linear con ductors some or all of which contain electromotive forces, be at potentials V P , V Q respectively, and let the resistance of the whole network when the current enters at one of these points and goes out at the other be r , then if P and Q be connected by an additional wire W of resistance r, the cur rent in this wire will be ( V P V Q ) / (r + r) in the direction from P to Q. For if (1) W contained an electromotive force ( V P V Q ) directed from Q to P, the rest of the network 238 ELECTROKiNEMATICS. being unchanged, no current would pass through W, and the other currents would not be altered by the introduction of W ; and if (2) W contained the electromotive force ( V P V Q ) directed from P to Q, and if all the other electromotive forces in the original network were annihilated, leaving the resist ances unchanged, a current ( V r V Q } / (r + ? ) would flow through W from P to Q : the given arrangement can be regarded as formed by superposing case (1) upon case (2). 74. The Heat developed in a Circuit which carries a Steady Current. Given, in a region not exposed to magnetic changes, a chain of n conductors, each in itself homogeneous, and at a uniform temperature throughout ; let a portion A of the surface of the first be kept, by means of some external agency, at potential V A , and a portion B of the surface of the last at a lower potential V R , while the rest of the outer surface of the chain abuts upon non-conducting media. S kt k+l , the surface of separation between the &th and the (k + l)th conductors, may or may not be equipotential, but if these conductors are of different materials, we must expect to find at all points of this surface a uniform discontinuity, E k >A + i> of potential. In following down from A to B an infinitesimal tube of flow which carries the steady current A (7, we start at potential V A , leave the first conductor at potential F/ , enter the second conductor at potential F 2 , leave it at F" 2 ", enter the third conductor at F 3 , and so on. Every second in the kfh conductor, A C absolute units of electricity are lowered from potential V k to potential F A; " and &C(V k V k ") units of work (representing loss of electrostatic energy) are done by the electrostatic field upon the electricity which moves with the current : this energy appears as heat in this conductor. The work thus done in the whole chain is A<7(F. t - F/ + F; - TV + JY ~ V" + + F. 1 - V B ), or, since P* + i ~~ ^V = E kt * + i> \C(V A - r B -Mi, 8 + E^ + + ^ H -i, w ) = A6 (F 4 - V K 4- E). ELECTROKINEMATICS. 239 This energy all appears as heat in the conductors which form the chain. At the surface S ttt+l , AC units of electricity are raised every second from potential V k n to potential V k+1 . The work thus done every second is ^C-E ttk + lf and, by virtue of similar processes at all the surfaces of discontinuity, the electrostatic energy is increased in this way every second by \C E. The net loss in electrostatic energy in the chain per second is, therefore, which is otherwise evident. Taking into account all the cur rent filaments which go to form the steady current C, we see that an amount of energy equivalent to C ( V A V B -f- E) appears as heat in the conductors which form the chain, and that an amount of electrostatic energy equal to EC is fur nished to the chain. If the chain is closed and if, going around it in. the direction of the steady current C, we denote by E the algebraic sum of the discontinuities of potential, counting a step up as positive, we shall find that the energy EC appears as heat in the conductors and that since the circuit is at the same temperature throughout, this is fur nished by chemical action in the chain. If r is the total resistance of the chain, C = E/ rand EC = C 2 r. This result represents ergs or joules, according as E, C, and r are meas ured in absolute electrostatic units or in volts, amperes, and ohms : a joule is equivalent to 10 7 ergs. If the chain contains a battery of electromotive force E in the direction of the steady current C, and if there are in the chain outside the battery discontinuities of potential which. reckoned against the current, amount algebraically to J5", E=E,-E , C = (E, - E ) / r, and the energy used in heating the chain is (E E ) C = C*r : when we wish to regard the battery as the source of this energy, it is convenient to write the last equation in the 240 ELECTROKINEMATICS. form EfiC = C 2 r + JK C, and to say that of the whole energy, E Q C, furnished by the battery, (7V, which appears as heat in the conductors which form the circuit, is used in maintaining the current, and IS C, in overcoming the counter-electromotive force E . If a cell of electromotive force E Q be joined up with a number of metallic conductors all at the same temperature to form a simple circuit of total resistance r, the current will be C Q = E Q /r, and the whole energy, E Q CQC^r t furnished each second by the battery, will appear as heat in the circuit. If, however, while the total resistance of the circuit remains unchanged, the battery be called on to do each second an amount W of outside work of any kind (such, for instance, as that involved in decomposing an electrolyte in the external cir cuit), the steady current will have a value C smaller than C OJ the whole energy E Q C furnished each second by the cell will be a fraction of E C , and the portion of it (7 2 r, which appears as heat in the circuit, a smaller fraction of C ( fr. The differ ence between E Q C and C 2 r will be equal to W, and this equation determines C. If a given steady current C is to be conveyed partly by a conductor of resistance r l and partly by a parallel conductor of resistance r 2 , and if the portions carried by these conductors are d and C 2 respectively, the amount of heat developed per second in the conductors will be u = C^i\ + C/r 2 . If C lt and consequently C 2t be changed so as to keep their sum equal to the constant C, u will, in general, change, and we shall have D c u = 2 Cj\ + 2 C 2 r, D C C, = 2 ( C.r, - <7 2 r a ) : u, which is sometimes called the dissipation function, will, therefore, be a minimum if the current is divided between r x and r 2 as it would be if the conductors were connected at the ends. It is easy to prove that if a given steady current be led into a given network of metallic conductors, at a uniform temperature, from without, the distribution of this current in the network will be such as to make the dissipation function as small as possible. If, for instance, a steady current C be ELECTROKINEMATICS. 241 led into the network represented by ABDF in Fig. 58 at the point A and out again at B, we have c r = c-c,, c p = c, + c-c,, c, = c.-c, and u is equal to c- cy + q(C 9 - c g y + r(c- cy + s-c* + ff .c g *. If we equate to zero the partial derivatives of u with respect to C s and C g , we shall get two necessary conditions for a minimum : the equations thus obtained are (P + 9 + ?) O g - (p + q) C s = -pC, -(p + *)e,+(jP+t + r + *)C.*=(j + r) C, whence C g /C = (qr ps) / (gq -f sp + sg + sq + pg + rp + rg -f rq), C /C=(gp + rp + rg + rq) / (gq + sp +sg + sq +pg + rp + rg + rq), etc., which are equivalent to equations already found. If the conductors r^ r 2 , ?* 3 , r n which form any network, complete or not, and carry currents C v C 2 , C SJ C n , contain electromotive forces JE U E 2 , E& - E n which have the direc tions assumed for the currents, the currents are such as to make, not the dissipation function, but W =u-( C,E, -f C 2 E, + C 3 E 3 C H E n ) a minimum. In the case of the complete Wheatstone s Net, and the equations formed by equating to zero the partial derivatives of W with respect to C g , C s , and C r yield the values for the currents given in the last section. 75. Properties of the Potential Function inside Conductors which carry Steady Currents. If at any time t, positive elec tricity is passing through a linear conductor in one direction at the rate P, and negative electricity in the other direction 242 ELECT HOKINEMATICS. at the rate N, the current strength is P -f- N in the first direc tion. Since there is no free electricity inside a homogeneous conductor which carries what we have called a steady current, it is customary to assume, when one uses the language of the " Two Fluid Theory," that such a current consists of a flow of positive electricity in one direction at every point, and an equal flow of negative electricity in the opposite direction. We shall avoid much circumlocution, however, and we shall introduce no error into our numerical computations if we speak as if the whole current were due to the motion of posi tive electricity. If the value of the potential function within a conductor which bears a steady current is given, all the cir cumstances of the flow in the conductor are fixed. Positive electricity flows into the conductor from without through all parts of the surface where the derivative of the potential function, taken in the direction of the exterior normal, is posi tive, and out of it through all parts of the surface where this derivative is negative. At all points where the conductor abuts on an insulating medium, the derivative is zero : it may be zero at other points also. There can be no closed equi- potential surface lying wholly inside a conductor which carries a steady current, unless there is some constant source of posi tive or of negative electricity within this surface, for the whole flow of electricity algebraically considered, per unit of time, through such a surface from within outwards, is equal to k times the surface integral of the intensity of the com ponent of force in the direction of the exterior normal, and this is not zero. There must then be such a constant source of free electricity within the surface as shall furnish just as much per unit of time as the current carries away. Although it is not very easy to prove analytically that given a homogeneous conductor and certain portions A, B of its surface which are to be kept at potentials V A , V B , while at all other portions the value of the derivative of the potential function taken in the direction of the exterior normal is to be zero there exists a function which (1) satisfies these ELECTKOKINEMATICS. 243 surface conditions, and which (2) inside the conductor satis fies Laplace s Equation, and with its first space derivatives is continuous and single-valued, it is nevertheless clear from physical considerations that one such function exists, namely, the potential function inside the conductor when A, B are kept at the given potentials and the rest of the surface is exposed to an insulating medium. For practical purposes we need to prove that this is the only function which satisfies the given conditions. Suppose for the sake of argument that two such functions, V and W, exist, and call their difference u. The function u, then, satisfies condition (2) and is itself equal to zero, or else has its derivative in the direction of the exterior normal equal to zero at every point of the surface. Applying Green s Theorem in the form of Equation 151 to u, we find that the quantity (D x uY + (D y u)* -f (D s u)*, which can never be negative, must be zero at every point within the conductor, so that D x u, D y u, and D z u must vanish and u be a constant throughout the space within the surface. Xow at portions of the surface itself, u is zero, hence it must be equal to zero everywhere inside the conductor, and V W. If by any means, then, we find a function which satisfies the surface conditions and the general space conditions characteristic of the potential function inside a certain conductor carrying a steady current under given surface conditions, this function is itself the potential function. Any surface supposed drawn in a conductor which carries a steady current in such a way that the derivative of the potential function taken normal to this surface is zero shall be called a surface of flow. If a conductor which under given surface conditions carries a steady current be cut in two by means of a surface of flow, and if the two parts be separated while the surface conditions on what was the bounding surface of the old conductor remain the same as before, and the fresh surfaces now abut on an insulating medium, the state of flow at every point inside each part of the conductor will be just the same as before, for the 244 ELECTROKINEMATICS. values of V and D n V on the surface of the new conductors are what they were before separation, and V must have its old values at all inside points. When a conductor is cut in two by a surface of flow the fresh surfaces exposed receive a statical charge of free elec tricity, and the charges on what was the bounding surface of the original conductor are in part changed so that it is only within the parts of the old conductor that the effect of the separation is nil after the currents have become again steady. If two mutually exclusive closed surfaces S l and S 2 , kept, respectively, at uniform potentials V l and F" 2 , are the elec trodes of an infinite homogeneous conductor 7f, of specific conductivity k, which fills all space outside these surfaces and is at potential zero at infinity ; if, moreover, the steady flow outward through ^ or inward through S 2 is equal to C, the current vector in K is everywhere equal to what the elec trostatic force would be if K were air and if S l and S. 2 had charges C/kirk and C/ktrk so distributed as to bring them to potentials V l and F 2 respectively. In most of the preceding discussion we have tacitly assumed the separate conductors considered to be homogeneous, and we shall continue to do so in the following sections unless the contrary is stated. We have to consider briefly, however, in the remainder of this section isotropic conductors which have in different parts different specific resistances. If the specific conductivity k of an isotropic conductor which carries a steady current can be represented by a posi tive scalar point function, and if the components, parallel to the coordinate axes, of the vector q which represents the current strength, are u, v, and w, we may state the fact that there is no growing accumulation of free electricity in any portion of the conductor bounded by the surface S by the equation ( \ ELECTROK1NEMATICS. 245 I I q cos (q, n) dS = I ( q [cos (x, n) cos (x, q) + cos (y, n) - cos (y, q) 4- cos (2, w) cos (s, q)~\dS [u cos(#, n)+ v eos(y, ?z) + w cos (2, 7i)]rf = C C C[D x u + ZV + D z w\dxdydz = 0. Here the double integrals are to be extended over the whole of S, and the triple integrals over all the space included by S. Since S is arbitrary, the integrand of the triple integrals must be equal to zero at every point within the conductor, so that D t u + D y v + D z w = [198] and q is a solenoidal vector. At every point within the conductor, u = - x , v = - y so that 2 (k.D z V)=Q, [199] or k.\ 2 V + (D 3 k.D x V + D y k.D y V+D z k-D z V)=Q. [200] If k is constant, V satisfies Laplace s Equation, and in this spe cial case, as we already know, none of the free electricity which gives rise to the potential function V is within the conductor. Given an analytic, scalar, positive point function k and a closed analytic surface S, it is easy to prove by the help of [149] that there cannot be two different functions, Fi and F" 2 , which (1) with their first derivatives are continuous within S and at every point in this region satisfy the equation (2) on the given portions Si and S 2 of S have at each point equal values, and (3) on the rest of S have at every point equal normal derivatives. The differential equations of the current lines are dx _ dy _ dz u v w 246 ELECTKOKINEMATICS. At a surface of separation between two conductors which carry a steady current the normal components of the current and the tangential components of the electrostatic force are continuous. If X and 2 are the angles which the resultant electrostatic forces f\ and F 2 make with the normal on the two sides of such a surface at any point, ktFi cos Oi = k 2 F 2 cos 2 and F : sin 6 l = F 2 sin 2 , whence, by dividing the members of the first of these equations by the corresponding members of the second, - - = - -, K\ K 2 an equation which shows how the current lines are refracted at the surface. At a surface of separation between copper and manganin where the ratio of the conductivities is about 30, 0! = 27 42 when 2 = 1, and L = 69 09 when 6 2 = 5. If % and n 2 represent normals drawn from any point of the surface of separation between two conductors which are carry ing a steady current into the first and the second conductor respectively, k l D ni Y l + k 2 D n2 V=0. [201] 76. Method of finding Cases of Electrokinematic Equilib rium, If w is a single-valued, generally continuous solution of Laplace s Equation, Aw -\- B, where A and B are constants, is another such function which has the same level surfaces as w. If an area be chosen on one of these surfaces, it is pos sible to draw through every point of its perimeter a line, defined by the equations dx / D x w = dy / D y w = dz/D z w, which shall cut orthogonally all the level surfaces of w which it meets. All these lines form a tubular surface such that the normal derivative of w at every point of it is zero. If I 7 is a portion of space bounded by such a tube and by portions, S f , S", of two of w s level surfaces on which it has the values w 1 and w" respectively, w is identical with the potential function that would govern the flow within any homogeneous conductor of the form T if the surface S and S" were kept at potential w ELECTROKINEMATICS. 247 and w", while the rest of the boundary was a surface of flow. Moreover, Aw + B, where A and B can be chosen at pleasure, must be the potential function within a homogeneous conduc tor of the form T, if the surface S and S" were kept at poten tials Aiv -f B, Aw" + B respectively, the rest of the boundary being a surface of flow. By using different pairs of level sur faces of w and tubes of different forms, it is possible with the help of this one function to study the laws of steady flow inside conductors of many different shapes and to obtain results some of which may happen to be practically inter- esting. For instance, w = - -f- d, where c and cl are constants and r the distance from a fixed origin to the point (a*, y, ), gives the value of the potential function inside a conductor bounded by two spherical surfaces of radii a and b having as their common centre when these surfaces are kept respec- C- G tively at potentials - + d and - + d. In this case the whole amount, per unit of time, of positive electricity which enters the conductor through the surface r a crosses every equi- potential spherical surface within the conductor and leaves it by the surface r = b is 4 trek, where k is the specific conduc tivity of the material out of which the conductor is made. The resistance of the conductor is, by definition, c _ c a b b a 4 -n-ck 4 -n-kab a quantity independent of c and d. It is evident that any conical surface the vertex of which is will be in this case a surface of flow, and that the function w = - + d governs the flow in any piece cut out of the spheri cal shell just considered by such a surface. It is easy to see that if w is the solid angle of the cone, the resistance of the portion of the conductor cut out will be - kuab 248 ELECTKOKINEMATICS. (1 1 = c[ Vl ?* 2 Again, the equation V= c( ) + d, where r and r 2 are VI r 2/ the distances of the point (x, y, z) from the fixed points O x and 2 , gives us the potential function inside an infinite conductor bounded in part by the surfaces = a and T \ r 2 = b, when the first is kept at potential ac + d, the r \ r 2 second at potential be + d. In this case the surface V = d is a plane bisecting at right angles the straight line 0^0^ Larger and smaller values of V than this give closed surfaces, each of which surrounds one of the points and leaves the other outside. For very large values of V, if c is positive, the equipotential surfaces are very small, nearly spherical surfaces surrounding 0^ To find the amount of positive electricity which enters the conductor under consideration, per unit of time, through the surface V = ac + d, where ac shall be positive, we must integrate over this surface kD n V or kc\ D n ( \ D n . L \ r i/ r 2j According to Green s Theorem, the resulting integral is exactly the same as that taken over any other closed surface, large or small, which surrounds : and leaves 2 outside. Let us consider, then, a spherical surface of radius e < L 0., whose centre is at Oj. The required integral in this case is 4 -rr^k times the average value of D r V taken over the spherical surface ; or, since ^ for all points on this surface is equal to e, 4 7T 2 kc average value of D r ( ) .a) If, now, e be made smaller and smaller, D r ( ] always has 1 V2/ some finite value for every point on the surface of the sphere surrounding 0,, and the expression just given approaches the limiting form 4 -n-kc. Hence, 4 -n-kc units of positive electricity ELECTROKINEMATICS. 249 enter the given conductor through the surface V = ac + d in every second, whether this surface is large or small. The resistance of the conductor between the surfaces V = ac -f d and V = be + d is, by definition of the term, 4 7T/C If a and b are made very large and equal, with opposite signs, the two surfaces through which electricity enters and leaves the conductor become very nearly coincident with spherical surfaces of radius e = drawn about and 2 respectively. The resistance of the conductor in this case is - Considerations of symmetry show that any plane which ^ TTK contains the line L 2 is a surface of flow. If we cut the conductor in two by such a plane, we shall have an infinite conductor with two nearly hemispherical electrodes sunk in its plane surface. The resistance of this part of the whole conductor is , a quantity independent of the distance apart 7TrC of the electrodes. This is nearly the case of two poles of a battery sunk in the earth. Again, the expression where r l and r 2 are the distances of a point P in space from any two parallel straight lines, A and B, is a solution of Laplace s Equation which, with its derivatives, vanishes at an infinite distance from these lines and which is constant all over any one of a double system of circular cylindrical surfaces (Fig. 59), some of which surround one of the given lines and some the other. This function, then, when c and d are properly determined, is the potential function within an infinite lamina, either thick or thin, when that lamina is per forated perpendicularly to its plane by two circular cylindri cal holes, the curved surfaces of which are kept at given 250 ELECTROKINEMATICS. constant potentials. 2 irkc units of positive electricity per unit of time per unit of thickness of the lamina enter the con ductor through one of the cylindrical surfaces, and the same amount leaves it by the other surface. The resistance of the lamina is then the difference between the values of the poten tial function at the electrodes divided by 2 irkc times the thickness of the lamina. These examples will serve to show how we may discover an indefinite number of cases of kinematic equilibrium by assum ing some function, in general finite and continuous, which FIG. 59. satisfies Laplace s Equation, and then taking as a conductor one inside which the given function is everywhere finite, and which is bounded by surfaces over each of which either the function is constant or its normal derivative zero. If we transform Equation 199 to orthogonal curvilinear coordinates defined by the scalar point functions u, v, w, where w satisfies Laplace s Equation, and assume V to be expressible as a function of w only, we shall obtain (see page 182) the equation DJV+ D W V D w k / k = 0. If the specific conductivity of a body occupying the space T men tioned at the beginning of the section were not constant but a given function of w, this equation would determine V. ELECTROMAGNETISM. 251 HI. ELECTROMAGXETISM. 77. Electromagnetism. Straight Currents. If a steady electric current be sent through a long straight wire, the space in the neighborhood of the current becomes a field of magnetic force. If the medium about the conductor is homogeneous, the direction of the field is such that a small magnetic needle freely suspended by its centre tends to set itself perpendicular to the wire and to the perpendicular dropped from the point of suspension upon the wire, so that " if a person be imagined as swimming in the current which flows from his feet to his head, and if he face the needle, the north pole will be turned towards his left hand." The field is symmetrical about the wire and, according to the rule just given, its direction at any point is normal to the plane drawn through the point and the wire, so that the lines of force are circumferences forming right-handed whirls about the current. To investigate the law of the change of the intensity of the force with the dis tance from the wire, we may imagine a rigid frame free to turn about the vertical wire as a hinge, and suppose a magnet to be rigidly attached to this frame. It will be found that in this case the frame will have no tendency to rotate under the action of the electromagnetic forces, so that the sum of the moments about the wire, of the forces which the field exerts upon the magnet, must be zero. If i\ and r z are the distances of the poles from the wire, and if F(r) is the inten sity of the field at a distance r from the wire, the equality of moments shows that, however the magnet be placed on the frame, or, in general, r F(r) = a constant, k. The value of k is found to be dependent upon the strength, C, of the current in the wire, and can be used to define this strength. We may write, therefore, F(r) = A- C/r, where A is a constant depend ing upon the units in which C is measured. If we use the 252 ELECTROMAGNETISM. absolute electromagnetic c.g.s. units, defined below, in deter mining C, it will presently appear that A is 2. If we take the plane of the paper for the xy plane, and imagine the wire which carries the current to cut the paper normally at the origin, then, if the current comes from below, the components of the field at the point (x, y) are X = - 2 C sin (x, r) /r and Y = 2 C cos (x, r) /r, or X = - 2(77// 2 -fy 2 ) and Y = 2Cx/(x 2 + y 2 ). Here D V X = D X Y and the magnetic force is, in general, a lamellar vector, so that it has a potential function which, since the lines of force are closed, must be multiple-valued. This potential function is evidently 26 tan -1 (y/x) + constant, or 2 CO + constant, and it satisfies Laplace s Equation. The plus or the minus sign is to be chosen according as we wish to use the derivative of the potential function taken in any direction, or its nega tive, as a measure of the component of the field in that direc tion. The line integral of the tangential component of the force taken around any curve in the xy plane which sur rounds the origin is 4?r(7, so that we infer from Stokes s The orem that at the origin the magnetic force is not lamellar. If a magnetic pole of strength m be moved around any closed path, the work done on it by the magnetic field will be 4 irm C if the path link right-handedly once with the wire, or zero if the path do not link with the circuit. These results are found to be independent of the inductivity of the homogeneous medium about the wire. Since D X X + D y Y = 0, the force in the medium about the wire is solenoidal, and the whole flux of force from within outward through any closed surface is zero. If two straight lines parallel to the wire are distant a and b centimetres from it respectively, ELECTROMAGNETISM. 253 the flux of force (Fig. 60) through the unit length of any cylin drical surface bounded by the lines is 2 C -log (b/a). Since we have assumed that a finite quantity of electricity is carried by a conductor of zero cross-section, it is not surprising that this useful analytic result becomes infinite if either a or b is zero. If two infinitely long straight wires parallel to the z axis carry equal steady currents of strength C in opposite directions, FIG. 61. and if they cut the xy plane at the points A i} A 2 , which have the coordinates (a, 0), ( a, 0) respectively, the scalar potential function, O, of the field has at the point (x, y, z) the value 2 C.tan-iO/t* - a)] - 2 C tan- 1 [y / (x + )], or 2 C tan- l [2 ay / (x* -f if - a 2 )]. The conjugate function, <l>, is + 2 C-logfa/rs), where i\ and >- 2 are the distances of the point (a-, y, z) from A^ and A z 254 ELECTROMAGNET1SM. respectively. The lines of force and the traces in the xy plane of the equipotential surfaces are shown in Fig. 61. D X O = D y $, D y l = D x $, and the derivative of O at any point in the xy plane taken in any direction in the plane is equal to the derivative of 3> at the same point taken in a direc tion in the plane at right angles to the first. If, then, a curve c is the trace in the xy plane of a cylindrical surface S, the generating lines of which are parallel to the z axis, and if n represents a direction in the plane perpendicular to c, the line integral of D n Q, taken along c represents the flux of mag netic force across S per unit of its height, perpendicular to the xy plane. This integral is equal to the line integral of the tangential derivative of < along c or to the difference between the values of <I> at the ends of the curve. If this difference is nothing, the corresponding flux is nothing ; if $ is constant all along c, this curve is a line of force. From the results just obtained, it is evident that if two straight lines parallel to the z axis cut the xy plane in the points J5 1? B z respectively, the flux of magnetic force through a cylindrical surface bounded by these lines, per unit of its length, parallel to the z axis, is 2 C log [(^A - A Z B Z ) I (A,B, - AiBj)-]. This represents the flux of force per unit of its height, through a circuit s z , consisting essentially of two infinitely long straight wires, parallel to the -z axis, cutting the xy plane at B 1} B z when the steady current C traverses the circuit s 1? consisting essentially of the two wires already mentioned, which cut the xy plane at AI and A 2 . Symmetry shows that this expression would also give the flux through s 1; due to a steady current C in s z . If an infinitely long cylindrical conductor, the generating lines of which are parallel to the z axis, and which is sur rounded by a homogeneous medium, carry a steady current in the direction of its length, and if the current density at the ELECTROMAGNETISM. 255 point (x , y , z ) be </ , a function of .* and y but not of z , the intensity of the magnetic field // within or without the con ductor can be obtained by imagining the conductor made up of separate current filaments, each of which has a field like that about a fine straight wire, unaltered by the presence of the others. If L, M, N are the intensities of the components of If parallel to the coordinate axes, L and M are functions of x and y while N is zero. _ r rZg ds-^dx dy r r 2 g (x - x ) dx dy JJ (x-x T+(y-!/y 2 JJ (x-x<y+(y-y<y where the double integrals extend over the section of the con ductor made by the xy plane. If the whole amount of cur rent in the conductor is C, and if u represents the distance of the point (x, y, z) from the axis of , and <f> the angle tan" 1 (*//#), uL and i(M approach the limits 2 C sin <f> and 2 C cos < when u increases without limit. The line integral of the tangential component of the field, taken around any curve, which surrounds the conductor, is equal to the corre sponding integral taken around a circle in the xy plane of infinite radius, with centre at the origin. The value of this last integral is obviously kirC. Except for points in the mass of the conductor, the integrands of the expressions for L and J/ are continuous functions of x and y for all values of x and y within the limits of integration, and D y L = D X M and D*L + D y M = 0. At all points in empty space near the conductor, therefore, the field is solenoidal and lamellar and there is a potential function which satisfies Laplace s Equation. In the special case where the conductor is in the form of a right circular cylinder (or of concentric shells bounded by cylindrical surfaces of revolution), and where the current den sity is a function only of the distance from the z axis, which 256 ELECTKOMAGNETISM. coincides with the axis of the conductor, the field is evidently symmetrical, and the direction of the force at any point is perpendicular to the perpendicular to the axis drawn through the point. Everywhere in empty space in the vicinity of the conductor a potential function, O, exists, and, since D r l = 0, Laplace s Equation degenerates into Z>/O = 0, or O = aO + I. The work done by the field when a magnetic pole of strength m moves around a circumference, the axis of which is the z axis, is evidently equal to .kirCm, where C is the sum of the currents in all the current filaments which the path encloses. Since the line integral of Z> S ,O taken around any such path in empty space in right-handed direction around the current is 2 TT&, a is equal in absolute value to twice the whole current carried by so much of the conductor as lies within the path. If the direction of the z axis is such that, if the eye is in the positive x axis looking at the origin, a counter-clockwise rotation of the positive axis of y through 90 would make it coincide with the positive z axis, and if O = 2 CO -h b, the force at any point not in the mass of the conductor, in any direction, is the derivative of O at that point taken in the direction in question, and the resultant force is Dfl I r or 2 C / r. This is the same as if all the current nearer the z axis than the point in question were flowing through a fine wire coincident with the axis of z. If the infinitely long cylindrical conductor is a uniform tube, the axis of which is the z axis, O = aO -f- b in the empty space within the tube, and, since (on account of symmetry) the resultant force a/r must vanish on the z axis, a is zero and the intensity of the field within the tube is everywhere zero. We may easily find the intensity of the electromagnetic force at any point P within an infinitely long, round con ductor carrying, in the direction of its length, a steady cur rent with intensity the same at all points equally distant from the axis of the conductor, if we imagine a cylindrical surface, S, of revolution coaxial with the conductor drawn through P. The magnetic force at P, due to so much of the ELECTROMAGNETISM. 257 current as lies outside *S , is nothing; the force due to so much of the current as lies within S is evidently the same as if this portion of the current were concentrated in the axis. Tf, there fore, a straight conductor in the form of an infinitely long cylinder of revolution of radius a carries a steady current C in the direction of its length, and if the intensity (q) of the cur rent is a function only of the distance (r) from the axis of the conductor, the intensity of the magnetic force (Jf) is 2C/r without the cylinder and - - I xqdx within. The flux of V */0 induction per unit length of the cylinder across so much of any plane through the axis as lies within the conductor is Xa c j r f*r p I xq dx. ) /o If q does not involve r, the current is uniformly distributed through the conductor, the strength of the field within the cylinder is 2 CV/a 2 , and Q is equal to pC. If the axis of the cylinder is the z axis, the force components at any inside point distant r from the axis are L = 2 Cy / a?, M = 2 Cx/a 2 , so that H is solenoidal, as it would be if q were any analytic function of r. Since H is not lamellar within the conductor, it is at the outset clear that there can be no scalar poten tial function O there ; it is well to notice, however, that, if the derivative of a scalar function, Q, at any point in any direction were required to show the force at that point in the given direction, it would need to satisfy, within the conductor, the two incompatible conditions, D,ti = 0, (D 9 Q) /r=-2 Cr/a*. Since H is solenoidal even at inside points, we may ask whether its components are not the components of some vector, Q, which may be regarded as a vector potential function of //, and it is clear that a vector of intensity u-gr 2 , directed at every point parallel to the z axis, satisfies all the conditions, as do many other vectors. The component, at any point within the conductor, in any direction, of the curl of the 258 ELECTROMAGNETISM. vector (0, 0, Tr^r 2 ) shows the component of the magnetic force // at the point in the given direction. The abscissas of Fig. 62 represent distances from the axis of the conductor, FIG. 62. and the ordinates the corresponding values of the resultant magnetic force in the case just considered. If a uniformly distributed current C be brought up normally through the plane of the paper by an infinitely long cylinder of revolution and down through a similar cylinder parallel to the first, the lines of force without the cylinders are of the same shape as those shown in Fig. 61. The curve in Fig. 63 shows the intensity of the field at points in a straight line which cuts the axes of the cylinders perpendicularly. If two infinitely long, coaxial, cylindrical surfaces of revo lution carry symmetrically equal and opposite currents, each FIG. 63. of strength C, parallel to their common axis, the space between the surfaces is a field of electromagnetic force of strength 2 C /r, where r is the distance from the axis. There is no force within the inner surface or without the outer one. ELECTROMAGNETISM. 259 In the case of a long, straight wire of radius a surrounded by a coaxial tube of radii b and c, and carrying uniformly distributed a steady current C which returns through the tube, the electromagnetic force is evidently zero on the axis of the wire and continuous at every distance r from the axis. If Wi and w 2 are the intensities of the current in the wire and in the tube respectively, C = w^a 2 = u\ir (c 2 # 2 ), and if we apply the formulas just proved, we shall learn that the strengths of the fields w r ithin the wire, between the wire and the tube, in the body of the tube and without the tube, are given by the expressions 2 Trw-p, 2 Tra?u\/r, 2 TTW Z (c 2 r 2 )/r, and 0. It is to be noted that the strength of the magnetic field due to a given electric current is, in the homogeneous medium which surrounds the current, wholly independent of the per meability of this medium, whereas the field due to a given magnet would be inversely proportional to the inductivity. If the fields of a given circuit and a given magnet were the same in one homogeneous medium, they would not be the same in another homogeneous medium of different magnetic inductivity. The induction due to a current circuit in a homo geneous medium filling all space is proportional to the induc tivity, as is the energy in the medium. The induction due to magnetic matter surrounded by a homogeneous medium is independent of the inductivity of the medium. The action of a distribution of magnetic matter in an infinite homogene ous medium on a circuit carrying a steady current is not altered by changing the inductivity of the medium. 78. Closed Circuits. Experiment shows that if a steady current of strength C runs in a simple linear circuit of any form, there is a magnetic field in the neighborhood of the conductor and the lines of the field are all linked right- handedly with the circuit. If a unit magnetic pole be carried round any closed path which does not link with the circuit, 260 ELECTROMAGNETISM. FIG. 64. the work done by the field on the pole is zero, whatever the character of the medium near the circuit, so that a potential function exists in the so-called empty space about the wire. This potential must be multiple-valued, since the lines of force are closed. If the pole be carried round a closed path which links once with the circuit, the work done on the pole by the field is 4 TT (7, whether the medium intersected by the path is homogeneous or not. We infer, therefore, that no scalar potential function exists in the wire which carries the current. It follows from the experiments of Ampere that the field of magnetic force, due to a steady current of C electromagnetic units flowing in a closed linear circuit in a homogeneous medium, is iden tical with the field of magnetic Induction due to a simple magnetic shell (Fig. 64) of strength C bounded by the circuit. This statement defines the electromagnetic unit of current. The magnetic force, due to a current of C electromagnetic units flowing in a closed linear circuit in a homogeneous medium of inductivity /x, is the same in magnitude and direc tion at any point P as the force due to a simple magnetic shell of strength C/A bounded by the circuit.. The shell may be of any form, provided that it does not pass through P and that its positive side is such that the current surrounds right-handedly the direc tion of polarization. To make the potential function single-valued, we may cover the circuit by a cap or dia phragm, fix at pleasure the value O of the potential function at some one point in the field, and define the value at any other point Q to be the line integral of the magnetic force taken from to Q along any path which does not cut the diaphragm. FIG. 65. ELECTROMAGNETISM . 261 At any point P on the axis of a circular current of radius a, at a distance x from the plane of the circuit, the circuit subtends the solid angle o> = 27r(l - cos ff) = 2 TT (1 - x / Va 2 + x 2 ). If the strength of the current in the circuit is C, the magnetic force at P is directed along the axis of the circuit (Fig. 65) and is numerically equal to the negative of the derivative with respect to x of Co>. The intensity of the force is, therefore, and at the centre of the circuit, where x = 0, it is 2-jrC fa. This result evidently agrees with the awkward statement sometimes used to define the electromagnetic unit of current. " If one centimetre of a linear circuit which carries the unit current be bent into an arc of one centimetre radius, the strength of the field at the centre of the arc, due to this portion of the circuit, will be one dyne." The ampere, which is the practical unit of current intensity, is one-tenth of the unit just defined. If for convenience we denote the quantity a/x by u and its recip rocal by v, the potential function (Ceo) just found may be written in either of the forms 27rC Jl I/ Vl + u*\ or 2 TrCjl v/Vl + v 2 J, and, according as x is greater or less than a, we may use one or other of the developments FIG. 66. 2 w2 ~2.4" 4 + 2.4.6" 6 "\ If, then, PU P 2 , P s , represent zonal harmonics expressed in terms of cos a, and if u l and i\ represent a/r and r/a, the 262 ELECTROMAGNETISM. value of the potential function at a point distant r from the centre of the circuit, in a direction (Fig. 66) making an angle a with the x axis, is given according as r is greater or less than a by one or other of the developments If an infinitely long straight wire which carries a steady current, (7, forms part of a plane closed circuit, all the other parts of which are at infinity, and if the plane of the circuit be used as the xz plane and the wire as the z axis, the solid angle subtended at the point (x, y, z) by the circuit is 2(7r 0), where tan y/x. The force components at the point are, then, the negatives of the derivatives with respect to x and y respectively of 2 C(ir 9 ), that is, 2 Cy/(x 2 + ?/ 2 ) and H- Cx / (x 2 + ?/ 2 ), as we already know. 79. The Law of Laplace, Mechanical Action on a Con ductor which carries a Current in a Magnetic Field. It will be evident from the discussion on page 218 that the strength of the magnetic field, H, due to a steady current of C electro magnetic units in a rigid linear circuit may also be computed, whatever the inductivity of the homogeneous surrounding medium, on the assumption that every element ds of the cir cuit (Fig. 67) makes a contribution numerically equal to C sin (r, ds) ds /r 2 , to the force at a point P, where r is the distance of ds from P. The direction of the contribution is normal to the plane of P and ds, and such that a north magnetic pole at P tends to whirl right-handedly about a straight line drawn through ds in the direction of the current. For a simple illustration of ELECTROMAGNETISM. 263 the use of this rule, which is sometimes called "Laplace s Law," let P be a point at a distance r from an infinitely long straight wire which carries a current C, and let s be the distance of ds from the foot of the perpendicular dropped from P upon the wire. If the angle (r, ds) be denoted by 6, s = r ctn 0,ds= r esc 2 6 d6, r = r esc 0. All the elements of the current conspire to produce at P a magnetic force per pendicular to the plane of P and the wire. The magnitude of this force is C * sin ds o r< smOdO = > as before. If a circuit is not plane, the different elements of the current will contribute to the magnetic force, at a point P, elementary forces which do not all have the same directions. In this case it is necessary to compute sepa- rately the components L, M, X of H. If the coordinates of the beginning of ds are Xj, y l} z l} and those of the end #! + dxi, y + dy^ z l -f- dz lf while those of P are or, y, z, the direction cosines of r and ds are (x l x)/r, (!/i~y)/ r > Oi--)A> and dxi/ds, di/i/ds, dz^/ds, and, if the direction cosines of dH, the contribution to the force at P made by the current element ds, are /, m, n, then, since this direction is perpendicular to r and to ds, l(x l x)+ m(yi -y)+n (z^ - z) = 0, ldx v + mdy l + ndz l = 0, I 2 -h m 2 + n 2 = 1. If we represent the expressions (y l y) dz (^ z) dt/i, (z l - z) dx 1 - (o-i - x) dz ly fa - x) dy^ - (y - y) dx v by 8 , 8", FIG. 67. 264 ELECTROMAGNETISM. 8 " respectively, and 8 2 + 8" 2 + 8 " 2 by 8, we learn from these equations that I = 8 /8, m = 8"/8, n = 8 r "/8, cos (>!, ds) = [(a?! - a?)^ + (y t - y)dy + (! - z)dz l \/rds ) and sin(r, ds) = 8/rds. If, then, the components of dH are ^L, dJf, dN, we have the equations dL = (78 /r 3 , <Of = <7S"/r 3 , dN = C8" /r 3 , and from these, by integration over the circuit, the force at P may be computed. Since action and reaction are equal and opposite, a unit magnetic pole at P would exert upon the element ds of the conductor which carries the current a mechanical or " pondero- motive" force the components of which would be CS /r 8 , - CS"/r 8 , - CB "/r\ These components, written in terms of the components An = (*1 - a)/**, M m = (2/1 - y)l*> ^m = (1 ~ of the magnetic field at rfs due to the pole at P, are and, since so far as this force is concerned the origin of the magnetic field is immaterial, these expressions give the com ponents of the mechanical force which act upon the element ds of a circuit carrying a steady current C in any magnetic field which at ds has the components L m , M m , JV m . If the magnetic field at ds l an element of a linear circuit s l which carries a steady current C\ is due to a steady current C 2 in another circuit s 2 , the element ds 2 of the second circuit at the point (x 2 , y^ z 2 ) contributes to the magnetic field at ds^ at the point (x 1} y^ x) components numerically equal to [202] ^ - [(2/1 - 2/2) ^ 2 - (#1 - 3 2 ELECTROMAGNETISM. 265 so that the x component of the mechanical force exerted upon the circuit element ds by the circuit element ds z is dXi = -^ \ [(?/!- i/ 2 ) dx, - (*! - * 2 ) dy*~\ dy l - [(^1 - * 2 )rf 2 - (-1 - or C* (7 i z i _ or 2 - - [cos (x, r) cos (ds l} ds 2 ) - cos (x, ds 2 ) cos (r, rfjjj J [ 203 ] where r is the distance of c?s 2 from rfs^ The x component, X lt of the whole mechanical force exerted upon the rigid circuit s l by the rigid circuit s 2 is to be found by integrating the expression just found over both circuits. The resulting integral will evidently not be changed if we add to the integrand any quantity which disappears when integrated about either circuit, and this fact makes it possi ble to find many other expressions * for the mechanical force exerted upon an element of one circuit by an element of another, which will account mathematically for the observed forces between two rigid closed circuits. According to Ampere s analysis, the resulting action between the two elements ds l} ds 2 is an attraction in the line joining them of intensity * For exhaustive treatments of this important subject the reader should consult Ampere, Gilbert s Ann., 1821 ; Ampere, M6m. de V Academic. 1823, 1827; TV. Weber, Ges. Werke ; Grassmann, Fogg. Ann., 1845; F. E. Neumann. Abh. BerL Akademie, 1845 ; TViedemann, Lehre von der Elektricitdt ; Maxwell, Treatise on Electricity and Magnetism, 502-527 ; Webster, Theory of Electricity and Ifagnetism, 217-221. For conven ience of reference I have followed Professor Webster s order, and in part his notation hi the brief treatment of the Electrodynamic Potential given in Section 80. See Problem 307, page 452. 266 ELECTKOMAGNETISM. On this assumption two elements in the same straight line repel each other with a force CiC 2 ds l ds 2 /r 2 , while two parallel elements perpendicular to the line which joins them attract each other with a force 2 C^C^ds^ds^/r 21 . These expres sions, like those which precede, hold good whether the ele ments ds u ds 2 belong to the same circuit or to two different circuits. If two infinitely long straight wires (sj, s 2 ), parallel to each other at a distance a apart, carry in the same direction the steady currents C l} C 2 respectively, the mechanical force exerted on ! by s 2 is evidently CiC 2 1 I [cos(x } r)/r 2 ]ds 1 - ds 2 , J\ */2 or (2 OiC 2 /a) ( ds i9 so that every unit length of s l is attracted towards s 2 with a force of 2 CiC 2 /a dynes. If each of two closed circuits (s l5 s 2 ) which carry steady cur rents, Ci, <7 2 , consists essentially of two infinitely long wires parallel to the z axis, if the currents come up through the xy plane in the two circuits at the points (0, a), (c, b) respectively, and go down at the points (0, a), (c, 5), the first circuit experiences a force tending to urge it in the direction of the x axis, and the intensity of this force per unit length of both wires of 8l is 4 cdCi {!/[( - 6) 2 + c 2 ] -!/[( + &) 2 + c 2 ] }. It is evident from the discussion of the properties of mag netic shells in air given on page 217 that the mechanical action on a rigid linear circuit carrying a steady current C in a magnetic field (caused either by permanent magnets or by other currents or by both) may be mathematically accounted for on the supposition that every element ds of the circuit is urged by a force equal to C ds times the component (F), perpendicular to ds, of the total magnetic induction. The direction D of this elementary force is perpendicular to the plane of C and F in the sense shown in Fig. 68. ELECTROMAGNETISM. 267 The same assumption will account for the phenomena observed when a deformable circuit is placed in a magnetic field. According to this theory the component in any direction u of the force on the element ds is Cds B sin (B, ds) cos a, where a is the angle between n and the normal to the plane of B and ds, and this is numeri cally equal to the volume of a parallelepiped, adjacent edges of which are represented in magnitude and direction by Cds, B, and a unit length in the direction u. This volume may also be represented by Cds sm(u f ds) -B , F where B is the component of the induction B, normal to the plane of u and ds, and this expression for the force component is occasionally useful. If (I, m, n) are the direction cosines of the element ds and if the components of the induction B are B x , B y , B zJ sin (B, ds) = \ (m B 2 - n B y ) 2 + (n B x - I B x ) 2 + (I B y - m B^\\/ \ B* + B* + B?\t and the resultant electromagnetic force on the circuit element ds has the value C \ (m B z - n B y y + (n B x - I B s )* + (l-B y -m. B x ) 2 \ I , ds. If ds is an element of a current filament of cross-section o> in a massive conductor in which the current vector is q or (u, v, w), we have qu = C, in* = 1C, vw = mC, W<D = nC, and the electro magnetic force may be written u\(v B t -w Byf + (w -B x -u. B;) 2 + (u B y - v B x ) 2 ^ , ds. The components parallel to the coordinate axes of the electro magnetic force per unit volume of the conductor are, therefore. (v-B 2 -iv-B y ), (w-B x -u.B s ), (u.B,-v-B x ). If the element ds be moved parallel to itself through the distance du, the mechanical work done on it by the forces of 268 BLECTROMAGNETISM. the field can be represented numerically by the volume of a parallelepiped, conterminous edges of which are C ds, B, and du ; this volume is numerically equal to C times the number of lines of induction of the field cut by the element during the translation. If an observer be imagined to lie in the element in such a way that the current enters at his feet and goes out at his head, and if he faces in such a direction that he can look along the lines of force, the work done by the translation will be positive if these lines appear to pass him from left to right, that is, if the displacement is to his left. It is easy to see, moreover, that if the element ds be revolved about any axis through a small angle, the work done upon it may be represented by C times the number of lines of induction cut by the element during the displacement ; we may infer, there fore, that the electromagnetic work done by the field upon any portion s of a circuit during any displacement is measured by the product of the current strength and the number of lines of induction cut by s. The direction in which a rigid closed linear circuit carrying a steady current C in a magnetic field of any kind will tend to move may be inferred from the fact that the circuit will behave in this respect like the equivalent magnetic shell. It is easy to see from the discussion on page 216 that the mutual potential energy of an external field and the mag netic shell mechanically equivalent to a given circuit, that is, the mechanical work that must be done to bring the shell already formed into the field, is equal to CN, where N is the whole number of lines (unit tubes) of induction of the field which the current surrounds right-handedly. The cir cuit will tend to move, therefore, so as to make N as large as possible. If, for instance, a plane circuit of area A carries a steady current C in a uniform field of induction of intensity B y any motion of the circuit parallel to itself would not change the induction through it, and there is no tendency to any such motion; if the normal to the plane of the circuit makes an angle with the direction of the field, a couple, ELECTKOMAGNETISM. 269 of moment C A B sin 0, acts on the circuit and tends to decrease 6. If into a magnetic field F Q which has the components X QJ F , Z Q a linear circuit carrying a steady current be intro duced, and if the electromagnetic field due to the current alone is FU or (JCi, Pi, ZJ, the whole field is (JT + A 1? F + F 1? ^ 4- 3i), and the whole magnetic energy in the field is r or ^ + + F o r i + The first integral is the magnetic energy of the original field, the second that of the field of the circuit alone, and the third the magnetic energy due to the introduction of the circuit when formed into the field. "We may now show that this last term, which may be written cos is equal to the product of the strength of the current and the flux of induction of the original field in the positive direc tion through the circuit. Since all the equipotential surfaces of the field F l are bounded by the circuit, we may cap the circuit by a whole series * of such surfaces and write the * A. Gray, Treatise on Magnetism and Electricity, Vol. I, p. 293. 270 ELECTKOMAGNET1SM. total induction through the circuit due to the outside field in the form M = (IX, + m r o + nZ ) dS = ^o cos (F , F,) dS, where the integration is to be taken over any one of these caps and where /, m, n are the direction cosines of the normal to the cap. If a unit magnetic pole were carried around any line of force Sj of the field F l} the work done on it would be 4?r times the current C in the circuit, so that 4 irC = I F^- ds. If we multiply each side of this last equation by M, we have C CC p(lX9 + mY cos Since the caps are equipotential, F l ds has the same value for all lines of force between any two caps, and since the induction JJ,FQ is solenoidal, the first integral factor of the second mem ber has the same value for all the caps. We may find the value of the second member, therefore, by imagining space divided up into elements which are portions of tubes of force of the field F bounded by equipotential surfaces of this field, multiplying the volume of each element by the value in it of /J.FQ - FI cos (F , F-L), and finding the limit of the sum of all these quantities divided by 47r. The value of the volume integral must be, however, independent of the shapes of the elements, and we have, in general, cos (F , F$ dr, ELECTROMAGNETISM. 271 The magnetic energy in the medium is often called the "elec- trokinetic energy." That portion of the electrokinetic energy which is due to the introduction of the circuit already established into the given field is evidently the negative of the mutual poten tial energy, corresponding to work done against mechanical forces, of the equivalent magnetic shell and the field. If a portion s of a circuit electrically connected through mercury cups with the rest of the circuit, which is fixed, be rotated and finally brought back to its original position*, elec tromagnetic work will be done on s if it cut lines of the field in positive direction during the motion, but the whole circuit may be represented by . the same magnetic Q L* -L shell at the begin- _-/ ning and at the end -=F- of the process, and -Er the mutual poten- -p tial energy of the p / ^ ^ circuit and the field is unaltered by FIG. 09. the displacement. Under these circumstances, as will appear in the sequel, cur rents are induced in s by the motion. If in the case of the circuit shown in Fig. 69 the conductor AB is free to slide on the rails DA, GB in such a way as to be always parallel to DG, it will move in the direction indicated by the detached arrow, the circuit will be made to embrace in the positive direction a greater number of lines of induction, and the electrokinetic energy will be increased. If the motion take place without external help, the necessary energy must be furnished at the expense of chemical action in the battery. Let E be the electromotive force of the battery, r the resistance of the circuit at any instant, and C the current which then passes through it : the energy furnished by the chemical action in the battery during the time dt will be ECdt, and of this a 272 ELECTROMAGNETISM. / A FIG. 70. part, at least, C 2 rdt, appears as heat in the conductors which make up the circuit. If AB be held still, C will have such a value, CQ, that EC Q = C *r. If, however, AB be moving toward the right, the current will be smaller than C OJ EC will be a fraction of EC Q9 B / _ C 2 r a smaller frac- tion of C V, and EC will, therefore, be greater than C 2 r. The difference (EC - C*r) dt now represents the work done during the time dt in moving AB : a part of this work is used in overcoming friction on the rails, a part in communicating kinetic energy to AB, and a third part in increasing the energy of the medium. If for convenience we denote (EC (7V) dt by C dp, we shall have E D t p = Cr, and the current is the same as if there were in the circuit an electromotive force D t p opposed to that of the battery. If an external force were applied to AB tending to move it to the right, the velocity might be increased so much that the current would be reduced to zero or caused to flow in the opposite direction. If, however, AB were forced to move to the left by external forces, the current in the circuit would become greater than C and would have the same direction as E. Fig. 70 illustrates a case where the resultant magnetic field is, as before, normal to the plane of the circuit, though the field lines thread the circuit in the negative direction ; in this case AB will tend to move toward the left. Fig. 71 represents Faraday s metal disc, mounted on a metallic arbor and free to turn about a horizontal axis. At FIG. 71. ELECTROMAGNETISM. 273 any instant the current flows in the disc from the centre to the brush P and the conductor which carries the current is urged to turn in the direction indicated by the arrow. The energy in the medium is not increased by the motion of the disc, and the work done by the battery is spent in heating the conductors in the circuit, in overcoming friction and the resistance of the air, and in increasing the kinetic energy of the disc. If the field is uniform, if S is the area of one face of the disc, and if the media are of unit induc- tivity, the work done on the disc each turn is CHS, and if it is making n turns per second, we have EC = C*r + CHSn. If the disc be used as a motor to overcome resistance of some kind, and if the energy required per turn is/(?*), CHS =f(?i), and from these two equations n and C may be found, if f be a known function. In the arrangement shown in Fig. 72 a rigid wire free to turn about the axis of a fixed vertical magnet makes electrical contact with the magnet at its middle. The current from a battery flows through a circuit made up of the wire, the mag net, and a supplementary fixed conductor forming a prolongation of the axis of the magnet. In this case the wire will turn continuously in the direction indicated. It is easy to show that a fixed magnetic field cannot cause continued rotation of a complete rigid circuit about a fixed axis. 80. The Electrodynamic Potential. If while a linear cir cuit, s 2 , which carries a steady current, C Z) remains fixed, a neighboring linear circuit, s^ which carries a steady current, <?!, is deformed or moved without being stretched, so that every element ds l is unchanged in length but the coordinates FIG. 72. 274 ELECTROMAGNETISM. of the beginning of the element receive increments Sx l} 8y lt Bz l} which are analytic functions of x ly y ly z 1} the work done by the forces which s 2 exerts upon s l is approximately equal to dZi or to 2 -f- [dx 2 Sec! -f ^2/2 8i/i + c? 2 8i]- The first factor under the integral signs in the second integral of the last expression is equal to J) (l/ r ) -ds^ and if we integrate the whole integrand by parts with respect to s 1? we get [(da, Sa?! + dy* 8^ + d 2 8i) /r] taken between limits where the expression in brackets, having the same value at both limits, can be omitted. The expression for the elemen tary work done on s l during its displacement is, therefore, -L dx 2 + dyi dy z + dz l dz z ~\ + CiC 2 f C(dx 2 d8 Xl + dy z - dfyi + dzi d&zj /r, c/l*/2 and this is obviously equal to the variation of the integral Ci C, f C(d Xl dx. + dy l dy* + dz,. - dz 2 ) /r, */l/2 ELECTROMAGNETISM. 275 caused by the elementary displacement. This last integral written in the form gives what is often called F. E. Neumann s Expression for the Elect rodynamic Potential. The increase in the value of this function caused by any finite displacement of s^ evidently represents the work done on s l by the field due to s 2 during the displacement: this work depends only upon the original and final configurations. The Electrodynarnic Potential corre sponds to that portion of the electrokinetic energy which is due to the mutual proximity of the circuits. Its negative is equal to what is sometimes called the mutual potential energy due to the mechanical forces acting between the circuits. It is impor tant to notice that although the ponderomotive forces which urge a rigid circuit carrying a given current, (7, in a magnetic field can be correctly found from the expression for the mutual potential energy of the field and a magnetic shell of strength C bounded by the circuit, this may be regarded from one point of view as merely a convenient mathematical device. If the shell were to move under the action of the field alone and acquire kinetic energy and overcome external resistance, this work would be done at the expense of the mutual poten tial energy of the field and the shell. If, C being kept constant, the circuit were to move under the action of the field in exactly the same way, the work would be done at the expense of the generator which maintains the current. In other words, there is no sensible mutual potential energy of the field and the circuit, the exhaustion of which measures the work done by the forces of the field during any displacement of the circuit. The integrand in the expression given by Neumann can be increased at pleasure by any quantity which disappears when integrated around either s x or s. 2 . Such a quantity is X D gi D^i , or A[cos(r, ds^ -cos(r, ds z ) cos(ds l} ds^/r, where A is any 276 BLBCTROMAGNETISM. constant. The corresponding form of the Electrodynamic Potential is Ci C 2 \ \ \\ cos (r, ds-t) cos (r, ds 2 ) \(l/r) ds^ - ds^ i - x ) cos ( ds &) C 1 /r ) \ ds i ds r A form sometimes convenient is obtained by putting X = 1. In the case of two vertical, coaxial, circular wire circuits of radii r x and r z , at a distance a apart (Fig. 73), we may denote by fa and fa the angles which radii, drawn from ds lt ds 2 respectively to the centres of their circuits, make with the vertical and put x t TI cos fa, x 2 = r 2 cos fa, y l r sin fa, y z = r 2 sin fa, r* = a* + r, 2 + r 2 2 - 2 r^ cos (</> - ^ a ). The expression P = d <7 2 f f(^! ^x 2 + %! ^2/ 2 -h dz l dz 2 ) /T then becomes cos(fa-fa)dfa or C^Qdfa. */o ELECTROMAGXETISM. 277 That the definite integral Q is not a function of <f> 2 follows from the fact that the definite integral which represents its partial derivative with respect to < 2 is the limit of the sum of elements which destroy each other in pairs : we may there fore give to < 2 in the expression for Q any convenient value (say zero) and write P 2-n- C^r^Q. We may conveniently transform the integral which represents Q by putting 2 = and get or - -\ E\, where K and E are the complete elliptic integrals of the first and second kinds. The numerical values of these integrals for various values of A* are to be found in " A Short Table of Integrals " (Ginn & Company, Boston). It is to be noted that if in this analysis we imagine finite currents to be carried by conductors of zero cross-section, and -t\ and r 2 to be equal, then, if a approaches zero, A- approaches unity and P grows large without limit. The derivative of P with respect to a gives in general the mutual attraction of the two circuits. If the external field about a linear circuit s 1} carrying a circuit CD is due to a current C 2 in another linear circuit s 2 , we have two different expressions for the mutual potential energy of the magnetic shells which correspond to the two circuits. These are C^Y, where N is the number of lines of induction due to C 2 which thread s l positively, and the negative of the Electrodynamic Potential of the two circuits. When Ci and C. 2 are both unity the Electrodynamic Potential measures the magnetic induction through either circuit when the unit current traverses the other. The number of lines of magnetic induction which thread either of two simple linear circuits, made of non-magnetic material and removed from the neighborhood of other currents 278 ELECTROMAGNETISM. and permanent magnets, when the unit current passes through the other circuit, is called the coefficient of mutual induction or the mutual inductance of the two circuits. The numerical value of this coefficient depends upon the character of the media in the neighborhood of the circuit. If two exactly similar linear circuits, S L and s 2 , carrying steady currents of unit intensity, lie side by side, and if one of them (.9 2 ) be imagined to move up towards coincidence with the other, the value of the integral which represents the Elec- trodynamic Potential approaches the form C C GOS ( ds n ds 2 ) ds-L - ds 2 Li = I I ) where the integration is to be extended twice over the same circuit. If the circuits are supposed to be mere geometrical lines, the value of this integral will be in general infinite ; if, however, s x and s 2 are made of wires of small but definite cross-sections, the finite limit, as s 2 is moved into close contact with s ly of the flux of magnetic induction caused by the unit current in s 2 through a diaphragm bounded by s l is practically the flux through the diaphragm due to the unit current in s^ The number of lines (unit tubes) of magnetic induction which thread a simple fine wire circuit made of non-magnetic material, which carries a steady current of unit strength when there are no other currents and no permanent magnets in its neighborhood, is very nearly equal to what is called the coefficient of self-induction or the self -inductance of the simple circuit, under the circumstances. The numerical value of this coefficient, which we shall soon be able to define more accu rately, depends very much upon the nature of the media about the circuit. 81. Coefficients of Induction. If two fine wire closed cir cuits of non-magnetic material, exactly alike in size and shape, and carrying in the same direction steady currents of intensity C and C" respectively, are placed as nearly as possible in BLECTROMAGNETISM. 270 coincidence, the coefficient of mutual induction of the two is practically the same as the coefficient of self-induction (L) of either, and the work required to separate the two circuits to an infinite distance from each other is C C"L. If, then, a fine wire closed circuit which carries a steady current C be supposed made up of infinitely slender closed circuit filaments lying freely in contact, it is easy to get an expression for the work that must be done in removing these filaments one after another out of the field. If at some stage in the process the remaining filaments carry altogether the current C C", the work required to remove another filament carrying the cur rent dC" would be (C- C")dC"-L, and this integrated with respect to C" between and C yields % C 2 L, which is an expression for the intrinsic energy of the original collection of filaments. Again, if a current C be set up and kept steady in any closed circuit in a medium of any kind which contains no permanent magnets and no other currents, the medium becomes polarized by induction and is a field of force. The electrokinetic energy is equal to the volume integral taken over all space of /x(7 2 J? 2 /8 TT, where R is the intensity of the field due to a unit current in the conductor. It is easy to see that this reduces in the case of a linear circuit to % C 2 times what we have called the coefficient of self-induction of the circuit, and we are led to define the coefficient of self-induction of a circuit, made up of conductors of any form surrounded by media the susceptibilities of which are independent at every point of the intensity of the force at the point, as twice the energy in the magnetic field when the circuit carries a current of one electromagnetic unit and there are no other currents and no permanent magnets in the neighborhood. If, for instance, a uniformly distributed current C be carried lengthwise in a homogeneous, infinitely long cylinder of revolu tion, of radius a, and be brought back in a thin cylindrical shell of inside radius b and outside radius c, coaxial with the cylin der, there is no field without the shell ; the intensity of the field is 2 Cr/a* within the cylinder, 2 C/r between the cylinder 280 ELECTROMAGNETISM. and the shell, and 2 C(c 2 r*)/r (c 2 b 2 ) in the shell itself. Neglecting the space occupied by the thin shell, which would contribute little to the result, the whole energy in the field per unit length of the cylinder is ^ f a* Jo 2 i*dr + - 4 C 2 vr dr. 8 7T If the medium between the shell and the cylinder has the uniform inductivity /x, 2 , this energy is i/xjC 2 + n 2 C z Iogb/a. The coefficient of self-induction of the circuit per unit length is, therefore, when the shell is thin, \^ + 2 p 2 logb/a. The coefficient of self-induction, in electromagnetic absolute c.g.s. units, of a circular ring of circumference I, made of non magnetic wire of radius r and surrounded by air, is, according to Kirchhoff, 2 I [log (l/r) 1.508], and that of a square circuit of perimeter I, made of similar wire, 2 I [log (l/r) 1.910]. Regarding the coefficient of self-induction from the point of view of the energy in the field, it is possible to prove that the coefficient of a part of a circuit consisting of a straight wire of length I and radius r is approximately 2 I [log (2 Z/r) + i /* 1], where /x is the magnetic permeability of the wire. For addi tional examples, the reader is referred to Winkelmann s Hand- buck der Physikj Vol. Ill, Maxwell s Treatise on Electricity and Magnetism, Vol. II, and to Gray s Absolute Measurements in Electricity and Magnetism, Vol. II. If X lf YI, Zi are the components of the electromagnetic field which a unit current flowing in a given circuit s x of self-induc tance L 1 would cause if the surrounding space contained no other currents and no permanent magnets, and if this space is already the seat of a magnetic field X, Y, Z, caused either by currents or by permanent magnets, or by both, then if a steady current Ci be set up and maintained in s lt the electrokinetic energy is <^f ff 1*1(0^ + xy ELECTROMAGNETISM. 281 The integrand can be split up into three terms, and 2nd \XX 4 Y l Y 4 Z^Z], and the corresponding integrals represent respectively -J- C-fL-^ the energy of the original field, and that part of the electro- kinetic energy due to the introduction of the current into the field. If the external field is due to a steady current C 2 in a second circuit s 2 of self-inductance L 2) the second integral is J- C^LK and if the third be written dd^f, the whole energy becomes CfL^ 4 Mdd 4 i CJL* The quantity M, which in the case where s^ and s 2 are linear is the coefficient of mutual induction of the two circuits, serves to define this coefficient in the case of circuits which are not linear, surrounded by media which have susceptibilities independent of the strength of the field. If n circuits which have self-inductances L lf L 2 , L S) and carry currents C 1? C 2 , C 3 , exist together in a soft medium, and if the mutual inductance of the pth and kth circuits is M pk , the elect rokinetic energy T is equal to f 4- L 2 d* 4- L 8 d* 4- 4- L n C^ 4 MdC + MdC 4 4 M ln d where the values of the inductances depend upon the configura tion of the system. If this configuration is determined by a number of generalized coordinates q v q 2 , q^ , the electro- dynamic force, in the Lagrangian sense, which tends to increase any one of these coordinates (leaving the rest unchanged) is the partial derivative of T with respect to this coordinate. If every circuit is rigid, the L s are constant during any change of configuration. 82. Maxwell s Current Equations. Various Current Sys tems. We may infer from experiment that if a unit magnetic pole be moved about a simple closed path in any steady electro magnetic field, whether the medium in which the part lies is 282 ELECTHOMAGNETISM. homogeneous or not, the work done on it by the field is equal to 47r<7, where C is the whole current which passes in positive direction through any surface or diaphragm which caps the path. If u, v, w are the components of the current intensity, the flux through the cap may be written in the form J J [u- cos (x, ri)+v> cos (y, n) + w> cos (z, n) ] dS, and if L, M, N are the components of the magnetic force H, the line integral of H taken around the path is equal, according to Stokes s Theorem, to ~ D * M ) cos ( x > n ) + (P* L ~ D*N) cos (y, n) + (D X M - D y L) - cos (, It follows that the integral Jj [(4 - D y N + D Z M) cos (x, ri) + (4 TTV - D Z L -f D X N) cos (y, n) + 4-7rw- DM + Z)L cos z must vanish whatever the shape of the cap and, therefore, that at every point 47m = =D Z L-D X N, = D X M - DyL. [205] These are Maxwell s Current Equations, which can be stated in the single vector equation 4 Tn/ = curl of H. This has been called by Heaviside " the first circuital equa tion" of the electromagnetic field. It states that 4 TT times the resolved part of the current intensity at any point within a conductor, in any direction, is equal to the resolved part in the given direction of the curl of the magnetic force. The equation holds even in a non-homogeneous medium. ELECTROMAGNETISM. 283 Maxwell s Equations, with the characteristic volume and boundary differential equations which the magnetic induction, as we have seen, must always satisfy, completely determine a steady magnetic field in given media, when the cm-rent q is known. In any homogeneous soft medium the magnetic force H is solenoidal, and we may infer from the work of Section 69 that it has a vector potential function Q equal to Pot q. We have, therefore, H = curl Q, 4 irq = curl H, and, if the compo nents of H and Q are L, M, N and Q x , Q y , Q z respectively, L = D v Q z -D z Qv M=D Z Q X -D X Q Z , N=D x Q y -D y Q,. When in a steady field H is known, Maxwell s Equations, or their equivalent, give the current vector q directly. If, for example, the magnetic force is zero everywhere without an infinitely long cylindrical surface A B S of any shape, while within S the K I I - K* field has the uniform strength N, F "4 and is directed parallel to the gen erating lines of the surface, q is zero within and without S. To show that S itself is a current surface, let KK be a por tion of a generating line drawn in the direction of the field within, and let AB and CD be lines each of length I parallel and close to KK 1 , one within and the other without S, drawn so that AC and BD are normal to the surface. The line inte gral of the magnetic force taken around the perimeter of the rectangle ACDB is numerically equal to IN, so that, by Stokes s Theorem, the surface integral of the normal upward component of the curl taken over the area of the rectangle is IN, and this is equal to 4 TT times the steady flow of electricity through the rectangle. There is, therefore, a uniform flow of electricity in S perpendicular to its generating lines equal to N/4: TT per unit length of the surface. 284 ELECTEOMAGNETISM. This is practically the case of an electromagnetic solenoid, that is, an infinitely long cylindrical surface wound uniformly (and as nearly perpendicularly to the axis of the cylinder as possible) with turns of fine wire. If there are n turns on each centimetre of length of the cylinder (Fig. 75) and if each turn carries a steady current (7, N /kir = nC, or N = kirnC. This result is independent of the magnetic inductivity of the homogeneous soft medium within the cylinder. The induction in the medium is 47m/A(7, and the intensity of its polarization (magnetization) is kirnkC or nC(p 1). The coefficient of self-induction per unit length of the solenoid is 4 Trn^^A, where A is the area of the cross-section of the cylinder. If a part of the space within the solenoid be taken up with a homogeneous soft medium of permeability pi, and the rest by an infinitely long cylinder of another FIG 75 homogeneous soft medium of permea bility /z 2 , the lines of which are parallel to those of the sur face upon which the wire is wound, the lines of force are unchanged in form, the induction in the first medium is 4 -n-n^ C and in the other 4 -rrn^ C. If A l and A 2 represent the portions of the cross-section A of the solenoid occupied by the two media, the self-inductance of the solenoid per unit length is The coefficient of mutual induction of two infinitely long solenoids S-^ S 2 , one of which has n^ turns and the other n z turns per unit of its length, is zero, unless one, say $ 2 , is within the other. In this case the coefficient has the value 4 irn^n^A^ per unit length of the two, where A 2 is the area of the cross- section of $2- If two infinitely long, cylindrical surfaces, whatever their shapes may be, have parallel generating lines, and if one of ELECTROMAGNETISM. 285 these surfaces is within the other, the space between the surfaces will be a uniform field of magnetic force of strength Nj directed parallel to the generating lines, and the regions without the outer surface and within the inner one will be fields of no force, if a uniform current of strength N/TT per unit length flows in each surface perpendicular to the gen erating lines and if the directions of flow around the two surfaces are opposed. If the two infinite parallel planes x = a, x = b carry uniform currents parallel to the y axis, of strength N/kir per unit width of the planes parallel to the z axis, and if the directions of the two currents are opposite, the region between the planes is a uniform field of force of strength N parallel to the z axis. There is no force without the space included between the planes. The current in each plane evidently gives rise to a uniform field of intensity \ N on both sides of the plane. If a ring surface be formed by revolving about the z axis an area in the xz plane, and if electricity be supposed to flow symmetrically on the surface, in closed paths which lie in planes through the z axis, and coincide with perimeters of cross-sections of the ring formed by such planes, the field has the same intensity at all points of any one of the family (/) of circumferences, the centres of which lie in the z axis and have that line for their common axis. If, using columnar coordinates (?, 0, z), we denote the force components at any point, taken in the directions in which these coordinates increase most rapidly, by H, , Z, these components are independent of 0. Since the amount of work which would be done on a magnetic pole if it were carried around any closed path with out the surface, whether or not it linked with the surface, or around any evanescible path wholly within the surface, would be zero, we are led to guess that the field outside the ring is everywhere zero, and that the lines of force within the ring are circumferences of the / family. If a unit magnetic 286 ELECTKOMAGNETISM. V FIG. 76. pole were carried about one of these circumferences of radius r, the work done on it would be 2 Trr, and this is equal in abso lute value to 4 nE, where E is the whole amount of electricity which flows about the ring per second. We learn, therefore, that = 2E/r. We may now prove that if there is no field without the ring surface, and if the only component within is 2 E /r, the currents which give rise to the field must be those assumed above. The com ponents of the field within the ring, taken parallel to rectangular axes, are 2Ey/r 2 , 2Ex/r 2 , 0, so that the force is lamellar within and without the surface of the ring. To find what currents flow in the surface itself, we may use a circumference s of radius r, in which a plane perpendicular to. the z axis inter sects the surface, draw two arcs parallel and very close to .?, one on either side, so that one is within the surface and the other without it, and complete a narrow closed contour by drawing two radii (Fig. 76) which make with each other any convenient angle <f>. Only one side of the contour yields any contribution to the line integral (2 E<j>), taken about it, of the tan gential component of the force. This integral measures the work done on a unit magnetic pole car ried around the contour, and is equal to 4 TT times the strength of the current across the portion of s, of length r<f>, which the contour encloses. If the whole flux across s is F, the flux across this arc is ^F/Zir, and we have the equation, 2 E<f> = 4 7r<f>F/2 TT, or F E, The case here considered is approximately that of a ring of revolution wound uniformly with fine wire (Fig. 77) in turns which lie nearly in radial planes through the axis of the ring. If there are n turns on the ring, and if each turn carries the FIG. ELECTROMAGNETTSM. 287 steady current C, E = n f\ and the force within the ring is 2nC/r, whatever the inductivity of the homogeneous soft medium within the ring. The induction in the medium is 2pnC/r, and the intensity of its magnetization is 2knC/r. It is to be noted that the reasoning here employed might be applied unchanged if the inductivity of the medium were a function of r and z, but not a function of ; this would be the case, for instance, if into air space within the ring were intro duced a soft iron ring coaxial with this space. A slender magnetic filament within the ring surface, of length I and cross-section S, carries 2 pJES/r, or 4 irnC / (l/f*S) lines of induction. The line integral of the magnetic force taken along a magnetic filament in a soft medium is sometimes called the magnetomotive force in it, and the ratio of this quantity to the flow of induction in it the reluc tance of the filament. In the case before us k-n-nC is the magneto motive force, and ///i the reluc tance. This last expression bears a close resemblance to the formula for the electric resistance of a wire of length /, cross-section S, and specific conductivity p.. The reciprocal of the reluctance of a magnetic filament in a soft medium is sometimes called its permeance. If wire were wound part way around a soft iron ring, in the manner described above, most of the lines of induction would still be confined to the iron, though a few would emerge into the air at the ends of the coil. If a radial gap be cut in a soft iron ring completely wound with wire, the field is no longer symmetrical about the axis of 2, and the character of the problem is changed. The line integral of the tangential force taken around a circumference inside the ring, of radius / (Fig. 78), with centre on the z axis 288 ELECTROMAGNETISM. and plane perpendicular to that axis, is still kirE or but the portion of the path in air now contributes far more than its due proportion to the result, and the path in the iron much less than before. We know that at any surface of discon tinuity in the inductivity of a soft medium the normal com ponent of the induction is continuous, so that if the normal component of the force in the iron, just where the path is about to emerge into the gap, is If, that in the air near by is p.H, where /u, is the inductivity of the iron. Although the lines of force within the iron are no longer exactly circular, they are nearly so, and the line integral of the force about the circumference just mentioned is very approximately Hr(?nr <f>) -f- p_Hr<l>, or 4 TrnC ; and H= 2nC/\r[l + <(> - 1) /2 TT] J, where <f> is the angle subtended by the gap at the axis of the ring. If, in the case of the core used, p = 1201, and if only one per cent of the ring be cut away, the induction in the iron will be reduced to about one-thirteenth of its old value ; the reluctance of the path will be increased thirteenfold. If the lines of force in a steady electromagnetic field are all circles with centres on the z axis and planes perpendicular to this axis, and if the intensity of the force in a direction linked right-handedly with the z axis is/V^r 2 + y 2 or /(?*), it is evi dent that L = y -f(r)/r, M=x -f(r)/r, N=Q,so that u = 0, v = 0, mv = D X M- D,L=f (r)+f(r)/r = A !>/]/> According to this, if in any portion of the field f(r) = 0, in that portion w is ; if f(r) = c, w = c/4:irr , iff(r) = c/r, w = 0j and \if(r) = cr, 10 = 0/2-*. If, on the other hand, while u and v are zero, w is given as a function of r, /(?) can be obtained from the equation /(/) = J rwdr; when, therefore, w is equal to the constant w ,f(r)= 27rw r + d / r. If the cylindrical surface r = b separates two regions in a field of this kind where the laws of force intensity fi(r), / 2 (r) in the inner and outer of these regions are different, and if /a (&) /i (#) = &, it is easy to see, with the help of Stokes s ELECTROMAGXETISM. 289 Theorem, that the surface r b is itself a current surface in which there is a total flux parallel to the z axis across any right section of % kb. Up to this time we have considered only media which have inductivities independent of the magnetizing force. The per meabilities of the so-called magnetic metals do not in general satisfy this condition, and we ma} note in passing that some of our definitions have to B be restated when there are masses of such media near a circuit. If fine wire, carrying a steady current, be wound uniformly upon a cylin drical rod of soft iron, the length of which is at least 400 times its diam eter, the induction at the middle of the rod is sen sibly the same as if the rod were infinitely long, and if this induction be measured (in a manner to be described in a later section), the permeability of the iron may be determined. Curves D, E, F of Fig. 79, in which the abscissas represent the magnetizing force H in units, and the ordinates the corre sponding induction B = ^H in thousands of units, show the results of experiments upon specimens of very soft malleable iron, soft cast iron, and very hard steel, respectively. It is evident that so far from being constant, the permeability and the susceptibility of each of these specimens increase to a maximum at a value of H corresponding to a point where a tangent from the origin touches the curve, and then decrease. In the case of the curve D, for instance, the permeability, corre- -E FIG. 79. 290 ELECT BOMAGNETISM. spending to a value OM of the magnetizing force, is MP / OM, that is, the slope of the straight line OP joining the origin with the point on the curve which has OM for its abscissa. When the conductors which make up a simple linear circuit which carries a steady current C, and the soft media about it, have inductivities independent of the magnetizing force, and there are no other currents and no permanent magnets in the field, the coefficient of self-induction of the circuit may be defined indifferently as the ratio of the total induction through the circuit to C or as twice the ratio of the integral of />tZr 2 /8 TT, taken over the field of the current, to C 2 . In this case the magnetizing curves of all the substances in the field are straight lines, and these definitions lead to the same value whatever C is. If the magnetizing curve of any medium in the field were, like that of soft iron, not straight, the definitions would not agree, and each would yield different values for different values of C. Mechanically soft iron or steel cannot be regarded as magnetically soft, for if a piece of either of these metals be magnetized by FIG 80 induction, this magnetization does not wholly disappear when the magnetizing force is removed. If the magnetizing force be made to change contin uously from a given negative value to an equal positive value and back several times, the induction goes through a cycle which may be represented graphically by a curve somewhat like that shown in Fig. 80, in which the abscissas represent magnetizing forces, and the ordinates the corresponding values of the induction. Such diagrams make plain the fact that the induction in a piece of soft iron or steel is not a definite function of the magnetizing force, and that the energy in the medium, as defined by the volume integral of 1/8 TT times the product of the numerical values of the induction and the magnetizing force, may have, for the same force, very different values, depending on the previous history of the metal. When CURRENT INDUCTION. 291 the induction has passed through such a cycle as that indicated in Fig. 80, the energy in the field returns to its old value, but it is easy to prove that an amount of work represented by 1/4 TT times the area of the cycle per unit volume of the substance had to be done on the metal during the cycle, and that this appeared as heat. The reader will find the subject which has been just touched upon here admirably treated under the head of " Hysteresis" in E wing s Magnetic Induction in Iron and Other Metals, and in Fleming s The Alternate Current Transformer. IV. CURRENT INDUCTION. 83. Electromagnetic Induction. If one of two circuits (* lf s 2 ), so situated that their coefficient of mutual induction is not zero, contains a galvanic cell and a key, and the other (s 2 ), which is permanently closed, a galvanoscope, a momen tary current appears in s 2 when the key is depressed so that a current circulates in s l ; and another momentary current, opposed in direction to the first, runs through s. 2 when the key is opened again. A current in either S L or s. 2 gives rise to a magnetic field and causes lines of magnetic induction to thread s 2 : the direction of the transient current in s. 2 in each of the cases mentioned is such that the lines which it threads through s 2 oppose the sudden change in the flux of induction through s. 2 which the change in the current in s 1 tends FIG. 81. to cause. Thus, if the relative position of the two circuits and the direction of the current in s v are correctly indicated in Fig. 81, the transient induced current in s 2 will flow from B to A when the key is depressed and from A to B when the key is again opened. In general, if a rigid, closed circuit s is in a magnetic field caused either by perma nent magnets or by electric currents in neighboring circuits, 292 CURRENT INDUCTION. or by both together, and if the positive flux Q of magnetic induction through any cap or diaphragm bounded by s be changed in amount, either by moving s or by changing the field in any way, a temporary current is induced in s in a direction which tends to oppose the change in Q. The phe nomenon is quantitatively explained, when s is unchanged in form, by assuming that, superposed upon such electromotive forces as the circuit may already contain, a temporary electro motive force numerically equal to the time rate of change of Q is induced in s in the proper direction. Transient currents are usually induced also in any circuit in a magnetic field when the circuit is deformed or extended in any way. These currents, like those already considered, are mathematically accounted for on the supposition that there is induced in every circuit element ds, which moves in a magnetic field so as to cut across the lines of induction during the motion, an c electromotive force numerically equal to the time rate at which the element cuts these lines. FIG 82 This electromotive force is directed from the feet to the head of an observer who, lying in the element and looking along the lines of force, sees these lines move past him from right to left. The induced cur rent at any instant in either direction around the circuit is equal to the ratio of the algebraic sum of the electromotive forces induced at that instant, in that direction, to the whole resistance of the circuit. If in Fig. 82 OC represents the direction of a circuit element at the point 0, OM the direction in which the element is moved, and OF the direction of the whole field at 0, the induced electromotive force will have the direction OC, not CO. The direction of the current, induced by the motion, in the direction OM, of a circuit element at in a magnetic field which has there the direction OF, may be found by choosing that direction, OC, in the element which will cause the three directions OC, OM, OF to be related like those of the x, y, z axes of a Cartesian system. It is to be noticed CURRENT INDUCTION. 293 that the direction of the current induced in an element is such that the mechanical action of the field upon the element carrying this current alone would hinder the motion ; a circuit element carrying a current in the direction OC in a field having the direction OF in Fig. 83 would be urged in a direction ON perpendicular to the plane of FOG and would move in the direction MO, if free to do so, rather than in the direction OM. The reader will do well to compare, in this connection, Figs. 68 and 82. If (a, b, c), (a, (3, y) are the components of two vectors, I and A, the vector which has the components (eft by, ay ca, ba aft) is sometimes called their vector product and the quantity (aa + bft + cy) their scalar product. The vector product of I and A has a direction perpendicular M to the plane of these vectors : its tensor is the product of their tensors and the sine of the angle between their directions. The electro motive force induced in or impressed upon an element ds of a linear conductor moving in a magnetic field is evidently equal to the product of ds and the component in its direction of the vector product of the induction and the velocity of the element. If (B x , B y , 7?,) are the components of the induction, (> *7> those of the velocity of the element relative to the field, and if the induction does not change with the time, the absolute value of the electromotive force induced in the element is l(B s -n - B y . cos (x, s) + (B x - B, ) cos (y, s) + (B, .S-Bx ij) cos (*, s) ] ds. [206] The whole electromotive force induced in the conductor is the integral of this expression : if the conductor is not closed this electromotive force gives rise to a statical distribution of electricity on the ends of the conductor, and hence to a 294 CURRENT INDUCTION. difference of electrostatic potential which tends to destroy itself by causing a current in the conductor in the direction opposite to the impressed electromotive force. If the induction (B x , B y , B z ) of the magnetic field in the neighborhood of a fixed linear circuit changes with the time, the induced or impressed electromotive force e in the circuit is equal to the negative of the surface integral, taken over any cap S bounded by the circuit, of \_D t B x - cos (x, n) + D t B y cos (y, n) + D t B z cos (, w)]. If, then, a vector can be found of which the vector (D t B x , D t B y , D t B z ) is the curl, then the line integral, taken around the circuit, of the tangential component of this new vector increased, if we please, by any lamellar vector will be equal in absolute value to the induced electromotive force. If (F m F y , F 2 ) is any vector potential (Section 69) of the induction, (D t F x , D t F y , D t F z ) is a vector potential function of (D t B x , D t B y , J) t B z ), and if ^ is the scalar potential function of any lamellar vector, the inte gral, taken around the circuit in positive direction, of - \_(D t F x + DJ) cos (*, s) + (D t F y + Drf) cos (y, s) + (D t F z -f- Djf) cos (, s) ] [207] will be equal to e. This value of the whole electromotive force induced in the circuit will be obtained if we assume that every circuit element ds is the seat of an electromotive force equal to the product of ds and the tangential component of the vector - [D t F x + D^, D t F y + Drf, D t F z + D z fy If a closed linear circuit s in a magnetic field be deformed or moved according to any law so that during the time dt the coordinates (aj, y, z) of the beginning of an element ds receive increments (&e, 8y, 8y) which are analytic functions of x, y, z, and dt, and if, during the interval dt, the scalar point func tions which represent the magnitudes of the components of the magnetic induction in the field change from the values CURRENT INDUCTION. 295 B x , B y , B z to the values B x , B y , B s , the flux of induction through the circuit has been increased by the amount C C [BJ cos (x, n) + , cos(y, n) + B z cos (2, n] \_B X cos (#, ri)+ B y - cos (y, n) + B z cos (2, ?i)] dS, where /S and $ are any surfaces which cap the circuit in its final and initial positions respectively. In moving, the circuit traces out a narrow surface S", each element ds of the circuit generating the surface element dS", and we may take for the cap S the surface made up of S and S". We have therefore d& = dtC C[D t B x cos (x, n) + D t B y cos (y, ti) + D t B z cos (2, n)~]dS In the second integral, cos (2, n) dS" measures the area of the projection of dS" on the xy plane and is, therefore, equal to (8x dij By - dx) plus terms of higher order ; the sign being positive, if the direction in which ds moves, the positive direc tion of the element, and that of the normal to dS" are arranged like the x, y, z axes of a Cartesian system. We may substitute in the integrand B x , B y , B z , dt, r, dt, - dt for E x , B,, , B z , So-, 8?/, 82, without changing the value of the integral, and then write D& = C\_P cos (a-, s)+Q> cos (y, s) + R cos (2, 5)] ds, where P = - D t F x - D^ + B. rj - B,, , Q = - D t F y - DJ, + B x .^- B s -t, [208] P, Q, R are said to be the components of the induced electro motive force at the element ds. We may note in passing that we cannot generally assume that the motion of the electricity in a circuit which is the seat 296 CURRENT INDUCTION. of an induced current is governed by a potential function due to an electrostatic distribution on the surfaces of the conductors, or elsewhere. If a magnet, the axis of which coincides with the axis of a plane circular ring of wire, be made to approach or to recede from the plane of the ring, a transient current is induced in the wire, but no imaginable electrostatic distri bution would furnish the multiple-valued potential function needed to account for the current. If a circuit at a distance from other circuits and perma nent magnets carries a changing current (7, the ratio of the numerical value of the intensity of the electromotive force induced by the change of the current in the circuit to D t C is sometimes used as a definition of the self-inductance of the circuit. The mutual inductance of two circuits may be denned in a similar manner. It is evident that all the definitions of self and mutual inductance which we have mentioned are equivalent when all the media in the neigh borhood of the circuits concerned have susceptibilities inde pendent of the intensity of the field. The definitions of this section are often used when there are masses of soft iron or other magnetic metals near the circuits, or when the circuits themselves are made of soft iron conductors. If a number of circuits s v s 2 , s n , carrying currents C 19 C 2 , C n , have self-inductances L u L 2 ,--- L n , and if the mutual inductance of s k and s t is M kl , the total electrokinetic energy T is of the form + M 12 C,C 2 + M^C^ + + M 2S C 2 C. + M 24 C 2 C 4 -f ... + M 8 ,C s C t + ... + M H __ lt n C n _, C n , where the L s and the M s are independent of the C"s and are to be considered as functions of a set of geometrical coordi nates equal in number to the degrees of freedom of the system. If p k denotes the electrokinetic momentum of s k , that is, the partial derivative of T with respect to C k , if r k represents the CURRENT INDUCTION. 297 resistance of s k , and E k the internal electromotive force in this circuit, -~ measures the intensity of the induced electromo- (JLif tive force, and T E "~~dt = TkCk If the relative positions of two rigid circuits s lt s 2 , which carry currents C : , C 2 , and are surrounded by a soft medium in which there are no other conductors, be altered by changing under their mutual action the geometrical coordinate q by the amount dq in the time interval dt, leaving the other coordinates which determine the configuration unchanged, the electrokinetic energy T=%L V C\ 2 + MC l C 2 +$L 2 C} will receive the increment dT= L l C l dCi + LiCt dC a + M( C. 2 d\ + C, dC 2 ) + C& dM. The electrodynamic force (in the Lagrangian sense) which tends to bring about this change of configuration is the partial derivative of T with respect to q, taken under the assumption that the other coordinates and the currents are constant : the work done during the change by this force is d W = C^C Z dM. Within the circuits we have -EiCi dt - C l d(L l C l + MC 2 ) = C?r dt, E^C.-dt- C z -d (L 2 C 2 + MCI) = C 2 2 r. 2 dt, so that the work done against the inductive electromotive forces by the applied electromotive forces (besides the amount C\ r \ + C^r 2 dissipated in heat) is + MC 2 ) + C 2 d(L 2 C 2 L.C, dC, + L Z C Z - dC 2 -h M(C Z dC l + Q dC 2 ) +2C 1 C 2 - dM, or dW+dT. If, starting from rest, the circuits come again to rest and the currents regain their steady values before the end of the interval dt, we have ^ 2 dW. 298 CURRENT INDUCTION. B/ The principles just laid down enable us to infer that, if the conductor AB of length I in either of the circuits repre- B/ sented by Figs. 84 and 85 be moved parallel to itself along the rails CB, DA, in the direc tion indicated by /A the arrow attached F IG . 84. to it, with constant velocity v, and if the field have the direction shown, an electromotive force will be induced in AB in the direction pointed out by the arrow by its side. If the component of the total induc tion normal to the plane of the circuit have the constant value H all along D AB, and if r be the resistance of the whole circuit ABCD, the induced current will be IHv/r in abso lute units. The volt, ohm, and ampere are equal respectively to 10 8 , 10 9 , 10- J times the absolute elec tromagnetic c.g.s. units of electromotive force, resistance, and current strength; if in this example, therefore, I = 1 metre, v = 1 metre per second, and H = 1, the induced electromotive force will be 10,000 units, or 10 ~ 4 volts. If a Faraday s disc (Fig. 86) which has a radius a be rotated in a uniform field, in which the component of the induction normal to the face of the disc is H, with uniform angular velocity w in the direction indicated, the number of lines of induction cut per I /A FIG. 85. FIG CURRENT INDUCTION. 299 second by OP is % a-Hw. If r be the resistance of the circuit, the current in it is a z lfat/2r and the disc is a very simple form of constant current generator. Fig. 87 represents a circuit a part of which consists of a rigid wire free to turn in the air about the axis of a magnet. This wire makes elec trical contact, by means of brushes, with the magnet at its mid-section and with a conductor which forms an extension of the axis of the magnet. If the wire be rotated with uniform angular velocity <o, and if m be the strength of one pole of the magnet, the electromotive force induced in the circuit will be 2 m<D. If a thin coil (Fig. 88) closely embracing a magnet be suddenly slipped from one position to another, the electromotive force induced in the coil is proportional to the amount of induction which emerges from the surface of the magnet between the two positions. 84. Superficial Induced Currents. Although a mathematical treatment of the currents induced in a massive conductor of any form, in a magnetic field varying in a given manner, is beyond the scope of this elementary text-book, we may give a very simple proof (taken essentially from Prof. J. J. Q N ^ | s ) Thomson s admirable Elements of Elec tricity and Magnetism) of the fact that the currents due to a sudden, finite change in the field lie at the first instant wholly on the surface of the conductor. Let n linear circuits, the resistances of which are r lt r 2 , r 3 , ., and the self-inductances L lt L. 2 , L 3 , -, lie near each other in a magnetic field so that the coefficient of mutual induction of FlG 300 CURRENT INDUCTION. the ith and jth circuits is M tj . Let the flux of the external field through the circuits be N 19 N^ JV 3 , -, and assume that the currents are originally zero and that no one of the cir cuits contains any battery or other generator. If, then, the external field experiences a finite change during the extremely short time interval r and thereafter remains constant, the flux through the kih conductor becomes changed from N k to N k . Transient currents, C 19 C z , C 3 , , flow through the circuits and at the end of the time r attain the values Ci, (7 2 , C 8 j -. During the given interval we have in the first conductor, which will serve as a general example, (M lk - D t Ct) + D t N, + T& = 0, and if this be integrated with respect to the time between and T, the last term of the result will be less than ^C /r, which is negligible, so that the result may be written in the form LI Ci + S (M lk C k ) + -ZVi = NI. The second member repre sents the whole induction flux through the first circuit before the change and the first member the whole flux at the end of the time r, so that the currents generated by the sudden change in the field are such as to keep unchanged the whole flux. Imagine a compact mass of metal divided into such cir cuits as we have just considered and it will be evident that the flux through every circuit in the metal is the same just after the sudden change in the field as it was before. The work done in carrying a magnetic pole about any closed path in the metal is unaltered by the change: it is zero before the change and zero after. No such path can enclose any current filament and, therefore, all the induced currents are initially on the surface, though afterwards transient currents are excited within the metal. It is easy to infer from this that, if the external magnetic field is a very rapidly alternat ing one, the induced currents never penetrate very far into the mass of the conductor. For references to the literature of this important subject, the student may consult Winkelmann s CURRENT INDUCTION. 301 Handbuch der Physik, Vol. Ill, p. 403. Various problems are discussed at length in J. J. Thomson s Recent Researches in Electricity and Magnetism. We shall confine our attention in the three sections which follow to circuits made up of long slender conductors like wires. 85. Variable Currents in Single Circuits. When a simple inductive circuit of resistance r, containing a constant elec tromotive force E, is suddenly closed, the current in the circuit grows gradually in strength and in a short time prac tically attains a maximum value C = E/r, after which it remains constant. While the current is increasing in inten sity, the electromagnetic energy in the surrounding medium if there are no permanent magnets and no other currents in the neighborhood increases also from zero to %LC 2 , and electrostatic charges are established which account for the electrostatic potential differences in the conductors which make up the circuit. After the current has attained the value CQ the energy (C^E watts or C E- 10 r ergs per second) given up to the circuit by the generator in it is used in heating the conductors in the circuit, and EC = C Q 2 r. Before the current C has become steady CE is only a fraction of C Q E, and the rate C 2 r, at which energy is used in heating the cir cuit, is a still smaller fraction of C 2 ?-; hence CE (7V is positive, and in the time interval dt the energy (CE C 2 r)dt joules is used partly in increasing by dw the energy of the electrostatic distribution on the surface of the conductors and elsewhere, and partly in increasing by d (^ LC 2 ) or LC dC the electrokinetic energy in the medium. Unless something iu the nature of a condenser is attached to the circuit, dw is usually of no practical importance, and we may write (CE- C 2 r)dt = LC-dC, or Cr = E - L D t C, or L.D t C+Cr = E, or C = E/r + A -e~ rt/L . 302 CURRENT INDUCTION. It appears from the equation Cr = E L D t C that the counter-electromotive force cannot be greater than E while the current is positive ; D t C> therefore, is not greater than E I L and, unless L = 0, the current cannot jump at the instant to a finite value. We must assume, then, that C = when t = 0, so that C = E(l - e~ t/r ) /r, where r = L/r. The quantity (1 - e~ f /T ) has the values 0, 0.3935, 0.6321, 0.7769, 0.8647, 0.9179, 0.9502, 0.9817, 0.9933, 0.9975, 0.9991 when the ratio of t to T has the values 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0. The difference C - C or - Ee~ t/T /r, which we may call the induced current, has the value C at the beginning and sinks to 1/eth of this value in T seconds, which is sometimes called the relaxation time of the circuit. The induced electromotive force has the value Je~ t/T and becomes insignificant in a short time after the circuit is closed. The between and oo, of FIG. 90. integral, with respect to the time the induced current, is EL/ r 2 . If, now, the electromotive force in the circuit be suddenly changed to E , we have at any time t seconds after the change E C-dt= C 2 r-dt + LC-dC or C = E / r + (E E )e~ t/T /r. The induced current is now the second term in this expression for C and the induced electromotive force is never larger than E- E . The quantity e~ t/T has the values 1, 0.6065, 0.3679, 0.2231, 0.1353, 0.0821, 0.0498, 0.0183, 0.0067, 0.0025, 0.0009 when the ratio of t to T has the values 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0. It is to be noted that if L is expressed in terms of the practical unit (the henry) of self- induction, which is equal to 10 9 absolute units, r must be measured in terms of the ohm, which is equal to 10 absolute units of resistance. The relaxation time of a circuit, which is sometimes called also its time constant, is usually a fraction CURRENT INDUCTION. 303 of a second. The ordinates of the curve in Fig. 90 represent the strength of the current in the circuit just described, on the assumption that the electromotive force is kept constant for 5 T seconds after the circuit is closed and is then suddenly annihilated. If, starting with no current in the circuit, the electromotive force have the constant value E$ for the time interval a, then the value zero during the interval b, then the value JE again during an interval a, then the value zero during an interval I, and so on, and if we denote e~ a /T and e~ b ir by a and /?, the current at the end of the ?ith period of interruption, n (a H- b) seconds from the beginning, will be E j3(l - a) (1 + aft + a 2 /3 2 + + a 1 ^- 1 ) A> and the limit of this, as n increases, is Starting with this value (7 , the current during the next period a, while the electromotive force is equal to E Qt would be E (l - e~ t/r )/r + C e~ t/r , and during the next interval b, when the electromotive force is zero, E (l - a)e-" T /r + C (t ae~ t/T . At the end of this interval the current is again <7 and the state is final. If E in the equation L D t C + r C = E is a given function of the time, L- C=e- t/T (A + Ce" T - E-dt). If the resistance of an inductive circuit containing a con stant electromotive force E and carrying a steady current CQ = E/r be suddenly changed from r to r , we have at any time after the change EC-dt = CV dt + LC dC, or, if C = E/r , C= C +(C - C )e~ r t/L . The induced electro motive force is now r (C C )e- r>t/L , and if r 1 is large this 304 CURRENT INDUCTION. may be at first enormous. Although it is very difficult in practice to increase the resistance of a circuit thus instan taneously, the rate of change in ; may easily be made very rapid, and the spark which is often visible when a circuit is broken bears witness to the fact that the induced electromotive force is sometimes large. If the terminals of a battery of internal resistance r and electromotive force E be connected by a _|_ coil of resistance r l and self-inductance L lt 9 : \ in parallel with a non-inductive resistance J J r 2 (Fig. 91), and if (7, C lt C 2 represent the FIG 91 strengths of the currents in the battery and in the two branches of the external surface respectively, If the value of C 2 from the equation before the last be substituted in the last equation, we get r 2 ) = Er z /(r + r a ), where H = rr^ + rr 2 + r^, so that C v = Er^/ R + A e~ kt , where k = R/L l (r + r a ). C 2 = (E - C,r) / (r + r 2 ). If the main circuit be suddenly closed when t 0, we have If, after the circuit has been closed for some time and (7, has attained the value B, the battery be suddenly detached, C l and C 2 become suddenly equal numerically, and d = Be~ mi , where m=(r l + r 2 ) /L lf CURRENT INDUCTION. 305 If the poles of a constant battery of resistance b are con nected by two coils in parallel (Fig. 92) which have resist ances r D r. 2 and self-inductances L lt L 2 , we have b-C\ +(i + >x) C, + b-C, = E, L. 2 .D t C,+(b + r,) a - E, or b-C l +(L. 2 -D t r,) C, = E. FIG. 92. If we perform the operations (L 2 - D t -+- b -+- r 2 ) and 6 upon the two equations respectively, and subtract one result from the other, we shall get the equation + r 2 ) + L, (b + L, - L, - D?C\ + whence C l = r 2 E/ (^ + br. 2 + /V 2) -h ^ e A -h where A and /t are the roots of the quadratic + [A (6 + n) + , (6 + 7^)] x + (br, + = 0. Fig. 93 represents a Wheatstone s Net which has self- inductance in all members except that which contains the cell. Using, as far as it goes, the notation of Section 73, let us call the coefficients of self- induction of the branches which have the resistances p, q, ?, s, g ; L p , L q) L r , L s , L g respectively. Let ps = qr, so that, when the current has become steady, there is no flow through g, while the current P O EEE C(q + 8)/(p + q + r + s) flows through p and r, and the current FIG. 93. 306 CURRENT INDUCTION. through q and s. If, now, the branch b be suddenly broken, transient currents C p , C ffl C r , C s , C gJ which have the initial values P , Q QJ P w Q w zero respectively, and the final value zero, will flow through the members of the rest of the net. KirchhofPs Laws give at every instant p C, + Lr.D t C p + ff.C 9 + L g .V t C g -q-C q - L q .D t C q = 0, T. C r + L r -D t C r -s.C s - L s -D t C s -g-C g - L g -D t C g = 0, If we multiply each of these equations by dt, integrate between t = and t = oo, and write P = = - Cc q -dt, R= Cc r -dt = */0 */0 G= f*C g .dt, /0 we shall get the equations (p + q)P + P - R - G = 0. Whence, or, since ps = qr, c = Cps(L p /p - L r /r + L./s - L Q /g) g (p + q + r + s) + (p 4- q) (r + s) If Z g and i s are both zero (Fig. 94), it is possible to choose r and^> subject to the condition ps = qr, so that there shall be no transient current through g, and in this case L tt /L r =p/r. If L qt L r , L s are all zero, and if the steady current C and FIG. 94. the quantity G be measured, L p can CURRENT INDUCTION. 307 be found. This method of determining coefficients of self- induction is described at length by Lord Rayleigh in the Philosophical Transactions for 1882. If at the time t the positive plate of a condenser of capacity A", which is being charged by a battery of constant electromo tive force E (Fig. 95), has a charge Q-, if r is the resistance of the "circuit," L its coefficient of self-induction, and C = D t Q, the charging current, we have E- Q/K- L-D t C=rC or L -D?Q + r . D t Q + Q/K= E. The general solution of this equation for Q is the sum of any special solution (for instance, KE) and the general solution of the equation formed by equating the first member to zero. If. therefore, X l = - r/2 L -f R and X, = - r/2 L - E, where E 2 = r 2 / 4t L 2 1 1 KL, the solution required is of the form KE -f- ae* lt H- be**, where a and b are constants to be determined from the initial conditions. If the absolute value of the quantity under the radical sign in the expressions for X l and Xo taken positive, whatever its real sign may be is ra 2 , the value of the radi cal will be m or mi according as r 2 is greater or less than 4:L/K. If at the time zero, when Q =Q 0) the circuit be suddenly closed, - KE) (X, e^ -A , The current has the value XjX, ( Q - KE) (e^ f - e&) / (X, - X^, and if X l and X> are real, it has the same sign for all values of t. If, however, X l and X 2 are imaginary, the expression given above for Q may be more conveniently written in the formA^-f(^ - KE)e~ rt/ - L (cos mt + r/2Lm- sin w),and the sign of the second term is alternately positive for ir/m seconds and negative for TT /m seconds, so that the current is sometimes positive and sometimes negative. The curves 308 CURRENT INDUCTION. in Fig. 96 exhibit Q and C in terms of t in a case where r 2 > &L/K, QQ = 0, and the condenser is being charged; the curves in Fig. 97 correspond to a case where E = and the condenser is discharging FIG. 96. FIG. 97. itself through the circuit. In each case the absolute value of the current starts at zero, attains a maximum, and then FIG. 98. decreases. Fig. 98 shows Q in terms of t when E = and the condenser is discharging itself through a circuit such CURRENT INDUCTION. 309 that r 2 < L/K) the curve, the ordinates of which are EK minus the ordinates of this curve, shows Q at any time while the condenser is being charged by the battery. The shape of the curve may be seen by looking at Fig. 98 through the back of the leaf and upside down. If we differentiate the equation E Q/KL-D t C = rC with respect to t, we get L D t -C + r D t C + C/K= D t E = 0, and we might determine C directly from this last equation. If a condenser of capacity K, originally charged to poten tial QQ/KJ be discharged through a circuit (Fig. 99) which consists of a non-inductive resistance r l and an inductive resistance ?* 2 , arranged in mul- X tiple arc, and if the currents at the time t through the branches of the external circuit [I] I be Ci and C. 2 respectively, C l + C 2 = - D t Q. I _ If we take into account the induced elec- FIG. 99. tromotive force, we may apply Kirchhoff s Laws directly to this circuit and learn that Q / K C^ = 0, and that or Q/K+LfD t (D t Q + (7,) + r. 2 (D t Q + C,) = 0. If the values C\ = Q / Kr D t C l = D t Q / Ki\, obtained from the first of these equations, be substituted in the last one, it becomes L, D?Q + D t Q(L^/Kr^ + r,) + Qfa + r^/Kr^ = 0, and the solution of this is of the form ae M -f be* , where X and /A are the roots of the equation L#? + (Lt/Kri + ?- 2 ) x + (>-! + r a ) / Kr, = 0. After a and b have been determined in accordance with the given conditions, C l and C z can be found directly. The equa tion <7 2 r 2 = Q/K L<D t C z shows that, if C 2 is positive, D t Ci cannot be greater than Q / KL, and that C 2 cannot jump suddenly from zero to a finite value as soon as the condenser circuit is closed: the initial value of (7 2 is therefore zero, 310 CURRENT INDUCTION. while that of Ci is Q /Kr l} and under the conditions of this problem Q = Qo [O^n + 1) eP - (XKr, + 1) e^/Kr, (/x - X), /. - X), \n - X). If X and /x are real, Ci decreases from the value Q / Kr^ to zero, Ca starts at zero, increases (accompanied by a self-induced counter-electromotive force L 2 - D t C 2 , so that <7 2 r 2 < Cft) until it attains a maximum at the time = (logX-lo g/ x)/(>-X), at which time D t C z vanishes and C^ = C z r 2 , and then con tinually decreases, accompanied by a self-induced positive electromotive force, so that <7 2 r 2 > Cfo If we integrate C\ with respect to the time from t = to 2 = oo , and remember that X + it = - (Lt + JTr^j,) /^^"n, and that we shall obtain the whole flow Q Q r 2 / (r, -f r a ) through r p This is the same (whether or not X and p are real) as if r z had no self -inductance ; but if the circuit be broken before the dis charge is complete, a greater portion of the electricity will have gone through r x than would be the case if L z were zero. If the condenser connections have a considerable resistance b, the differential equation becomes Kbr + br + If the resistance of a circuit made up of a generator of electromotive force E=f(t\ a condenser of capacity k and necessary leads, is r, if its self-inductance is i, and if we CURRENT INDUCTION. 311 denote D t E by E\ we have L D?C + r D t C + C/K= E . If R = V^/f 2 - 4 ,fir, and a = (r/f -R)/2 LK, (3 = (rK + 7?) /2 LK, the general solution of this equation is C = (K/K) (eP Cerf" - E dt - e at Ce- at JE dt) If the poles of a battery of constant electromotive force E and internal resistance b are connected by a coil of resistance i\ and self-inductance j in parallel with a condenser of capacity (Fig. 100), we have L D t C l -f (b -f- 1\) C l 4- b C 2 = and E- Q/K=bC l +(b + r a )C a , or 6 D,C, +(b + r 9 )D,C, -f CJK = 0. If we perform on the first and last of these equations the operations [(b + r 2 )D t + I/ A"] and b respectively, and sub tract one result from the other, we shall learn that r + br. + > and that C^ is the sum oi E /(b + r^ and the general solution of the equation formed by putting the first number equal to zero. If the arms p and r in the Wheatstone Net contain con densers of capacity K p , K r respectively, the steady current through q and s will be C = E / (b -f q + s), and the charges of the condensers will be CqK p and CsK r . If, now, the bat tery circuit be suddenly broken, transient currents will appear 312 CURRENT INDUCTION. in the remaining members of the net and the condensers will be discharged. The whole flow through p will be CqK p , and that through r will be CsK r . At any instant during the discharge C p = C r + C gJ and if we multiply this equation by dt and integrate between and oo, it will appear that the whole flow through g is C(sK r qK p ). This will be zero 86. Alternate Currents in Single Circuits, In many prac tical applications of electricity it is necessary to deal with inductive circuits which contain periodic electromotive forces. In the simplest case the electromotive force is harmonic of / \ FIG. 101. the form E m sin (pt a), or the form E m cos (pt a) ; the amplitude is then E Q ; the periodj T = 2?r /p\ the frequency, n = p/2 TT ; and the phase lag, a. Two harmonic electromotive forces, of the same period, A - sin (pt a), B sin (pt /3), which conspire in a simple cir cuit, are equivalent to a single simple harmonic electromotive force C sin (pt - y), where C 2 = A 2 + B 2 + 2 AB cos (a - ft) and tan y = (A sin a + B sin /?) /(A cos a + B cos /?). If a parallelogram be constructed with adjacent sides equivalent on any scale to A and B, and with the included angle equal to (a /?), a diagonal of the parallelogram will represent C, and the angles which this diagonal makes with adjacent sides CURRENT INDUCTION. 313 of the parallelogram will be equal to (a y) and (y (3) respectively. If, starting at the time t = from the position P , a point P be made to move uniformly with angular velocity^? in counter clockwise direction around the circumference of a circle with centre and radius E m , if Q be a fixed point in the plane of the circumference such that P OQ = a, and if y be any straight line in the plane perpendicular to OQ, the projec tions of OP on OQ and on y will be equal, at any time , to FIG. 102. E m cos (pt a) and E m sin(pt a) respectively. If OQ be used as an axis of real quantities, QOP will represent the argument, and the length of OP the modulus of the complex quantity E m - e (pt ~ a)i ; the real part and the real factor of the imaginary part of this quantity will be represented by the projections of OP on OQ and on y. If while P is moving in the circumference, y moves parallel to itself away from O with constant velocity a /Tor ap/2ir, the projection of P upon y will trace out a sinusoid (Fig. 101) the length of the base of which is a. We shall frequently find 314 CURRENT INDUCTION. it convenient to imagine diagrams as generated in this way. If in Fig. 102 the lines OA, OB, OC revolve uniformly in the plane of the diagram about with angular velocity^, if the angle AOB /?, and if the lengths OA, OB are equal to the amplitudes of two simple harmonic quantities a = a m -sinpt, b = b m - sin(^ + )3), the projections of A, B, and C on y give the curves a, b, and c ; every ordinate of the sinusoid c is the sum of the corresponding ordinates of the sinusoids a and b. If in Fig. 103 the independent lines OA, OB, OC revolve about in the plane of the diagram with the same constant r D angular velocity p, the lengths of the pro jections of these lines upon any fixed line (z) in the plane will represent harmonic quantities of the same frequency (p/2tr), but with phase differences equal to the angles between the lines projected. The sum of these harmonic quantities may be represented by the projection upon z of QD, which is equivalent to the geometric sum of OA, OB, OC, if QD revolve about Q with angular velocity p, starting to move at the same time with the original lines. If a circuit s which has a resistance r and a self-induct ance L contains an electromotive force E m cos pt, we have L D t C + rC = E m cospt, if the capacity of the circuit is neg ligible. The complete solution of this equation is the sum of any special solution and the complete solution, Ae~ rt/L , of the equation formed by writing the first member equal to zero. To find the special solution needed, we may consider first the equation L - D t C + r C = E m (cos pt + i siupt) = E m e pti , which is in some respects simpler; if any solution of this new equation has the form u 4- vi, u is a solution of the given equation. Since the first member of the new equation is linear in terms of (7 and D t C, and since D t e pti = pie pti , it is clear that a spe cial solution of this equation, of the form Be pti , must exist. CURRENT INDUCTION. 315 Substituting this form in the equation, to determine , we learn that the solution is E m e^/(r + Lpi), or E m (r - Lpi)e*"/(i* + of which the real part is E m (r cos pt + Lp sin pt) / (r 3 + L*p*), or = cos (pt a), where tan a = Lp/r. The current in s is, therefore, E m cos(> - a)/ Vr 2 but after a comparatively short time the first term becomes negligible, and then the current becomes harmonic with the same period, 2ir/p, and the same frequency, p/2 TT, as the electromotive force, but with a retardation in phase of a. The amplitude is ^ OT / Vr 2 + y. The radical Vr 2 + iy = Z is called the impedance of the cir cuit and Lp = x its reactance, or inductive resistance, under the given circumstances; the self-induction of the circuit reduces the amplitude of the current in the ratio of r to Z. The relation between the elec tromotive force and the current strength may be represented by corresponding ordinates of two curves like those shown in Fig. 104. The counter-electromotive force of self-induction, sometimes called the back electromotive force of self-induction, is equal to L-D t C,OT P z m . cos (pt - a - i TT) ; it lags 90 behind the current in phase. The electromotive force necessary to over come self-induction is the opposite of this ; it has the same numerical value, but its phase is 90 in advance of that of the FIG. 104. 316 CURRENT INDUCTION. FIG. ]05. current. If we denote the amplitude, E m / Z, of the current by (7 m , the amplitude of the electromotive force necessary to over come self-induction will be LpC m . Cr is called the apparent electromotive force or the instantaneous energy component of the electromotive force; its ampli tude is rC m . The amplitude of the applied electromotive force J m cospt is ZC m . If a right triangle be drawn (Fig. 105) the legs of which represent r and Lp on any scale, the hypotenuse will represent Zou the same scale and the angle between the r and Z sides will be a ; this triangle is the triangle of resistances. A triangle OPQ (Fig. 106) similar to this, the sides of which are equal to rC m , LpC m , and ZC m , may be called the triangle of electromotive forces. If the figure OQPR be made to rotate positively about with constant angular velocity p, the projections at any time of OQ, OP, OR upon any line in the plane of the diagram parallel to the original position of OP will give the electromotive forces at that instant. The activity o? the energy spent in the circuit, during any time interval, at the expense of the generator is the time integral, taken over that interval, of EC = E n ? - cos pt cos (pt a) / Z. The mean value of the activity for any num ber of whole periods is fi m 2 r/2(r 2 -f L*p*) t and this is the same as if a steady current of intensity EJ A/2 (r 2 + zy 2 ) had passed through the circuit during the interval; for this reason E m / A/2 (r 2 + L 2 p* ) is said O <? to be the virtual or effective current. The FIG. 106. mean values for any number of whole periods of the current and of the square of the current are zero and 2 I */2Z*; the effective current is, therefore, the square root of the mean square of the current, and this is sometimes called the quadratic mean. The effective applied CURRENT INDUCTION. 317 electromotive force is E m / V2, and the effective apparent elec tromotive force is E m r/^/2-Z. The apparent electromotive force would yield the current C if applied to a circuit of ohmic resistance r and inductive resistance zero. The activity, or "power in the circuit," is equal for any number of whole periods to the product of the effective current and the effective apparent electromotive force. For this reason the effective apparent electromotive force is frequently called the effec tive energy component of the electromotive force. The first term E m 2 r-cos 2 (pt a)/Z 2 of the second member of the equation EC = C 2 r + LC D ( C shows the rate at which heat is being dissipated in the circuit ; the second term, E m *Lp sin (pt a) cos (pt a)/Z 2 , the rate at which power is used in increasing the energy of the electromagnetic field. It is evident that the average value of this last quantity for any number of whole periods is zero. The effective impressed electromotive force is often called simply "the electromotive force." Such voltmeters and ammeters as are commonly used in alternating circuits usually indicate effective electromotive forces and currents ; their read ings must be multiplied by V2 to obtain the maximum values of these quantities. It is often convenient, as Prof. C. A. Adams has pointed out, to regard the values, at any instant of the impressed elec tromotive force and of the current, as the projections, on the real axis, of the radii vectores which join the origin to the two points on the complex plane which represent at that instant the quantities E m e pti , E m e pti / (r + Lpi). This last expres sion is the simple solution already found for the differential equation L D t C + r C = E m e pti . If in the problem just considered we reckon the time from an epoch J T earlier, we shall have E = E m -sm pt, C = E m -sw(pt - a)/Z , 318 CUKKENT INDUCTION. these quantities may be regarded as the projections on the axis of imaginaries of the moduli of E m - e pti and E m e pti / (r + Lpi). The quantity (r -f- Lpi) has been called the complex impedance, but some writers give this name to r Lpi. If a linear plane circuit of area A, resistance r, and self- inductance L, in a uniform magnetic field in air of intensity H, be made to rotate about an axis perpendicular to the lines of the field with angular velocity p, and if at the time t the plane of the circuit is parallel to the field, the flux of the field through the coil at the time t is AH sin pt, and the current C satisfies the equation L- D t C + Cr pAH cos pt, so that after a few seconds C = HAp cos (pt a) / Z. The whole flow of electricity through the circuit during a positive half revolution is 2HA/Z. The mechanical action between the circuit and the field is equivalent to a couple the moment of which is C times the rate of change with respect to pt of the flux AH sin pt through the coil. This moment is CHA cos pt, or H 2 A^p cos pt (cos pt a)/Z, its average value is H 2 A 2 pr/2Z 2 , and the work done against it in a single revolution is H^A^r-n-p / Z*. External work must be done to turn the coil against the resistance of this couple, and the equivalent of this work is all used in heating the circuit. If the rate of rotation is so rapid that the ratio of r to Lp is small, a is nearly equal to -J- rr, and C is nearly equal to HAsmpt/L , CL is the flux through the circuit of the lines of its own field, HA sin pt is the corresponding flux of the lines of the external field, and in this case the sum of the two is nearly zero. If two points A and B in an inductive circuit be joined by an additional (non-inductive) conductor which carries an elec tromotive force of such a value at every instant and so directed that no current ever passes through the extra conductor, the electromotive force may be taken as a measure of the differ ence of potential between A and B. If between A and B in a CURRENT INDUCTION. 319 simple circuit which carries the current C m sin (pt a) there is an ohinic resistance r and a self-inductance L, the difference of potential between the points is evidently where tan 8 = (r sin a Lp cos a) / (r cos a + Lp sin a). If the terminals of an alternating current voltmeter were attached to A and B, the instrument would measure C m Z / V& If a circuit which carries a current C m eospt contains three coils in series which have resistances ?* 15 r 2 , r 3 and inductances LU Lo, L s , we may lay off on a horizontal line in succession (Fig. 107) the lengths 0.4 = r 1 C aa AS = r a C m> SQ = r t C m . Erect at Q a vertical line and lay off on it the lengths QD = L l pC na DF=L z pC na FP = L 3 pC m . Then OP will represent the amplitude of the difference of potential between and P, and QOP will be the angle of advance of its phase over that of the current. The lines , b, c rep resent similarly the amplitudes of the differences of potential of the ends of the separate coils, and the angles which these lines make with the horizontal the phase differences FIG. 107. between these potential differences and the current. Starting at the time zero, let the triangle OQP revolve about O, in the plane of the diagram, with con stant angular velocity p, and let the initial position of OQ be denoted by OQ . Let the points of intersection of the lines a, b and &, c be denoted by G and H, and the projections of A, B, Q, D, F, P, G, J7 upon OQ by corresponding accented letters ; then the lengths at any time of the lines OG\ G lT, H P represent the instantaneous values of the " electromotive forces " applied to the several coils, and the lengths of OA , A B , B Q the corresponding apparent electromotive forces. 320 CURRENT INDUCTION. If the terminals of a generator of electromotive force E m -cospt and internal resistance r are connected by two conductors in parallel (Fig. 108) of resist- ance r^ r 2 and of self-inductance L l} L 2 respectively, FIG. 108. If r is negligible, rC, + (r + r 2 ) C 2 - E m . cos j*. = E m cos ( jp* - ai ) / Vn 2 4- Ay = A cos and = - cos = cos - ttl ), - a 2 ), - a 2 ) / Vr 2 2 + where tan c^ = Lip/r lt tan a 2 = L 2 p/r. 2 . Ct+ C 2 = C m COS(pt-a), where C w 8 = ^ 2 + B* + 2 J7? cos ( ai - a 2 ) and tan a = (A-sin.a l + I> sin a 2 ) / (A cos a t 4- B - cos a 2 ). If in Fig. 109, OP = E m and QOP = a l , OQ = Ar v QP = A^p, and A can be represented by a length laid off from on OQ. A similar construction, represented by the dotted lines, may be made for B. The diagonal OR of the par allelogram, two sides of which are the lines which represent A and B, represents C m . If OR cuts the semi- circumference in 6r, OG represents the product of C m and the resistance of the divided circuit. If a simple harmonic difference of potential E m cos pt be applied to two points A and B which are connected by n simple conductors of resistances r lt r 2 , r &) LV L 2 , L 3 ,-", and impedances Z lt Z 2 , Z 3 , the n fractions of the form r k /Z k * be denoted by F and that of the fractions of the foimpL k /Z k 2 by G, the n conductors are 109 - , self-inductances and if the sum of CURRENT INDUCTION. 321 equivalent to a single conductor of resistance R = F / (F~ + G~) and of reactance X = G / (F- -f- 6r). The sum of the currents in all the conductors is E m cos (pt a) / ^/ R 2 + X 2 , where tan a = X/R. If a non-inductive circuit of resistance r containing a con denser of capacity k and a generator of electromotive force E = E m -sin2 }t be suddenly closed at the time t = 0, and if Q is the charge on the positive plate of the condenser at the time t, E Q/k = rC, or, since C = D t Q, r-D t C + C/k=pE m -cospt. From this it follows that C = Ae~" * + E m sin (pt + /?) / V/- 2 + ra 2 , and Q = B- Arke~ t/rk + E m sin (pt + /3 - % where m = 1/pk, and tan ft = The exponential terms soon become negligible, and if we assume that Q is zero at the outset, we shall have eventually C = E m sin (pt + ft) / V>~ + m 2 , or pE m k - cos(pt 8) / VI + k 2 p 2 r, where tan 8 =prk; Q = E m - sin Here the phase of the current is in advance of that of the applied electromotive force E by the angle ft and in advance of Q by 90. The electromotive forces o of the condenser and generator conspire in direction when pt lies between nv and nir -f- a, where n is any integer and a = 90 ft ; these electromotive forces are opposed when pt lies between n-n- -f a and (n + 1) ir- The electromotive force (Q/k) necessary to overcome that of the condenser lags behind that of the generator by a. The apparent electromotive force FIG. 110. 322 CURRENT INDUCTION. is rC. The sum of the squares of the amplitudes (rC m and in C m ) of r C and Q /k is 7 m 2 , the square of the amplitude of E\ if, therefore, we draw a right triangle of which rC m and w(7 m are legs, the hypotenuse will be equal to E m , and the angle adjacent to the first leg will be /?. If such a triangle OPQ (Fig. 110) be made to rotate in counter clockwise direction, with constant angu lar velocity p about 0, the projections of OQ, OP, and OR upon any line per- FIG. 111. pendicular to the initial position of OP will give the apparent electromotive force, the applied electromotive force, and the electromotive force necessary to overcome that of the condenser. If a condenser of capacity ^ furnished with leads of resist ance TI be joined in parallel (Fig. Ill) with a condenser of capacity k 2 furnished with leads of resistance r z , and if the compound condenser thus formed be connected up with a generator of internal resistance r and electromotive force E m - sin pt, we have (r + r$D t Ci + r D t C, + C,/ k, = pE m . cos pt, If r = 0, we have eventually cos (pt - ai ) / Vl + A^ V, <7 2 = pEJt* COS (^ - a 2 ) / Vl + where tan a! = pr-Jc^ tan 0% = pr z k z . , If a circuit of resistance r contains (Fig. 112) /N lyl L a generator of electromotive force E m sin _^#, ^r^ - k a coil of self-inductance L, and a condenser of VTP 112 capacity k in series, and if Q is the charge on the positive plate of the condenser at the time t, C = D t Q and E m >s\-o.pt Q/k- L.D t C= Cr, or L-D?C+r-D t C+ C/k =pE m -co$pt. CURRENT INDUCTION. 323 The real part of any solution of the equation (and one evidently exists of the form Be pti ) will be a spe cial solution of the equation just formed. It is easy to find B by substituting Be pti in the new equation, and to prove that E m sin (pt a) / R, where R 2 = r 2 + (pL 1/fcp) 2 and tan a = Qj 2 kL 1) /pkr, is the result required. To obtain the complete solution of the equation for C we should need to add to this special solution the complete solution (found in the last section) of the equation formed by writing the first member equal to zero; this solution is exponential in form, with negative indices increasing in absolute value with the time, so that after a few seconds the current may be repre sented by the equation C = E m sin ( pt a) / R. It is to be noticed that the capacity of the condenser tends to offset in some respects the effect of the self-induction of the coil. Since R 2 = r 2 +p 2 (L-l/p*k) z and tan a =p(L - l/p*k)/r, it is clear that the current in the circuit is the same as if the con denser were removed and the self-inductance decreased by l/p*k. The maximum current is obtained when both self- inductance and capacity are absent, or when both are present and such that Lkp 2 = 1. If Q = $ when C has its maximum value, the difference of potential (Q/k) between the plates of the condenser is Q Q /k E m cos(pt a)/pRk, and if the denominator of the harmonic term is less than unity, this term will have an amplitude greater than that of the impressed force. If we make k infinite in these expressions, they become applicable to the case of a simple inductive circuit containing no condenser. The radical R, which is called the impedance of the circuit, becomes V?~ + L-p 2 when k is infinite. When an inductive circuit contains a generator of electro motive force E m sin pt and an electrolytic cell with polariz- able electrodes, we may assume that when the frequency is 324 CURRENT INDUCTION. fairly large the counter-electromotive force in the cell at any instant is approximately equal to 1/Jc times the quantity of electricity which has passed through the cell in the direction which the current then has, since the last reversal. On this hypothesis the cell acts like a condenser ; k depends only upon the electrolyte used and upon the size J l_J I B ? and material of the electrodes. Experiment * shows that if similar platinum electrodes of moderate size be used, the capacity, per square FIG 113 millimetre of the surface of either electrode, will be about 0.049, 0.089, 0.183, 0.049 micro farads, according as the electrolyte is a dilute solution in water of K 2 SO^ KCl, KBr, or KL If between A and B in a simple circuit (Fig. 113) which carries the current C = C m sin (pt a) there is a resistance r, a self-inductance L, and a condenser of capacity k in series with the self-inductance, the difference of potential between these two points is rC -f L D t C + Q/k. If Q = Q when C = C m , this is Q Q /k + C m - Vr 2 + (Lp - 1 /pk)* sin (pt - 8), rp sin a + (1 / k Lp*) cos a where tan 8 = rp cos a + (Lp* 1 /k) sin a If the ends of a coil of resistance i\ and self-inductance LI, which is joined up with a generator of resistance r and electro motive force E m sin pt, be connected by E j - leads of resistance r 2 with the terminals /J\ of a condenser of capacity k 2 , the coil and ^p the condenser are in parallel (Fig. 114), and L l D t C l + (r + r x ) C l + rC 2 = E t d + (r + r a )D t C 9 -f C 2 /k 2 =pE m cos pt. CURRENT INDUCTION. 325 If r is negligible, Ci - E M sin (pt - a)/ V/V 2 + Lfp* and C 2 = pEJtt cos (pt -ft)/ Vl+AV^V, where tan a = L^p/r^ and tan /? = p^ky. In many practical problems r, is extremely small, so that ft is negligible. If the terminals of a generator of electromotive force E = E m sin jit, of self -inductance L, and of resistance r, be connected (Fig. 115) to the ends of a coil of resistance r x and self-inductance L and if the coil ends are attached by leads of resistance r 2 to the coatings of a condenser of capacity & 2 , we have FlG - [(L + ,) A + (r + r,)] C, + (L D t + r) C 2 = ^ M sin ^, [.!>,* + r- DJQ -h [i 7), 2 -h (r H- - 2 ) A + 1 / A- J C 2 = 7?^ cos jtf. If ^ = 0, we have the case last considered. If the terminals of a generator of resistance r and elec- . - _ p - tromotive force E m sin _p< are connected (\\ ~ I I ( Fi g- H6) by two conductors in parallel ^rJ k L 1 having resistances r lt n, capacities k^ k 2 , I - : ^I and self-inductances L u L 2 respectively, FIG. 110. but no mutual inductance, L, DfC, + (r + rOACi + r - AC 2 + d/^ -^ m cos pt, L, - D/ 2 C 2 + r - AC X + (r + >,) AC 2 + C 2 /A- 2 - ^ m - cos . If we apply the operator \_L. 2 D* + (r +> 2 )A + 1/A-J to the first of these equations and the operator \_r A] to the 326 CURRENT INDUCTION. second and subtract one result from the other, we shall have eliminated C 2 and may solve for C l in the usual manner. In a case which sometimes occurs in practice, r is negligible and C l = H m sin (pt - a,)/ B 19 C 2 = E m . sin (pt - where -#i 2 = n 2 + (PL, - i/k lP y, R* = r? + ( P L 2 - tan aj = (p*k 1 L 1 )/pk 1 r 1J tan a 2 = (p 2 k 2 L 2 l)/pk 2 r 2 . The reader will find the subject of this section fully dis cussed in Bedell and Crehore s Alternating Currents, Franklin and Williamson s Elements of Alternating Currents, Steinmetz s Alternating Current Phenomena, Heaviside s Electrical Papers, and in many other books. 87. Variable and Alternate Currents in Neighboring Cir cuits. If the coefficients of self-induction of two neighboring circuits s lt s 2 > which contain constant generators the elec tromotive forces of which are EI and E 2 respectively, are LI, L 2 , and their coefficient of mutual induction M, if the resistances of the circuits are r lt r 2 , and the currents which pass through them at the time t are C l} C 2 , then L l D t d + M. D t C 2 + T& = EI, M- D t C l + L 2 D t C 2 + r 2 C 2 = E 2 . It is to be noticed that, since the electrokinetic energy or i must always be positive whatever the values or directions of the currents, if they exist at all, L^ 2 M 2 can never be nega tive. If we substitute in the differential equations just found C7/ and C 2 for C l and C 2 , where C, = C, - E./r,, C 2 = C 2 - JE 2 /r 2 , we get CURRENT INDUCTION. 327 or, symbolically written, If we perform the operation (L 2 D + r 2 ) on the first of these equations and the operation (M-D t ) on the second and sub tract one of the resulting equations from the other, we shall eliminate C. 2 and get the homogeneous linear equation (L,L, - J/ 2 ) DfCJ +(r*Li + r^D^ + r^C^ 0. The general solution of this equation is of the form A^ + B^e" , where X and n are the two roots of the equation that is, Lo) ( IV&! + jyL a ) - - 4 2 (L,L. 2 - M-) If we eliminate C/ from the original equations, we shall learn that CV = ^2 eA + B ^ where X and p have the values just given. " Both \ and /x are negative, since Z^ - M is positive, and both are real, since the expression under the radical sign may be written (L& - L^) - + Ir^M ~. The coefficients A v A v BV B, in the expression for CV, C 2 are not all inde pendent, for we find when we substitute these expressions in either of the original equations that the ratios A 2 /A l} B 2 / B must have the fixed values r 2 ) or - (L,\ + and - MI*./ (L^n + r,) or - (i^ + rJ/Mp. respectively. If we denote these ratios by a and /3, we have C, = E, / r/H- A^ + B^, C 2 = E/r 2 + a A^ + QB^ , where X, /A, a, ^8 depend only upon the forms of the circuits and the materials of which they are made and A^ B l are to be 328 CURRENT INDUCTION. determined from the conditions of the particular problem under consideration. If E 2 = (Fig. 117), and if s l be suddenly closed at the time t = 0, C v and C 2 are initially zero, FIG. 117. and A l -\-B l = -E l /r ly ^{1-i^ COv^- l ltt 4-^/3 = 0, 1] where R stands for the radical in the expressions for X and /A. The time integral of <7 2 from to oo is evidently and is the same whichever circuit contains the given electro motive force and is used as the primary. Fig. 118 shows the currents in s l and s 2 under these circumstances, when -JTJ f\ T 1 T __ 1 If E% = 0, and if s lt which has been closed for some time, has its resistance suddenly changed, when t 0, from r to r 15 we have initially C l = E l /r Q and C 2 = 0, so that A, = I3E, (r Q - and C 2 = ^ (a - 0), x - r CURRENT INDUCTION. 329 The integral of C 2 with respect to the time, from to <x>, is E^Ifa fo)/ r o r i r 2> and the limit of this, as -r L grows larger without limit, is E^I/rfa. C 2 attains its maximum value at the time (log X//A)/(/A X) : this fraction approaches zero when r L increases without limit. If t\ is infinite, we have C l = 0, 6 2 = Ae~ r */L 2 : the time integral of C 2 , between and oo, is AL 2 /r 2 , from which we infer that A = E^M/r^L^. If, then, i\ is infinite, that is, if the circuit is suddenly broken, the current in the secondary jumps instantly from zero to the value E^I/r^L.2 and then decreases after the manner shown in Fig. 119, which is drawn to scale on the assumption that in practical units E l = 2 ) L 2 = %, J/= I/ V8, r =l. The whole area between the time axis and the curve which represents the current in s 2 is the same in Fig. 119 as in Fig. 118, though the shapes of the curves are very different. If E 2 = 0, and if E 1 = E m - cos pt, we have FIG. 119. and M- D t C 2 L 2 D t C 2 ?- 1 C 1 = E m cospt rC = 0. If C l = x, C 2 = y represent any special solution of these equations, the complete solution may be found by adding x and y to the values of C l and (7 2 , found earlier in this sec tion, which completely solve the equations formed by equating to zero the first members. These last quantities, however, are exponential in form with indices intrinsically negative ; after a few seconds they are negligible, and we need use only the final forms of x and y. The real parts (?/ 1? i/ 2 ) of any solution, of the form C^ = M X -f v^, C 2 = n. 2 + ?y, of the equations 330 CURRENT INDUCTION. form a solution of the original equations. Applying the opera tor (L 2 D t -f- r 2 ) to the first of these new equations and the operator ( M D t ) to the second and subtracting one result from the other, we get (L,L 2 - M^DfC, + (r,L 2 + r.L^D.C, + an equation which evidently has a solution of the form B e pti , and if we substitute this expression in the equation, we learn that B = % m (r 2 + L 2 pi) I [n The real part (x) of Be pti is, therefore, E m 2 y + r 2 * cos (pt + 8 - ff) where tan 8 = L 2 p/r 2 and tan = (r-gLi + r^) p / [r^ (L r L 2 or x EEE ^4 cos (_p^ a), where, if L = L,- M 2 L 2l r I (L./p 2 + ? 2 2 ) and r = n + M*p>r s /(L 2 *p* + n 2 ), ^4 = E m / ~\ L 2 p- + v- 2 , and tan a = Lp /r. The primary, therefore, behaves like a single circuit (at a distance from all others) of resistance r, greater than r 1? and of self-inductance L, less than L r The presence of the secondary circuit makes the lag in the primary less than it would otherwise be. The corresponding value (?/) of C 2 can be found by substi tuting x in one of the original equations : it is cos (pt -ft)/ -^L/p 2 + r 2 2 , or MpA . cos (pt + TT - ft) / VZ 2 y + r 2 2 , CURRENT INDUCTION. 331 where tan (fi a) = r. 2 /L 2 p. The lag in phase of the secondary circuit behind the primary is TT + a /?, or \ IT + tan" l (L 2 p/r 2 ). The lag of the secondary current behind the electromotive force is tan- 1 [^ (L,L, - M*) - >y- 2 ] /|> (r 2 L, + r^)]. The average rate for any number of whole periods at which the generator furnishes energy to the primary is the average value of E m A cos pt cos (pt a), which is i E m A cos a or E m *r /2 (L*p* + r 2 ) ; this is greater when the secondary is closed than when it is open. The average rate for any whole number of periods at which energy is used in heating the secondary is the average value of C 2 2 ?% or r 2 M 2 p*A 2 /2(L 2 2 p* + r 2 2 ) ; the ratio of this to the power used in the primary is called the efficiency of the transformation and is equal to rJlty/r(Lf]P + r). The electromotive force induced in the secondary is FIG. 120. The problem here considered is in principle that of the alter nate current transformer (Figs. 120 and 121), and it is fre quently the case in practice that the ratio of /;, to L 2 p is very small. Under these circumstances L, r, the amplitude of C 2 , and (3 a are nearly equal to LI - M*/L 2 , r, + rJIP/io 2 , MA/L & and respectively. Both circuits are usually wound on a soft iron core (often a ring) of great permeability, and L-^L Z M* is very small compared with either L^ or L., ; in this case the lag of the primary is negligible, while, for high frequencies, that of the secondary is nearly two right angles. If L^L. 2 J/ 2 is practically nothing, the transformer is said to have no magnetic leakage. The ratio of L^ to L 2 FIG. 121. 332 CURRENT INDUCTION. is usually nearly equal to that of the square of the number of turns (n^/n^) of the circuits on the core, and under these circumstances B is approximately equal to n^A/n^. For exhaustive treatments of the problem of this section, which is of much practical importance, the reader is referred to such books as Fleming s The Alternate Current Transformer; J. J. Thomson s Elements of Electricity and Magnetism ; Nipher s Treatise on Electricity and Magnetism; and Steinmetz s Alter nating Current Phenomena. 88. The General Equations of the Electromagnetic Field. When a fixed, metallic, linear circuit s of specific conductivity X = l/o-, at a uniform temperature throughout, carries an induced current, positive electricity is urged around s in the direction of the current by " something of the nature of an electrostatic field," though we do not need to assume that this is always due to electrostatic charges. If we denote the com ponents of the field, at every point within or without the conductors which form the circuit by X, Y, Z, the line integral of [A cos (x, s) + Y- cos (y, s) + Z-cos (z, )], taken around s in the direction of the current, is the internal electromotive force and is equal to the negative of the time rate of change of the positive flux of magnetic induction through the circuit. If the circuit be covered by a cap S, if n denotes the direction of the normal to S drawn towards the positive side, and if B x , B y , B z are the components of the magnetic induction B, then, on the assumption that Stokes s Theorem may be applied to the vector (X, Y, Z), we shall have = ~ JJ Y) cos(x, n) + (D Z X - D x ^)cos (y, n) + (D X Y- D y X) cos (, 7i) ] dS cos (x, n) -f D t B y cos (y, n) + D t B z -COS(z,ri) ]dS, so that the expression - D z Y+ D t B x ) cos (x, n) + (D Z X - D X Z+ D t B y ) cos(y, n) + (D X Y- D y X+ D t B z ) cos (*, n)] CURRENT INDUCTION. 333 integrated over any cap bounded by s, whatever the forms of the latter, yields zero. We are led to assume, therefore, that at every point within or without any such circuit = D X Y-D V X, [209] and to say that the negative of the vector the components of which are the time derivatives of the component of the induc tion is equal to the curl of the electric field. If , 77, are the components of the curl of the magnetic induction B, and if the components F x , F y , F z of the vector F are defined by the equations 4 irF x = Pot , 4 irF y = Pot 77, 4 irF z = Pot , F is a vector potential function of B. By its aid we can transform the integral - CC[D t B x cos (x, ?i) + D t B y cos (y, n) + D t B z cos (z, n)~\ dS, in which the integrand is the component normal to S of the curl of D t F, into a line integral taken about 5 of the tangen tial component of D t F. We have, therefore, \ \_X- cos (x, s) + Y- cos (y, s) + Z cos (z, s)~\ds = - J"[ A^ cos (x, s) 4- D t F y cos (y, ) + D t F z cos (z, s)] ds, and the integrands can differ only by the tangential com ponent of some lamellar vector (G x , G y , G z ), which adds nothing to the integral taken completely around s. Since this is true whatever the shape of s, we assume that at every point X= D t F x -h G x , Y = D t F v + G y , Z = D t F. + G z . When the magnetic field is constant and the components of D t F vanish, X, Y, Z are equal to - D x F, - D y F, - D t F, and 334 CURRENT INDUCTION. the phenomena will be accounted for if we follow Maxwell and write Z=-D t F z -D e V. [210] The reader should compare these equations with [208]. Within the conductors which form s, the components (u, v, w) of the conduction current (q) satisfy Maxwell s cur rent equations 4 TTU = D V N - D Z M, 4 TTV = D z L - D X N, Ttw = D x M- D y L, [211] where L, M, N are the components of the magnetic field, and u = XX, X = cru, Y=<rv, 2= vw. According to Poisson s hypothesis, a dielectric consists of perfectly conducting molecules separated from each other by perfectly insulating spaces, the specific inductive capacity (K) depending merely upon the ratio of the volumes of the spaces occupied by the molecules and the intervening spaces. From this point of view, there is a transfer of electricity through every molecule when the dielectric is being polarized, one portion of the surface of the molecule becoming positively electrified by induction and another portion negatively elec trified. Every change in the polarization is accompanied by the passage of electricity through the mass of the molecule, and we are to assume that during the change every molecule acts electromagnetically like a current element. Whatever our theory, the appearance of the induced charges which account mathematically for the phenomena observed when a dielectric becomes polarized, involves the displacement of electricity, and corresponding electromagnetic effects. In his famous paper on "A Dynamical Theory of the Electromag netic Field," published in the Philosophical Transactions of the Royal Society in 1864, Maxwell assumed that whenever the polarization of a soft dielectric in which the electric induction has the components x , <,,, & z is being changed, electromagnetic CURRENT INDUCTION. 335 phenomena are to be looked for equivalent to those which would accompany the presence of currents, called displacement currents, in the dielectric denned at each point by the vector or (K-DtX/litj K.D t Y/ir, K- According to this assumption, u 1 = -#,<*>* /4 T H- \X, v = D& y /4 TT + A F, w = D t <S> x /ir + \Z, where u , v , w are the components of the total current, and we may write the current equations in the generalized form 4 TTU = D& x + 4 ,rw = D y X - DM, 4 TTV = D& y + 4 TTV = D Z L - D X N, 4 TTW = D& z + 4 TTIV = DJl - D y L, [212] in which u, r, w represent the components of the conduction current alone. In conductors the displacement currents are negligible, in a perfectly insulating dielectric the conduction currents vanish ; both are supposed to coexist in dielectrics which are slightly conducting. Within a conductor, since the curl of the magnetic force is solenoidal, D x u + D y v + D z w = 0. If at least that portion of the magnetic induction near the current which changes with the time, is induced in soft media, and if /x is the magnetic inductivity at the point (a?, y, z), we have D t B x = ^ D t L, D t B y = /i D t M, D t B z = ^ D t N, and [209] becomes -^D t L = D y Z - D Z Y, - /x D t M= D Z X- D X Z, -^.D t N=D x Y-D y X, [213] or, if the media are homogeneous, - fjt\ D t L = D y w - D z v, - ftX D t M = D z u - D x w, - p.\ - D t N = D x v - D y u. [214] 336 CURRENT INDUCTION. If we differentiate the equations of [212] with respect to t and substitute the values of D t L, D t M, D t N from [214] in the results, we shall get for homogeneous media three equations of the form DfX + 4 TT D t u) = V z u - D x (D x u + D y v + D z w) = V 2 w, that is, /*\(JST. 7> t a JT 4- 4 u - Z> <tt ) = VX M(^- A 2 y + 4 TT - Z> <V ) = vw, A*X(JT- Z>, 2 ^ + 4 TT - D,M;) = V 2 w. [215] Where there is no conduction current these become pK D*X=V*X, nK.D?Y=^Y, pK-D?Z=V*Z. [216] If we substitute in the equations [214] the values of u, v, and w from [211], we shall obtain for homogeneous media the equations 4 TT/A D t L = V 2 Z, 4 7r/xA - D t M = V 2 M, 4 TT^tA D t N = V 2 N. The energy of the field is W+ T where MISCELLANEOUS PROBLEMS. 1. The astronomical unit of mass in any length-mass- time system is the mass which, concentrated at a fixed point, would cause by its attraction unit acceleration in any particle at the unit distance. The astronomical unit of mass concen trated at a point at a unit distance from a particle of mass equal to the absolute unit would attract it with a force of one unit. Show that the astronomical unit of mass in the c.g.s. system is 15,430,000 grammes, while in the f.p.s. system it is 963,000,000 pounds. Show also that the mass which, concen trated at a point distant 1 centimetre from a particle of equal mass, would attract it with a force of 1 dyne, is only 3928 grammes. Prove that the earth s mass (Problem 9) in astronomical c.g.s. units is 3.98 x 10 20 . Show that a mass of 1 kilogramme must be raised about 3 metres at the earth s surface in order to reduce its weight by 1 dyne. 2. Prove that two equal marbles, each of 4 grammes mass, must be placed with centres a little over 1 centimetre apart, if the attraction between them is to be 1 microdyne, and find the attraction [5535 kir~] of an iron cylinder of revolution, of 10 cen timetres radius, 1 metre long, upon a marble of 100 grammes mass, with centre in the axis of the cylinder and distant 10 centimetres from the nearer base. If the specific gravity of iron is 7.5, the radius of each of two equal iron balls which, placed in contact, attract each other with a force of one gramme s weight is 88.5 centimetres. If the mass of each of two equal homogeneous spheres with centres 1 mile apart were 415,000 gross tons, the attraction between them would 337 338 MISCELLANEOUS PROBLEMS. be about 1 pound s weight. The force of attraction between two equal particles 1 foot apart and each of mass n times as great as that of a cubic foot of water, would be equal to the weight of about ?i 2 /(7.94 x IQ 6 ) pounds. 3. Assuming that a force equivalent to the weight of a mass of 1 gramme is equal to 4 2 7r 2 (98.95) 4 centimetre-gramme attraction units, find the radii of two equal homogeneous spheres which, made of matter of density 6, would attract each other with a force of 1 gramme s weight if they were placed in contact with each other. [98.95.] 4. Assuming that 1 dyne is equal to 15,430,000 absolute c.g.s. attraction units and that 1 poundal is equal to 13,825 dynes, show that if two equal homogeneous spheres of density p, when placed in contact, attract each other with a force of /dynes, the radius of each is about (43.3) y cm., and that two equal homogeneous spheres of the density of water when in contact will attract each other with a force of 1 dyne, 1 gramme s weight, 1 poundal, or 1 pound s weight, according as the radius of each in centimetres is 43.3, 242.2, 469.4, or 1118.5. 5. Show that, having found the value of the attraction unit of force in any length-mass-time system in terms of the absolute unit of force in this system, you may find the value of the attraction unit of force in any other system the ratios of the fundamental units of which to those of the old system 2 are A, /x, and r, by multiplying the found value by *~ A 6. Show that if two homogeneous spheres of mass m l and m a , starting from rest with centres at a distance a apart, move toward each other under their mutual attraction, and if at any time t, x represents the distance between the centres, A 2 * = v \ D* - -:. , X {** MISCELLANEOUS PROBLEMS. 339 = \.-> 7 / ( . - 7 1 Va^ (a - x) + tf cos- V- } *2 k (M! + ?>? 2 ) [ *aj = \L 7 , a , - r I Vz (a - x) + a tan^xp^ i ^ 2 & (&! + w 2 ) L * a: J Hence prove that if the spheres are each one foot in diam eter and of density equal to the earth s mean density, and if their surfaces are i of an inch apart at the start, they will come together in about five minutes and a half. In this con nection we may note that if M is the mass of the earth, R its radius, p its mean density, and k the gravitation constant for the particular units used, kM 3g "-^^^ = 4^k If the first sphere is fixed while the second, of mass m 2 , is free to move, ax \x a If in this case the radius of the fixed sphere is r, and if m z is comparatively small and a infinite, the velocity with which the second sphere reaches the surface of the first is some times called the final velocity for bodies falling to the fixed sphere. Its value is \- -, or V2/- /, where f is the force of gravitation at the surface of the fixed sphere. Show that if the diameter of the sun is 109.4 times that of the earth and its mass 331,100 times the earth s mass, the final velocity for bodies falling into the sun is 55 times the final velocity for bodies falling into the earth. The radius of the earth being 6.37 X 10 8 centimetres, show that the final velocity for bodies falling to the earth under the attraction 340 MISCELLANEOUS PROBLEMS. of the earth, only is nearly 11,180 metres (or about 7 miles) per second. 7. Show that if a meteor falls upon a planet with velocity equal to that which it would acquire if it fell from rest at an infinite distance from the planet under the planet s attraction, its kinetic energy will be proportional to the product of the radius of the planet and the force of gravity on its surface. 8. Given that a falling body reaches the earth s surface with a velocity v , compute the height through which it has fallen from rest, first, on the assumption that the force which urged it was constant, and, secondly, on the assumption that the force varied inversely as the square of the distance of the body from the earth s centre, and prove that the difference between the reciprocals of the answers you obtain is equal to the reciprocal of the earth s radius. 9. Given the radius of the earth in centimetres (6.37 X 10 8 ), the mass of the earth in grammes (6.14 X 10 27 ), the radius of the sun (6.97 X 10 10 ), the mass of the sun (2.03 X 10 33 ), and the mean distance between the centres of the earth and sun (1.49 X 10 13 ), find the time when the sun and earth would come together, if both were arrested in their pafchs. Prove that the acceleration due to gravity is at the sun s surface about 27.6 g. 10. A body of mass m falls from rest near the surface of the earth and is retarded by the resistance of the air, which is \v 2 dynes when the velocity is v. Show that if s represents the space passed over up to the time t, and if /x = X/m and c* = g/n,2 ^t = log [(c + v) /(c- v)^ 2^s = log [6 Y (c* - i, 2 )], v 2 = c 2 (l e" 2 **"), and ps = log cosh Qict). Show that if the body were thrown upward with initial velocity v , we should have tan (/xc) = C(V Q v) / (c 2 -f v v). If in the case of the falling body v is the actual velocity and v the velocity which would be required by falling through the same distance in vacua, v */ v <* = 1 - v 2 /c 2 + J v 4 /c* - ^v 6 /c* + - . . . MISCELLANEOUS PROBLEMS. 341 11. Show that the periodic time of a planet moving about a fixed sun of mass m in a circular orbit of radius r is 2 TIT 3 /V A: HI, where 1/& is the ratio of the absolute unit of force in the given length-mass-time system to the corre sponding attraction unit ; and, assuming that the diminu tion of gravity at the equator due to the earth s rotation is about ^^th of the whole, and that the mean distance of the moon from the earth s centre is about 60 times the earth s radius, compute the length of the month. 12. When a particle moves in any plane curve, the tangen tial and interior normal acceleration components are D t v and v 2 1 p, while the acceleration components, taken along and per pendicular to the radius vector which joins any fixed point in the plane used as the origin of a system of polar coordinates, to the particle, are D*rr(D t ff)* and D t (i*.D t G)/r respec tively. If the resultant acceleration is always directed towards the origin, D t (i*D t G) = and r 2 -D ( = h, so that the areas of the sectors swept over in any two time intervals by the radius vector are to each other as the lengths of the inter vals : if p represents the perpendicular let fall from the origin upon the tangent to the path, vp = rD t = h. The acceleration towards the origin is and, if u represents the reciprocal of r, this may be written AV (u + Dfu). Since v 2 = h 2 [V + (D 9 uf] , J D t (r) 2 = h*D t n (u + Dfu) = - R D t r. In the case of a planet describing a plane orbit about a fixed primary centred at the origin R = nW = AV ( u + D *^ . Qr D s s + = <), 2 2 where z = u ^ > so that u = r 2 + C sin (6 X). 342 MISCELLANEOUS PROBLEMS. This is the equation of a conic section referred to a focus as origin : if e is the eccentricity and m the distance of the focus from the directrix, C = 1 jm and A 2 /ft 2 = am. The angle \\i between the radius vector, drawn from the origin to any point on the orbit and the tangent at the point, is given by the equation, ctn \f/ = r C cos (0 A). Assuming that, when is zero, \f/ = a, r a, and v = v , show that h = v a sin a, and 1 e 2 = (2 ^ v 2 a) h 2 / an*. Discuss separately the three cases where v<? is respectively less than, equal to, and greater than 2 p 2 /a, and find the lengths of the semiaxes of the orbit. Show that, if a = 90 and if v*a = /* 2 , the orbit will be circu lar; show also that, if T is the periodic time of the planet and a the semiaxis major of its orbit, /x 2 T 2 = 47r 2 & 3 . 13. Assuming that the equation * (sin i a, *), where sin < sin J. a = sin ^ 0, and a is the angular amplitude on one side of the vertical, gives the time occupied by a simple pendulum of length a in going from the vertical position to a position in which the thread makes the angle 6 with the vertical ; and that the complete time of swing is 2 TT\~ [1 + i sin 2 J a + e\ sin 4 1 a + ] ; J assuming also that a rigid body swinging about a horizontal axis under gravity moves like a simple pendulum of length k z /h where h is the distance of the centre of gravity from the axis and k is the radius of gyration ; show how a pendulum may be used to measure the force of gravity at a point. If the earth were a homogeneous sphere, would a clock which at a given temperature keeps correct time on the earth s surface lose or gain at the same temperature at the bottom of a deep mine ? Assuming that if g^ and g Q are the accelerations due to gravity at sea level, in latitude X and MISCELLANEOUS PROBLEMS. 343 at the equator respectively y^ = y (1 -f .005226 sin 2 A) and y = 978.1 ; show that the lengths of the seconds pendulum at the north pole, in latitude 45, and at the equator, are about 99.6 centimetres, 99.3 centimetres, and 99.1 centimetres. A pendulum which beats seconds on the earth s surface gains n seconds per day in a mine h metres deep. Show that if p is the mean density of the earth and p the density of the surface stratum, 2 ) approximately. Po / 86400 14. Assuming that the earth is a homogeneous sphere, of radius 6.37 X 10 8 centimetres and of mass 6.14 X 10 27 grammes, rotating uniformly about its axis in 86164 seconds, so that the velocity of a point on the equator is about 463 metres per second, show that the angular veloc ity of the earth is 0.00007292 or about (13713)- l radians per second, and that the downward acceleration at the equator is by 3.39 centimetres per second, or about - , less than the acceleration, G, at the poles. Show also (Fig. 122) that the acceleration of gravity towards the earth s centre at the latitude A is G ( 1 - ^ V the devi- \ ^89 / / . \ ation of the plumb line tan -1 ( ^L_ l an( j ^he h or i_ \289 cos 2 Ay zontal component of apparent gravitation - sin A cos A. FIG. 122. 15. If in the case of any homogeneous spherical body rotating uniformly about its axis, the polar gravity accelera tion and the equatorial gravity acceleration be g p and g e 344 MISCELLANEOUS PROBLEMS. respectively, the acceleration of gravity towards the earth s centre in latitude A is (g* p sin 2 A -f g\ cos 2 A) and the deviation of the plummet from the geometrical vertical is tan """ Sin X COS X H- g e cos 2 X } 16. A bicycle and its rider weigh together 75 kilogrammes. Show that if the machine were driven first eastward and then westward in this latitude at a velocity of 10 metres per second, the difference between the pressures on the ground in the two cases would be about 16.5 grammes. 17. The centre of a planet of radius a moves around a sun of mass M in a circular, or bit of radius r. Compute the pressures exerted on the surface of the planet by two equal particles, each of mass m, situated respectively on the points of the planet nearest and farthest from the sun. Show that the difference between these pressures is small compared with the difference between the attractions of the sun upon these particles. What is the difference between the apparent weights of a body of mass m on the earth s equator about September 21, at noon and at midnight ? 18. Two rods AB and CD, both of line density p, are placed parallel to each other. Show that the force on either in the direction of its length is 2/1 AC + AD + CD BC+ BD+ CD} p I S AC + AD -CD g BC + BD-CD) The component of the mutual attraction perpendicular to the rods is 2p 2 (BC BD AC + AD)/r, where r is the perpen dicular distance between them. 19. The sides of a triangle are formed of three thin uni form rods of equal density. Prove that a particle attracted by the sides is in equilibrium if placed at the centre of the inscribed circle. [M. T.] MISCELLANEOUS PROBLEMS. 345 20. Every particle of three similar, uniform rods of infinite length lying in the same plane, attracts with a force varying inversely as the square of the distance : prove that a particle subject to the attraction of the rods will be in equilibrium, if it be placed at the centre of gravity of the triangle enclosed by the rods. [M. T.] 21. The attraction of the straight rod AB at a point P is the resultant of two forces, each equal to /, acting at P towards the extremities of the rod, where f=2m- AB/[(AP + BP) 2 - AB-~\. Find the value of / when P lies on an ellipse the foci of which are the extremities of the rod. [Routh.] 22. If the direction at the point of the attraction of every portion of a uniform plane curvilinear wire bisects the angle subtended at by that portion, the wire is either straight or has the form of a circumference with centre at 0. [Routh.] 23. If the law of attraction be the inverse square, two curvilinear rods in one plane exert equal attractions at the origin if the densities at points on the two rods on any radius vector drawn through the origin are proportional to the per pendiculars from the origin on the tangents. [Routh.] 24. Prove directly from the formula for the attraction of a slender straight wire, that the attraction at a point P, due to an infinite homogeneous cylinder of any form, is twice that of so much of the cylinder as is cut off by a double cone formed by the revolution about a line through P, parallel to the generating lines of the cylinder, of a line which cuts this line at P at an angle of 60. 25. A uniform wire AB in the form of a circular arc has its centre at 0. Prove that the component of the attraction, at any point P, in a direction perpendicular to the plane containing P and the normal at to the plane of the arc, is a f*(> i~ l ~~ r z~ l ) I h> where t\ = AP, r = BP, h is the projection of OP on the plane of the arc, and p. the line density of the wire. 346 MISCELLANEOUS PROBLEMS. 26. Prove that the attraction in the direction PO at a point P on the circumference of a circle the centre of which is 0, due to an infinitely long, straight filament of given density passing through a point Q in the circumference and perpen dicular to its plane, is the same wherever the point Q is. If the filaments of a homogeneous columnar distribution of given mass per unit length are so arranged that the cross-section is a circle passing through a point P, the attraction of the distribution on P will be a maximum. [Tarleton.] 27. A water tower in the shape of a cylinder of revolution is 100 feet high and 10 feet in diameter. The mass of the tower and contents is 8400 pounds per foot of height. With out the help of pencil or paper, guess, to within one per cent of the truth, the value in f.p.s. attraction units of the horizontal component of the attraction due to the tower at a point at its foot just outside it. 28. Prove that at a point on the edge of an infinite homo geneous cylinder of semicircular cross-section, the components of the attraction across the plane face perpendicular to the axis, and normal to the face, are irakp and 2 akp respectively, and show that gravity is diminished by the fraction at the middle of the surface of a long straight canal of semi circular section, a being the radius of the semicircle, r the radius of the spherical earth, p the density of water, p that of the surface stratum of the earth, and p the earth s mean density. The corresponding quantity in the case of a canal of rectangular cross-section of depth a and breadth 2 a is TT + 2 log 2 3_a p- p TT 4r po 29. An infinitely long homogeneous prism has a rectangular cross-section of length a and breadth ft. Assuming that flog (a = x- log (a 2 + x 2 ) - 2 x + 2 a tan~ l (x / a), MISCELLANEOUS PROBLEMS. 347 show that at any point on one of the edges the components of the attraction along the sides a and b of the cross-section through the point are kp \2 a tan- 1 (6 /a) + b log[(a 2 + b 2 ) /b 2 ^ and kp\2b tan- 1 (a/b) + a . log [(a 2 + b 2 ) /a 2 ] \. If the ratio of b to a is large, the first of these quantities is nearly equal to irapk. Show that the apparent latitude of a point on one edge of a long, deep, narrow crevasse of breadth a, running east and west, is altered by the angle 3pa/4p r, nearly, by the presence of the crevasse. [Thomson and Tait.] 30. Assuming that the attraction of a homogeneous cylinder of revolution, of density />, radius a, and height h, upon a unit particle at the centre of one of its ends, is a 1 1 " S 1 1 3 " 5 h 1-1 7/ 3 1-1-3 or according as a is small or large compared with h, and con sidering that the mean surface density of the earth is 3 times and the mean density of the whole earth 5.5 times the density of sea water, obtain Siemens s expression, ? for the dimi nution of gravity at a point on the ocean where the depth is h. Is the intensity of gravity at the centre of the mouth of a ver tical mine shaft 20 feet in diameter appreciably less than before the shaft was dug ? Show that if h = a, the attraction due to a cylinder of revolution, at the centre of one of its ends, is 2 trkpa (2 V2). The attraction due to the earth .,2 at a point P at a height h above the surface, is -^> or (r -j- A) g ( 1 - - J approximately, where r is the radius of the earth. If p is the earth s mean density, g = $ vkp r. If P is 348 MISCELLANEOUS PROBLEMS. at the centre of a wide plateau of height li made of matter of density p, the additional attraction due to the plateau is about 2 irkph, or 3 yph/2 p Q r, so that if p = ^ p , the whole attraction 5 h\ / 31. A vertical solid cylinder of height a and radius r is divided into two parts by a plane through the axis. Show that the resultant horizontal attraction of either part at the centre of the base is v + V^T^ is nearly fMw 1 32. A right circular cylinder is of infinite length in one direction and is homogeneous. Prove that if the finite extrem ity be cut off perpendicularly to the generators, the attraction on a unit particle placed at the centre of this end is - > where M is the mass per unit of length. If the cylinder be elliptic, of the same density and mass per unit of length as before, and of eccentricity e, then the attraction will be n times the former value, where dO Vl - e 2 sin 2 33. A homogeneous, right circular cylinder of density p stands on the plane z = 0, and is infinite in the positive direction of the axis of z. Show that the z component of its attraction at a point P of its base is kpl, where I is the perimeter of an ellipse having the base for the auxiliary circle and P for one focus. 34. Show that the attraction at any outside point P, due to a uniform plane lamina of any shape, yields a component normal to the lamina, equal to the product of the solid angle subtended at P by the lamina, and a quantity which does not depend upon P s position. MISCELLANEOUS PROBLEMS. o4i" 4 35. Show that the component perpendicular to its axis, of the flM"""* 5 "" of a thin, homogeneous, circular, cylindrical sheet of height 2k and radius a, has at any point on one of the circular bounding odges of the cylinder the value where ^= 9 . M 36. An infinitely long plane sheet of constant width has a ffa11 thickness & and is made of homogeneous matter of density p. This strip cuts a plane perpendicular to its long edges in the line AB : show that the attraction of the strip at any point P in this plane has a component 2 kp& log (PB/PA) parallel to AB, and a component 2 fyS - Z APB perpendicular toAB. 37. Every diameter of a certain circle subtends a plane angle 2$ at a ffrrtam point P on the axis of the circle ; show that the circle subtends at P the solid angle 2-(l cos 0). 38. Compare the attractions, at the vertex of a. homoge neous oblique cone which has a plane base, due to the whole cone and to so much of it as lies between the vertex and a plane which bisects at right angles the perpendicular drawn from the vertex to the base. 39. Prove the truth of the theorem which Xewton states in the f ollowing words : Si corporis attracti. ubi attrahenti contiguum est, attractio longe fortior est, quam cum vel minimo intervallo separantur ab invicem : Tires particularum trahentis in i liuiiiiu. eorporis aHiauli, degtesflunt in ratione plusquam dnptJcita distantianim a particulis. Si particula rum, ex quibus corpus attracti vam componitur. Tins in reoessu corporis attracti decrescunt in triplicata. vel plusquam tripli- cata ratione distantiarum a particulis, attractio longe fortior erit in contactu, quam cum trahens et attractum interrallo vel ^ separantur ab invicem." [PAiZ JVfi/, Prime. Sectio 350 MISCELLANEOUS PROBLEMS. 40. Two homogeneous solids made of the same material are bounded by similar surfaces. Show that the intensities of their attractions at two points similarly situated respec tively with regard to them, are in the ratio of the correspond ing linear dimensions of the solids. Hence prove that the attractions at points on a given diameter inside a solid homo geneous ellipsoid are proportional to the distances of these points from the centre. 41. Prove that the attraction, at very distant points, of any system which has an axis of symmetry, may be represented as emanating from two equal poles of the same sign situated on the axis. 42. Show that the component, at the origin, in the direc tion of the x axis, of a given particle m, is the same wherever on the surface m cos (x, r)/r* = c, where c is a given constant, the particle lies. If it is anywhere without the surface, the component will be less than if it were anywhere within. Hence prove that the attraction of a given mass M for a point on its surface will be greatest if the boundary of M, referred to the given point, is a surface of the family cos = \- r*. 43. If the earth be considered as a homogeneous sphere of radius r, and if the force of gravity at its surface be y, show that from a point without the earth, at which the attraction is the area 2 Trr 2 [ 1 \l- - - } on the surface of the n earth will be visible. 44. The laws of attraction for which the attraction of a homogeneous shell on any external particle is the same as if the shell were concentrated at its centre, are the " law of the inverse square " and the " law of the direct distance." 45. Whatever may be the law of attraction, the intensity of the force exerted by the smaller of two concentric solid homogeneous spheres at any point on the surface of the larger, is to the intensity of the force exerted by the larger at any MISCELLANEOUS PROBLEMS. 351 point on the surface of the smaller, in the ratio of the square of the radius of the smaller to the square of the radius of the larger. [Minchin.] 46. Prove that if / be an external point and C the centre of a sphere, the sphere on 1C as diameter, the sphere with centre / and radius 1C, or the polar plane of /, will divide the sphere into two parts which exert equal attractions at /, according as the law of attraction is the inverse square, the inverse cube, or the inverse fourth power of the distance. [St. John s College.] 47. Two sectors are cut from a homogeneous shell bounded by two concentric spherical surfaces of radii i\ and r 2 , by a conical surface of revolution of half angle and with vertex at the centre of the shell. The attractions at a point P without the shell on the axis of the cone, on its inner side, at a distance c from 0, due to the portions of the shell which lie respectively without and within the cone are F l and P z . Show that jp\ is equal to the difference between the values when r = r z and > = )\ of a quantity A, and that F 2 is equal to the difference between the corresponding values of a quantity B where rt 7 A = ^ [J r 8 - oii (i r 2 - f c 2 + c 2 cos 2 + i re cos 6) 4- c 3 cos sin 2 log (o>* + r c cos 0)] , B 2 = ^jf [i r 3 + ^ (J ,* - f c 2 + c 2 cos 2 + J w cos 6) c 3 cos B sin 2 - log (o>* + > c cos 0)] , and o> = c 2 + ?* 2 2 cr cos 0. The attractions of the halves of the shell farthest from P and nearest to it are 352 MISCELLANEOUS PROBLEMS, and respectively. If the mass of the whole shell is M and if the shell is thin, the attractions at P due to the sectors are kM ( ^ r c cos 0\ , kM f r c cos 0\ ~f\ o I -^- ""I 2c 2 V PL ) 2c 2 \ PL ) where L is any point on the common rim of the sectors. 48. Prove that the attraction due to a homogeneous hemi sphere of radius r is zero at a point in the axis of the hemi sphere distant f r approximately from the centre of the base. 49. A segment of height h, cut from a homogeneous sphere of density p and radius a by a plane distant a h from the centre of the sphere, attracts a unit particle on the axis of the segment at a distance b, greater than the radius, from the centre of the sphere, with a force 27T&J h + * |(2 c 2 + 3 ac)c - (2 c 2 + 3 ac + ah + ch) |_ o (c -\- (ij L Vc 2 -f 2 ch + 2 ah | , where c = b a. If c = 0, this becomes 2>n-kph 4 1 - \ - r Assuming this to be true, show that the attraction of a homogeneous hemi sphere upon a particle at its vertex is to the attraction of the circumscribing cylinder of the same density as 529 to 586, nearly. Show that the attraction, at its vertex, of a slice 2 miles thick cut from the earth, and the attraction of an infinite disc of the same thickness and density upon a point at the centre of one of its faces, differ by about one per cent of either. 50. Show that if the earth were made up of two homogeneous solid hemispheres bf densities p and p separated by the plane MISCELLANEOUS PROBLEMS. 353 of the equator, the deviation of the plumb line from the zenith at^ any point of the equator would be tan" 1 ( - f V 51. Show that the attraction at the origin due to the homo geneous solid bounded by the surface obtained by revolving one loop of the curve r 2 a 2 cos 2 6, is ^ irakp. 52. A mountain of the form of a surface of revolution with vertical axis and elliptic outline stands on a horizontal plane which contains the centre of the ellipse. Find the horizontal component of its attraction at a point of the base. Show that if the mountain is 2 miles high and 4 miles broad at the base, and if the density of the mountain and of all the matter in its neighborhood is half the mean density of the earth, the plumb lines close to its base on the north and south sides will make with each other an angle greater by about 51 seconds of arc than the corresponding difference of geocentric latitude. 53. The attraction at the point (0, 0, b) of so much of the homogeneous paraboloid x 2 -f- if = \z as lies between the planes z = 0, z = h is \h - V(6 + A) 2 + h\ + b - H-log(2& + * A) + A iog(V(7 + hy+ h\ + b + h - 54. If a body M be divided into two rigid portions, A and B, the resultant action of each portion upon itself is nil, and the attraction between A and B is the same mathematically as the attraction between M and B. To find, therefore, the attraction between two equal homogeneous hemispheres so placed as to form a sphere, we may integrate through either hemisphere the product of the density and the component normal to the flat face of the hemisphere, of the attraction due to the whole sphere. Show that the result is 3 kM 2 /16 a 2 . 55. Show that the resultant attraction between the two parts into which a homogeneous sphere is divided by a plane 354 MISCELLANEOUS PROBLEMS. is equal to the mass of either part multiplied by the intensity of gravitation at its centre of mass. 56. Prove that the pressure per unit of length on any normal section of a spherical shell of mass M and radius a, due to the mutual gravitation of the particles, tends to the limit 7cJf 2 /167r& 3 , as the thickness of the shell is indefinitely diminished. [M. T.] The mass of the unit length of an infinite homogeneous cylinder of revolution of radius a which is divided into two parts by a plane through its axis is M. Show that the pres sure between the two parts due to their mutual attractions is 4 kM 2 /3 Ira per unit length of the cylinder. 57. If R and S denote the components of attraction of a gravitating system symmetrical with respect to a straight axis, taken along and perpendicular to the axis, then where r and z are columnar coordinates. [St. John s College.] 58. If the point of application of a force F move by the path s from the point A to the point B, the force is said to do work during the journey, equal in amount to the line inte gral taken along s of the tangential component of F. If the components of F parallel to the coordinate axes are X, Y, Z, and if dx, dy, dz are the projections on these axes of an element ds of the path, we have the expressions W= C F. cos(s,F)ds J A XB F [cos (x, s) cos (x, F) __ __ + cos (y, s) cos (y, F}+ cos (, s) cos (z, F)~\ ds C E I \X- cos (x, s) -f Y- cos (y, s)+ Z- cos (, s)] ds */ A = C B Xdx + Ydy + Zdz. J A MISCELLANEOUS PROBLEMS. 355 If a function O exists such that X=D x tt, r=Z> y O, Z=D Z Q W= C* dV, = l B -l A : J A such a function is called a potential function or a force function of the given force. The work done by a force which has a potential function, when its point of application moves completely around any closed path, is zero, and such a force is said to be conservative. The work done by a conservative force as its point of application moves from A to B is inde pendent of the path s. Prove by actual integration along the different paths, that the work done by the force X= 3 x 2 + 2 y, Y = 4 y 3 + 2ar, Z = 0, when its point of application moves from the origin to the point (2, 2, 0), is 32, whether the path be a straight line, or the parabola if = 2 x in the xy plane, or a combination of a straight line from the origin to (2, 0, 0) and another straight line from this point to (2, 2, 0). Show that the derivative with respect to x of any function of the form x s + 2 xy +/(?/), where f is arbitrary, will yield X, and that, by a proper choice of/, the derivative with respect to y can be made equal to Y j so that a force function exists. Prove by actual inte gration along the paths that the work done by the force as its point of application moves from the origin to (2, 2, 0), is not independent of the path. In this case no potential function exists, since it is impossible to give such a form to /, in the general expression [z 3 + 2 xy +/(?/)], which has X for its partial derivative with respect to x, that the partial derivative of the expression with respect to y shall be Y. Since the order of successive partial differentiations of any analytic function is immaterial, or D y Z = D 2 F, D Z X = D X Z, D X Y= D y X. 356 MISCELLANEOUS PROBLEMS. Show that this necessary condition for the existence of a force function is also a sufficient one. 59. Prove that if we have matter attracted to any number of fixed centres with forces proportional to any function of the distance, or if we have matter every particle of which attracts every other particle according to any function of the distance between the particles, there exists a potential func tion the derivative of which in any direction at any point gives the intensity of the force which would solicit a unit quantity of matter concentrated at the point to move in the given direction. 60. If r represents the distance of any point Q on a sur face S from a fixed point P, and if a is the angle between PQ and the normal to S at Q, drawn always from the same side /OOS n 3 dS, taken over any portion of the sur face, gives in absolute value the solid angle subtended at P by this portion, and, in the case of a closed surface, this value is 4 TT, 2 TT, or 0, according as P is within, on, or without S. Prove that the volume of the solid enclosed by any surface S is the absolute value of % IT cos a dS taken over the surface, whether P is within or without S. Show that it is possible to find an analogous expression, % (r cos ads, for the area enclosed by a plane curve, and explain in this case the notation. 61. Show that the absolute value of the component parallel to the axis of x, of the force at a point P, within or without a homogeneous solid body of any form, due to the attraction of .,:.,* / cos (x,n)-dS, . .,- i this body, is p I *-* " , where n is an interior normal, taken all over the bounding surface ; and prove that the component parallel to the axis of x of the force, at a point P, due to the attraction of a homogeneous infinite cylinder MISCELLANEOUS PROBLEMS. 357 with generating lines parallel to the axis of z, is of the form 2 /JL I cos (x, n) log r ds, where the integral is to be extended around the contour of the section of the cylinder made by a plane through P perpendicular to the axis of z. 62. The space within a closed surface S is filled with homo geneous matter of density p. Prove that the value at the point P, of the potential function due to the distribution, is p \ cosadS, where a is the angle which the normal to the sur face, drawn inward at any point Q on it, makes with QP. 63. Two distributions of gravitating matter possess a com mon closed equipotential surface. Prove that if all the matter of both distributions be within this surface, the potentials at the surface due to the two distributions are to each other as the masses. 64. Prove that if two different bodies have the same level surfaces throughout any empty space, their potential func tions throughout that space are connected by a linear relation. That the level surfaces should be the same, it is only neces sary that the resultant forces due to the two bodies should coincide in direction. 65. Show that if two distributions of matter have in common an equipotential surface which surrounds them both, all their equipotential surfaces outside this will be common. 66. Show that if we have matter every particle of which attracts every other particle with a force proportional to the nth power of the distance, the attraction at any point within a quantity of the matter will be infinite if n + 2 < 0. [Minchin.] 67. Show that if u, v, and w are any three solutions of Laplace s Equation, V 2 (iivw) = u v 2 (yw) -f v v 2 (?/?) + w y 2 (uv). 358 MISCELLANEOUS PROBLEMS. 68. Show that the potential function due to a solid hemi sphere of radius a and density p, at an external point P situated on the axis at a distance from the centre, is the upper or lower sign being taken according as P is on the convex or plane side of the body. 69. A sphere with centre at the origin has a radius r and a density given by the law p = ax + by + cz. Prove that the value at any external point (aj, y, z), at a distance It from the origin, of the potential function due to the sphere, is 70. An infinite cylinder of radius a has a cylindrical cavity of radius b cut out of it. The axes of the cylinders are parallel but not coincident, and the surfaces do not intersect. Show that the equipotential surfaces are cylinders the equa tions of which are : (i) r u 2 r b 2 = Ci within the cavity ; (ii) r a 2 -2b* log ^ = C 2 within the mass ; (iii) a 2 log ( J 6 2 log f ~ J = C 3 in outside space ; where r a and r b are the distances from the axes of the cylinder and cavity respectively. 71. From a homogeneous sphere of density p and radius a is cut an eccentric spherical cavity of radius b. The distances of any point P from the centre of the sphere and the centre of the cavity are r x and r 2 respectively. Show that V P , the value of the potential function at P, is given by the first, second, or third of the subjoined equations, V + 27rb z - MISCELLANEOUS PROBLEMS. 359 , 2& 3 Q / , r r \ rf + - - - 3 f a- - - ) , r z \ 2-irpJ Qi7 (* l >*\ 3 !> P = 4 Trp ( ) , \ r i V according as P is within the cavity, within the mass, or with out the mass. Indicate by a rough drawing the form of a line of force within the cavity. 72. Show that the lines of force due to a uniform straight rod are hyperbolas which have the ends of the rod for foci. 73. Show that formula [59] might be written V P = IL- log (ctn PBA - ctn PAS). 74. A number (n) of equal, infinitely long, homogeneous, straight filaments, all parallel to each other, cut the xy plane normally in points which lie uniformly distributed on a cir cumference of radius a with centre at the origin. One of these points is at the point (a, 0). Show that the value of the potential function at the point (r, 0} is m log (;* 2n 2 aV cos ?iO + a 2 "). 75. If the law of attraction were that of the inverse wth power of the distance, we should have If the density had the same sign throughout a distribution of matter, the potential function could not be constant in any region of empty space unless n were equal to 2. 76. In the case of matter every particle of which attracts every other particle with a force proportional to the product of their masses and a function (/) of the distance, we have V 2 FEE f f f [2/(? )/? +/ (r)]pf/T. Show that F cannot satisfy Laplace s Equation unless /(r) = K/r*. 360 MISCELLANEOUS PROBLEMS. 77. If instead of the polar coordinates r, 0, <, the independ ent variables are r, ^ </>, where /w, = cos 9, Poisson s Equation becomes D r (r 2 - D r F) + I)* [(1 - /* 2 ) D M F] + D\ V/ (1 - /x 2 ) = - 4 Trpr 2 . 78. If instead of the spherical coordinates r, 0, <, the coor dinates u t w, </> be used, where u = 1/r, and w = log tan 0, Laplace s Equation becomes sin 2 (2 tan- 1 e w ) u 2 . DJ> V + iy F + DJ V = 0. 79. Show that if matter be distributed symmetrically about an axis, and if 4 a, Aa be the latera recta of the two confocal parabolas, with this line as axis, which meet at any point, Laplace s Equation may be written in the form 80. Prove that at the surface of an attracting body, Z> X 2 F, D y z V, D z 2 V are discontinuous in such a manner that if n repre sents an interior normal drawn to the surface, the values of these quantities at any point just within the attracting mass are smaller than at a neighboring point just without, by the quantities 4 TT/J cos 2 (x, ri), 4 irp cos 2 (y, n), 4 -n-p cos 2 (z, n), respectively. 81. A portion of a spherical surface is occupied by a thin shell of matter of uniform density <r, which attracts according to the Newtonian Law. Prove that the value, at any point on the remaining portion of the surface, of the potential function due to this distribution of matter, is a a- <o, where a is the diameter of the sphere and w the solid angle subtended at the point by the contour of the portion of the surface occupied by matter. 82. Show that in so far as a transformation from one set of rectangular axes to another is concerned, D*V+ D*V + D Z Z V and (D x Vy+(D y Vy + (D Z V) 2 are differential invariants. MISCELLANEOUS PROBLEMS. 361 83. The potential function at all points external to the s P here x 2 + if + * = a* a 5 (ax 2 + ft/ 2 + yz 2 + 2 a ljz + 2 /fe + 2 y x^/r 5 . Show that if there be no matter in this region, a, ft and y must satisfy a certain relation. Show that if inside the sphere the density be uniform, the value there of the potential function Wl11 be c + XJP* + M* + vz 2 + 2 a yz + 2 ft zx + 2 y xy, where c, A, p, and v are known. Find the condition that under these circumstances the equipotential surfaces inside the x 2 i/ 2 z 2 sphere should be ellipsoids similar to + ^ + ^ = 1- x ^ 84. Prove that if C<j> (r) dr = x (> ) and A-- x (> ) ^^ = A ( r )> r r and if <#>, x and ^ vanish at infinity and are finite for finite values of r ; mm x (r) represents (1) the work done under an attracting force mm <f> (?) in bringing a particle of mass m from infinity to a point distant r from another mass m ; (2) the component, parallel to the rod, of the attraction of a particle m on a straight slender rod of line density m , if the end of the rod is at a distance r from m and thr other end at infinity. Show also that 2 irarm ty (z) repreient^ (1) the work done in bringing from infinity to a point distant z from a thin lamina of surface density o-, a particle of marfs m\ (2) the attraction of a particle m, placed at a distance z from the plane surface of an infinite solid of constant density o-. 85. Show that if s represents a direction which makes the angles a, ft y with the coordinate axes, + D 2 Fcos 2 y + 2 D x D y Vcos a cos (3 + 2 D y D z Fcos (3 cosy + 2 DJ) Z Fcos y cos a. 362 MISCELLANEOUS PROBLEMS. 86. When the line of action of the attraction of a body at every point of external space passes through a point fixed in the body, the body is said to be centrobaric and is called the baric centre. The lines of force in external space are straight lines passing through 0, and the equipotential surfaces are spherical surfaces with centre at the baric centre. Show that the whole external field must under these circumstances be the same as that due to a mass equal to that of the body, concentrated at 0. Show that if at internal points also the line of action of the force always passes through 0, the density of the body is a function only of the distance from 0. The centre of gravity of a finite centrobaric distribution is the baric centre. A distribution cannot be centrobaric unless every axis drawn through its centre of gravity is a principal axis. If for any finite space outside it a body is centrobaric, it must be centrobaric for all the rest of outside space. A distribution which consists of a spherical distribution and a distribution the potential function due to which at all outside points is zero is evidently centrobaric. 87. Show that if is a fixed origin within or near a distribution M of attracting or repelling matter, if P is any point of M a (J /,l P any point without M more distant from O than any pci n t of M is, and if P = (x, y, z), P = (x 1 , y } z ), OP = r, OP == r f , f OP = <j> ; the value at P of the potential function due tc> M i equal to M - MISCELLANEOUS PROBLEMS. 363 Show that if A, B, C, and / are the moments of inertia of M about the coordinate axes and about OP respectively, A + B+C= CCC2r *dm and /= f C C r 2 sin 2 <. dm , and that if is the centre of gravity of M , the second term of the development vanishes so that If M is centrobaric and if is the baric centre, V is a func tion of r only and the coefficients of r in the general develop ment are to be considered as constants. 88. If the law of attraction is expressed by any function, < (?), of the distance, the intensity of the attraction of any homogeneous solid, estimated in a given direction, at any point P, is expressed by the surface integral ( <J>(r) cos A dS, where r is the distance from P of any point on the surface bounding the solid, dS the element of this surface, and X the angle made by the normal to the element with the given direction. [Minchin.] 89. The function f (x p y) can satisfy Laplace s Equation only if p 1, or 1, or 0. 90. The invariable line which joins the centres (A , BQ) of two homogeneous spheres, A and B, moving under their mutual attraction, revolves with uniform angular velocity, w, about the centre of gravity, C, of the two. One of the spheres, A, does not rotate, but every line in it remains parallel to itself during the revolution. Show that every particle of A moves in a circle of radius equal to the distance of A s centre from (7, and is at every instant at the end of a diameter parallel to B A . Under these circumstances a loose particle at D on A s surface must in general be constrained to keep it moving in its path. 364 MISCELLANEOUS PROBLEMS. If we denote the radius of A by a, the distances Q A 0) CA by d and r, and the mass of B by M, the resultant force on a particle of mass m resting on A at D [Fig. 123] has the intensity mw 2 r = kmM/d 2 and a direction DT parallel to A B , while the attraction of B upon the particle has the inten sity kMmf B Q D and the direction DB$. Show that if a is fairly small compared with d, a constraining force equal to 3 akMm (sin 2 0) /(2 o? 3 ), where = CAD, must be exerted on m in a direction perpendicular to ^1 > to prevent its sliding on A s surface. Assuming A to be the earth, of mass M and radius a, and B, the moon, of mass T T M , with centre distant 60 a from FIG. 123. the earth s centre, prove that the maximum horizontal lunar tide-generating force on the earth s surface is to the force of terrestrial gravitation as 1 to 11,500,000, nearly. Find approximately the " vertical tide-generating force " at the points on the earth s surface nearest and farthest from the moon. [The student is strongly advised to read in this connection Prof. G. H. Darwin s charming Lowell Lectures on the Tides.] 91. Supposing that a sphere of water is brought together by the mutual attractions of its particles from a state of infinite diffusion, and that the amount of work done by these forces is sufficient to raise the temperature of the sphere MISCELLANEOUS PROBLEMS. 365 1 C. Show that the radius of the sphere is about one- fortieth of the radius of the earth, if the earth s radius be 637 x 10 6 centimetres, and if one water-gramme-centigrade- degree be equivalent to 4.2 x 10 7 ergs. [Minchin.] 92. The value at any point (x, y, z) of the potential func tion due to any system of attracting matter at a finite distance is F, the forces due to the attraction of this matter at any point (x , y\ z ) is F , the value at this point of the potential function F , and the density p. Show that V - F *)dx dij dz F = ^-fff 2^JJJ nv- where the integration takes in all space. 93. Prove that the rise of sea level in a shallow sea caused by the attraction of a homogeneous hemispherical mountain of radius c rising from it with its base at sea level, is approxi mately p c 2 /2 pa, where p is the density of the mass of the mountain, p the mean density of the earth, and a its radius. 94. A fixed gravitating sphere is partly covered by an ocean extending over the northern side of a parallel of colatitude A. A distant fixed gravitating body M is situated on the north axis of this small circle. Prove that if the self-attraction of the ocean be neglected, M will cause a rise of water at the north pole approximately equal to * sin 2 JA, where * is what the rise would be if the whole sphere were covered. 95. Show that if a finite distribution consists of m units of positive matter and m units of negative matter, anyhow dis tributed, it is possible to draw, with any given finite point as centre, a spherical surface so large that the whole flow of force through it, reckoned arithmetically, shall be as small as we please. Prove that the lines of force are all closed. 96. Imagine any point P in empty space near a distribution of repelling matter to be taken as origin of a system of orthog onal Cartesian coordinates with axis of z coincident with the 366 MISCELLANEOUS PROBLEMS. normal to the equipotential surface which, passes through P. Fwill then be given by an equation of the form Vf(x, y, z), where D x f, D y f vanish at P, and D z f is the force F in the direction of the z axis. If Q is a point near P on the sec tion of the surface V = V P made by the xz plane, and if we denote the coordinates of Q by (Ax, 0, A), the radius of ~ curvature at P of this section is A *n 7TT~ ) an( ^ A z is in Ax = o general of higher order than Ace. + ^ Ax 2 - D x * V+ terms of higher order. Since V Q = F P , and D x V vanishes at P, D x 2 V = ~ Prove F 1 similarly that D y 2 V = and then, by Laplace s Equation, that n 2F= _*YI__A "^ 2. * I T~> I 7~> Illustrate these results by an example. 97. If a distribution of active matter is symmetrical about a straight line (the axis of x) and if r represents the distance of any point from this axis, the potential function involves r and x only and the equipotential surfaces are surfaces of revolution. Consider one of these surfaces, $ , on which V has the value F , and let the " flux of force " through so much of $ as lies between some fixed plane (x = je ) perpendicular to the x axis, and the plane x = x, be represented by the function 27r/x, then if ds is the element of the generating curve of S between x and x + Ax, and if r is the distance of ds from the x axis, the area of the strip of S between x and x -f- Ax is approximately 2 TT?-. ds, the flux of force through it is 2 irr . D n V- ds, and this flux is the change made in 2 TT/X by increasing x by Ax. We may write, therefore, -D g /x, = r - D n F, MISCELLANEOUS PROBLEMS. 367 and, if a is the angle which the exterior normal to ds makes with the x axis, D s fj. = Z^ sin a Z> r /x cos a, D n V= D x V- cos a + D r V- sin a, and the equation becomes sin a (D^ - r-D r V) - cos a (Z> r /n + r-D x V) = 0. If this equation is to hold everywhere on every equipotential surface, the coefficients of sin a and cos a must vanish and /x is determined (apart from an additive constant to be chosen at pleasure) by the equations D x /u = r D r V, D r ^ = r D x V. Show that the values of /x corresponding to the three familiar potential functions - X&, Mx/(t* + x 2 )*, M/(r> + x 2 )* are %X Q ?*, Mr 1 /^ + x 2 )*, and - Mx/(i* + x 2 )-. Discuss the physical meanings of these results. The function ^ defined above is sometimes called " Stokes s Flux Function." It is clear that the level surfaces of the functions V and /n, both of which are symmetrical about the x axis, cut each other orthogonally and that the generating line of any level surface of /n is a line of force. Although any func tion of /a equated to a constant would serve to represent the forms of analytic lines of force, a special advantage arises from the use of ^ itself from the fact that if ^ and /A 2 are flux functions corresponding to two different potential functions, V-i and V ZJ due to two distributions of matter, M l and 1T 2 , symmetrical about the x axis, ^ + /u, 2 is a flux function of Vi + Fo, the potential function due to M and M 2 existing together. If generating lines of the ^ surfaces be drawn in a plane, for the numerical values , a + 8, a + 2 8, a + 3 8, a + 4 8, etc., and the lines of the ^ surfaces for the values b, b + 8, 6 + 28, 6 + 38, 6 + 48, etc., 8 being any con venient interval, the intersections of the curves ^ = a + n&, yu, 2 = 6 + (m n) 8 will be points on the generating lines of the surface ^ + /x 2 = a + b + raS. If, then, we fix m for a moment and give to n in succession different integral values, 368 MISCELLANEOUS PROBLEMS. we may get points enough to enable us to draw the line Pi + /n 2 = a -f- b -f- wS with sufficient accuracy. This graphi cal method of drawing lines of force (or equipotential surfaces) has proved in the hands of Maxwell and others extremely fruitful. Draw accurately several of the lines of force due to a charge 20 and a charge 10 concentrated at points 4 inches apart. 98. (a) Show that if P, P are any definite pair of inverse points distant respectively r and r from the centre of a spherical surface S of radius a, the ratio PQ /P Q is equal to the constant a/r wherever on S the. point Q may be. Hence show that if V is the potential function due to a heterogeneous surface distribution on S, V P , = V P (a/r )zndD r ,(V P ) = - a s -D r V p ./r 3 - aV/r<*. (b) Prove that i* *>D r r 2 , + r * *.D r ,rj,= -(aV r > F P .) 1/2 . [Routh.] (c) Prove that as both r and r are made to approach a, limit (D r V P + D r > F = - V/a. [Stokes.] 99. If P, P are any definite pair of inverse points with respect to a right section of an infinitely long cylindrical sur face of revolution, and if Q be any variable point on the circumference, P Q/PQ is equal to the constant r /a. Show that if the cylinder be covered with a superficial distribution the density of which varies from filament to filament of the surface, V P , V P = 2 log (r /a) M, where M is the amount of matter on the unit length of the surface. 100. V is the potential function due to a volume distribu tion of density p in the region T and a surface distribution of density a- on the surface S. V is the potential function due to a volume distribution of density p in the region T 1 and a surface distribution of density a- on the* surf ace S . Using all space as the field of Green s Theorem, apply [145] to these functions and interpret the resulting equation as giving an expression for the mutual energy of the distributions. MISCELLANEOUS PROBLEMS. 369 101. If two systems of matter (If and M ), both shut in by a closed surface S, give rise to potential functions ( V and F ), which have equal values at every point of S, whether or not S is an equipotential surface of either system, then V can not differ from V at any point outside S, and the algebraic sum of the matter of either system is equal to that of the other. 102. Prove that the level lines of the function u = F^ (x, y, z) on the surface FI (a*, y, z) = have direction cosines which are to each other as and (D x f\ . D y F 2 - D y F, . D X F 2 ) and that if these quantities be represented by A, /*, and v, respectively, the direction cosines, at the point (x, y, z), of a curve which lies on F l and cuts orthogonally at that point a level line of u on the surface, are to each other as D z Fi - v D y F,) : (v D X F, - A D Z F,} : (A - D y F, - p. - D X F,}. In particular, if u = x/z and if F l (x, y, z) = x*(b 2 ?/ 2 ) 2 - 2 , the level lines of u on F l are straight lines, the direction cosines of which at any point P are in the ratio u p : : 1, and since the sum of the squares of these cosines must be equal tojunity, the cosines themselves are u / V*r + 1, 0, and 103. In the case of a columnar distribution the density of which varies only with the distance r from a fixed axis, the lines of force are straight lines radiating from the axis (Sec tion 34), and the potential function Fand the resultant force D r V are functions of r alone. If we apply Gauss s Theorem to a cylindrical surface of revolution S, coaxial with the distribu tion, we learn that 2 -n-r D r V = 4 TT times the mass M of the unit length of so much of the distribution as is enclosed by S. 370 MISCELLANEOUS PROBLEMS. Show that if the distribution is a solid homogeneous repell ing cylinder of radius a and density p, D r V = 2 irpr and V=TTp [r 2 a 2 + 2 a 2 log a], if r is less than a. If r is greater than a, D r V 2 7rpa 2 /r and V = 2 Trpa? log r. Show also that if the distribution is merely a surface charge of density a- on a cylindrical surface of radius a, V = 4 Ti-ao- log a within the cylinder, and V = 4 TTO-O- log r without. 104. If V is the gravitational potential function belonging to a given distribution M of attracting matter, and if k is the constant of gravitation, the force of gravitation at any point in any direction s, measured in dynes, is the value at that point of k D s For D s F , where F = k V ; and V 2 F = 4 irkp. Prove that if M be made to rotate about the axis of z with constant angular velocity w, if O = w 2 (x 2 + y 2 ), and if F = F + O, the apparent force at any point in any direction s is D s F and V 2 F = 4 irkp 4- 2 or. Prove also that if S represents the surface of M, and n a normal to the surface drawn inwards, if v is the volume of the distribution, and if p is its mean density, - 4 Trkvp, + 2 fy = - [E. S. Woodward, " The Gravitational Constant and the Mean Density of the Earth/ Astronomical Journal, 1898.] If M represents the earth, a the semiaxis major, and e the eccentricity of the generating ellipse of the earth s spheroid, <f> and \ the latitude and longitude of dS, we have v = ^a 3 Vl - e 2 , dS = a 2 (1 - e 2 ) cos and the acceleration D n F is given in centimetres per second per second by the equation D n V = a-f (3 sin 2 <j> + y cos 2 < cos 2 A, where a = 978, ft = 5.19, and y is a constant, so that MISCELLANEOUS PROBLEMS. 371 Show that if x = e sin <. 2ff 2 (i-e 2 ) r c df r- 17 _ e Jo (1 x-)-Jo TF and and, assuming that log e- = 3.83050, log a = 8.80470, obtain Professor Woodward s equation, Ay> = 36797 x 10 ~ n . For a discussion of the value of p , see Prof. J. H. Poynting s Adams Prize Essay on " The Mean Density of the Earth." 105. If u is single-valued and harmonic at all points of a region but one (the exceptional point being an interior point P), and if u becomes infinite for all paths along which the point (a;, y) approaches P, then u can be written in the form u a- log r -f- r, where v is single- valued and harmonic at all points of the region. [Bocher.] 106. If the superficial density of a mass distributed on a spherical surface is inversely proportional to the cube of the distance from a fixed point A, the distribution is centrobaric. If A is inside the surface, it is the baric centre ; if A is outside, its inverse point is the baric centre. 107. If the superficial density of a columnar distribution on a cylindrical surface of revolution varies inversely as the square of the distance from a given line parallel to the axis of the cylinder, there is a baric line within the distribution parallel to the axis. Where is this line ? 108. A certain distribution M has two mutually exclusive closed equipotential surfaces, S l and S 2 , upon which V has the different constant values Ci and <7 2 . Will the potential func tion in space without /S^ and S 2 be the same for the original distribution and for distributions on ^ and S of surface 372 MISCELLANEOUS PROBLEMS. density a- = D H F/4 ?r together with so much of M as lies without the surfaces ? 109. The straight-line tangents at a point to a tube of force which ends there, evidently form a cone of definite solid angle. A number of points, P ly P 2 , P s , etc., have charges, m 1} m 2 , w 3 , etc. Show that if at any one of these points there end two tubes the solid angles of the cones of which are o> and CD , the flow of force in the one tube is to the flow of force in the other as W : to . Show also that if a tube starts with solid angle w k at a point P k where the charge is m k) and ends with solid angle cu r at a point P r where the charge is m r) o> k m k is numerically equal to c> r m r . 110. All the masses of a certain distribution lie within two closed surfaces S 1 and S 2J which exclude each other and are equipotential. All the lines of force which abut on a con tinuous portion A of S 1 also abut on S 2 . All the lines of force which abut on S { outside of A are open, while none of the lines which abut on S 2 are open. Show that one of the equipotential surfaces is made up of two lobes, one of which includes S 2 alone and the other both S 1 and S 2 . Separating the closed lines of force from the open ones is a surface which passes through the point where the lobes of the sur face just mentioned are connected. All the equipotential surfaces are closed. 111. The potential function due to a certain distribution of matter has a value at any point Q which depends only upon the distance, r, of Q from a fixed point 0. This value is 2 a*\ -r 2 - J, 3r or according as < r < a, a<r<bj b<r<c, or.r>c. What is the distribution ? [p 0, p 1, p 0, a = 1, p = 0.] MISCELLANEOUS PROBLEMS. 373 112. The lines of force due to two similar, homogeneous, infinitely long, straight filaments of repelling matter, parallel to the z axis and cutting the xy plane at the points (a, 0), (a, 0) are hyperbolas of the family x* 2 \xy if = a 2 . 113. (a) Prove that when there is symmetry about the axis from which is measured, > . P m (cos 0) and Pm ^ , where P m (cos 0) is the coefficient of a m in the development, in ascend ing powers of a, of (1 - 2 a cos 6 + a 2 )"*, are particular solu tions of Laplace s Equation in polar coordinates ; that is, of r - D r \r S>.+gjj- A(sin D e V) = 0. Hence show that any expression of the form A + A,r P^cos 0) + A. 2 >* P 2 (cos 0) + - 4- Aj P n (cos 0) , B Q , JVP^cosfl) , ^ 8 .P,(cosg) m -P m (cos6) / ^ ~7~ r-M where J , ^ , ^ 1; J5 1? J 2 , P 2 , etc., are arbitrary constants, satisfies the equation. The P s here introduced are some times called Legendre s Coefficients, sometimes Zonal Surface Spherical Harmonics. (b) Show that P (/x) = 1, (c) Show that when (9-0, P m (cos tf) or P m (l), the coefficient of a m in the development of (1 a) - 1 , is equal to 1. 114. Prove that if in any case of symmetry about a line, a convergent series a + a^ + a 2 z 2 +a 3 z 3 -\ ---- represents the 374 MISCELLANEOUS PROBLEMS. value of the potential function at a point Q distant z from the fixed point 0, both and Q being on the line ; then the series a P (cos 6) + a^P^cos 0) + a 2 r*P 2 (cos 0) -f a s i*P s (co8 6) H , formed by writing instead of z m in the former series r P m (cos 6), will represent in polar coordinates with as origin and the given line as axis from which 6 is measured, a finite, single- valued function which satisfies Laplace s Equation and for all points on the given line on the positive side of 0, where 6 = and r = z, has the same values as the given series. Given the radius of convergence of the first series, within what limits can we safely use the second series ? If any portion of the given line traverses a region of empty space, does the new series represent the potential function at all points in this region within the limits of convergency of the series ? 115. Prove that if in any case of symmetry about a line, a convergent series + -f + -f -f represents the value of the potential function, the series 0) , ftgP^cos 6) a 3 P 2 (cos 0) formed by writing instead of -^ in the former series, " will represent, so long as the new series is convergent, a finite, single-valued function which satisfies Laplace s Equation and, for all points on the line on the positive side of 0, has the same values as the given series. 116. Prove that the potential function due to a uniform circular ring of mass M, of radius a, and of small cross-section, is equal to M( _1 ft 2 .P 2 (cos(9) 1-3 a 4 .P 4 (cos(9) _ \ r\ 2" r 2 2- 4" r 4 "/ if aO, and equal to r -P^cosfl) 1-3 r 4 -P 4 (cosfl) a" ~ + 2-4 ~ COS0) \ ) MISCELLANEOUS PROBLEMS. 375 if a > r, where the centre of the ring is the origin, and the axis of the ring the axis from which is measured. 117. Prove that the potential function due to a uniform circular disc of mass Jf, of radius a, and of small thickness, is equal to 2Jfcf/l a 2 1 a 4 .P g (cosfl) 1-3 ft 6 . P 4 (cos 6) a* \2 r 2 2 -2!" r 3 " f ~2 3 .3!" i* if a < ?-, and to 2 if a > r, when the centre of the ring is the origin. 118. Show that the expression (r 2 c 2 -f- y*) /y of equa tion [21], page 12, is numerically equal to the length, k, of the chord of the sphere, formed by a radius vector drawn from P to a point L on the surface, distant y from P. The sign is to be taken negative or positive, according as L is or is not visible from P. Hence find an expression, Trcra(k l AVj/c 2 , for the intensity of the attraction of an " annulus " of a thin spherical shell lying between two parallels of latitude, at any point P on the axis. 119. A thin spherical shell of radius a attracts an internal particle P at a distance c from its centre. If the shell be divided into two parts by a plane through P perpendicular to the radius, the resultant attraction of each part at P is 2 Trcra \_a Va 2 c 2 ] /c 2 . [Todhunter s History of Attraction.] 120. The equation of the surface of an infinitely long homo geneous cylinder of density p, the lines of which are parallel to the z axis, being r=/(0), a filament of the cylinder of cross-section rdrdO contributes to the components (X, T) of the attraction at the origin the amounts 2 p cos dr dO and 2 p sin dr - dO respectively. If the cross-section of the cylinder is an ellipse of semiaxes a and b and if the origin is on the surface and distant y , x respectively from the 376 MISCELLANEOUS PROBLEMS. principal planes, the equation of the surface may be written in the form r = 2 (b 2 x cos 6 + a z y sin 0) / (b 2 cos 2 + a 2 sin 2 0). Assuming that C f . =^r.- X /*.tan Ja + btSitfx a b |_ A c*- prove that in this case X = 4 Trpbx /(a + 5), F = 4 7rpay / (a + b) and that the resultant force has the intensity 4 Tlf/ (a + b), where M is the mass of the unit length of the cylinder. Prove also, by a method analogous to that of Section 12, that the attraction due to a homogeneous shell bounded by two concentric, similar, and similarly placed elliptic cylin drical surfaces is zero within the shell, and that the attraction components (X, Y) at any point within a solid homogeneous elliptic cylinder are proportional to x and y respectively. 121. If two confocal ellipses (s and s ) have semiaxes (a, b) and (a 1 , b ) t a point (cc, y) on s is said to correspond to a point (x j y) on s , if x/x = a /a and y/y = #/& . Show that if P! and P 2 are any two points on s and P/ and P 2 , the corre sponding points on s , P-^PJ = P 2 Pi . Hence prove (Section 51) that, if two homogeneous, solid, confocal, elliptic cylinders of the same density be divided into corresponding thin strips by planes parallel to the xz plane, the x component of the attrac tion of any strip of the first at a point P on the second, is to the x component of the attraction of the corresponding strip of the second at a point P on the first corresponding to P f , as b is to b . The two components of the attraction of the whole of the first cylinder at P are to the same components of the attraction of the second cylinder at P, as b to b and as a to a . MISCELLANEOUS PROBLEMS. 377 122. It follows from the results stated in the last two problems, that if the components at an outside point Q of the attraction due to a solid homogeneous elliptic cylinder of density p bounded by the surface s (Fig. 124) be X and F, if a surface s confocal to s be drawn through Q , and if X and Y are the components, at Q on s which corresponds to Q on s t of the attraction of a cylinder of density p bounded by s ; X/X=b/b\ Y/Y=a/a , where a and b are the semi- axes of s, and a and b those of s . Show that A", Y are the components at Q of the attraction due to a cylinder of FIG. 124. density p, bounded by a surface s" drawn through Q, similar to s . Show also that, if the coordinates of Q are x, i/, X = 7rpbx/(a + b ), Y = 4 irpay/(a + ). Prove that, if a = o, b = 3, x = 4, and ij = if- ; x = , y = V5, . = 6, b = V20, so that, approximately, A" = 12p, Y = 13 - 42 p. 123. Two parallel planes, the direction cosines of the nor mals to which are (7, m, n), touch two confocal ellipsoidal surfaces at the points PI, PI respectively. The serniaxes of 378 MISCELLANEOUS PROBLEMS. the surfaces are (a, b, c) and (a -\- da, b + db, c + do) where, since they are confocal, d(a 2 ) = d(b 2 ) = d(c 2 ). Show that if p and p -f- dp are the lengths of the perpendiculars dropped from the origin on the tangent planes, p 2 = a 2 l 2 + b 2 m 2 -f c 2 n 2 , and p.dp = l 2 .d(a 2 ) + m? -d(b 2 ) + n 2 d(c 2 ) = d(a 2 ), so that dp is inversely proportional to p. If the surfaces bound a homo geneous shell, this is called a thin focaloid. Show that the thickness of the shell at the point P differs from dp, if at all, by an infinitesimal of higher order, and that a superficial distribution on an ellipsoid with surface density inversely proportional to p is equivalent to a thin focaloid bounded internally by the surface. The thickness of a thin homoeoid at any point is directly proportional to p. 124. Show that if the potential function due to a distribution of matter has the value zero at all points outside the ellipsoid Lx 2 + My 2 + Nz 2 = 1 and the value /* (1 - Lx 2 - Mx 2 - Nz 2 ) at all inside points, the distribution consists of a homogeneous ellipsoid of density /x, (L -f M + N) / 2 TT and a superficial stratum on it of surface density n/Zirp, where p is the length of the perpendicular dropped from the origin on the tangent plane. Since the surface distribution is equivalent to a thin focaloid, it is clear that the potential function due to a homogeneous ellipsoid has at outside points the same values as the potential function due to a thin focaloid of the same mass coincident with the surface of the ellipsoid. Prove from this that confocal ellipsoids of equal mass have equal potential functions at points outside both. 125. Two homogeneous, solid, confocal ellipsoids of masses MI and M 2 attract any particle outside both with forces which have the same direction and are to each other as J/i to Jf 2 . [Maclaurin.] 126. Show that it follows from the reasoning on pages 123 and 124 that the components, taken parallel to the axes, of the attraction of a homogeneous ellipsoid S at any point P on the surface of a homogeneous confocal ellipsoid S 1 of the MISCELLANEOUS PROBLEMS. 379 same density, are to the corresponding attraction components due to S at the point P on S which corresponds to P , as the areas of the principal sections of S and S perpendicular to these components. [Ivory.] 127. We know from the equations of page 191 that, in the case of a prolate ellipsoid uniformly polarized in the direction of the long axis, the depolarizing force is Prove that if the ratio of a to b is large, this is nearly equal to - 4 7rA(b 2 /a 2 ) [log (2 a/b) - 1], and that when a/b = 4, this approximate result is in error by about 4 per cent. Show that if we denote the depolarizing force in an ellipsoid of revolution uniformly polarized in the direction of the x axis by \A, A has the values 12.57, 6.63, 5.16, 4.19, 2.18, 0.95, 0.25, 0.0054, 0.0016, 0.0004, according as a/b is equal to 0, J, j, 1, 2, 4, 10, 100, 200, 400. 128. If the quantity c on page 121 be supposed to increase without limit, the limits of the expressions for X and Y are the force components within a homogeneous elliptic cylinder of semiaxes a and b. Making use of the integral dx 2 (x + #)* show that these limits are 4 irbpx / (a + b) and 4 irapy / (a + b). Using the form of integral given on page 124 in the seventh line from the bottom, show that if c be made to increase without limit, the limit of X is 4*pftz/(a f -f- b 1 ) and that the corresponding limit of Fis irpay /(a -f b ). Show that the equipotential surfaces within an infinitely long, solid, homogeneous, elliptic cylinder, the semiaxes of which are a and b, are elliptic cylindrical surfaces, the ratio of the semiaxes of any one of which is va/ vS 380 MISCELLANEOUS PROBLEMS. 129, Using the integrals given on page 190, show that if a = b > c and if X 2 = (a 2 c 2 )/c 2 , we may write the expres sions for the attraction components within a homogeneous oblate ellipsoid of revolution, in the form (- 3 Jfz/2 A V) [tan- 1 A - A/ (14- A 2 )], (- 3 My/ 2 A-V) [tan" 1 A - A/(l + A 2 )], (- 3 Mz/\ 3 c s )(\ - tan- 1 A). 130. Show that if in the case of the prolate ellipsoid of revolution where b = c < a, we put A = ea/c, the components of attraction at the inside point (x, ?/, z) may be written (3 Jfaj/AV) [A/ Vl + A 2 - log (A 4- Vl 4- A 2 )], (3 My 12 AV) [log (A + Vl + A 2 ) - A Vl + A 2 ], (3Mz/2 A 3 c 3 ) [log (A 4- Vl + A 2 ) - A VT+7 2 ]. 131. If these force components be denoted by X, Y } Z, the quantity (X j x 4- Y /y 4- Z /z) is numerically equal to 4 ?r/o within any ellipsoid of revolution. This is true in the case of every ellipsoid, as Poisson s Equation shows. 132. If a = a 2 , (3 = b 2 , y = c 2 , and if G has the value given on page 122, a-D a G 4 p-DfiGt + y DyG, = - t G () and 2 (a - ft) D a D 8 G = D a G - D & G { , The potential function V satisfies the equation at every point within a homogeneous ellipsoid. If (a , b , c ) are the semiaxes of an ellipsoidal surface through a point (x, y, ,?), confocal with the ellipsoid E which has the semiaxes (a, b, c), and if the result of substituting a , b , c for a, b, c in the expression for G be denoted by G , the value of MISCELLANEOUS PROBLEMS. 381 the potential function due to E at an external point may be written f J/ \ G + 2 x 2 D t G + 2y*.D m G + 2z 2 . D n G \ where I = a -, m = b* 2 , n == c". [Tarleton.] 133. Show that if X, Y, Z are the components of the body forces applied to a mass M of liquid revolving with uniform angular velocity <u about the axis of z, and if p denotes the pressure at the point (x, y, z), dp=(X + o>-x)dx +(T+ <*ry)dy + Z dz, so that at a free surface (X + ?x)dx +(Y+ rt/)dy + Zdz = 0. 134. If the liquid be homogeneous and exposed to its own attraction only, and if the bounding surface be the ellip soid tf<?y? + a-(?if 4- a-b 2 z 2 = a-lrc 2 , we have X=- Y = | J/Z> ?/, Z \ J/J/o^, and at the free surface Ircrxdx + a-rydy -h arlrzdz = 0, so that Show that this condition is satisfied for a given value of to by an oblate ellipsoid of revolution (Example 129) for which X satisfies the equation, X = tan [(3 X + 2 a> 2 X 3 /4 irp) /(3 + X 2 )] ; but that a prolate ellipsoid of revolution is not a possible form of the bounding surface. [Besant s Hydromechanics, Vol. I; Laplace s Mecanique Celeste, Vol. III.] 135. Prove that if V be the potential function due to any distribution of matter over a closed surface $, and if a- be the density of a superficial distribution on S, which gives rise to the same value, F , of the potential function at each point of S as that of a unit of matter concentrated at any given point 0, then the value at of the potential function due to the first distribution is j V 1 - a- dS. 382 MISCELLANEOUS PROBLEMS. 1/136. Show that the derivative of the function x 2 + xy + z* at the point (1, 2, 3) in the direction denned by the cosines (j, i, $ V2) is J(5 + 6 V2). Find the angle between the vector differential parameter of this function and the direction just defined, at any point of the plane 3 x -+- y -f- 2 A/2 = 0, at every point of the line x + y = 0, x = 2z, and at the origin. Show that it is not possible to find a scalar function the level surfaces of which cut orthogonally the lines of the vector (x + y, z, y). Show that the normal derivative of the function x * + y + % with respect to the function x + y -f- 2 is zero at every point of the plane x = 1. Prove that if u and v are the distances of the point (x, y, z) from two fixed points, h u = A, - 1. , 137. A harmonic function which has a constant gradient different from zero cannot vanish at infinity like the Newtonian Potential Function due to a finite mass. 1 138. [f(x, y, s), 0, 0], [3>(x), 0, 0], [*(y,-), 0, 0], the first of which is neither lamellar nor solenoidal, the second lamel lar but not solenoidal, and the third solenoidal but not lamellar, are examples of vectors the lines of which are parallel straight lines, though the intensities are not constant. Prove that if in any region the lines of a vector which is both lamellar and solenoidal are parallel straight lines, the intensity of the vector is everywhere in that region the same. * 139. (2x/r, 2 y/r, 2z/r) and (sin y, A/3, cos y), the first of which is lamellar but not solenoidal and the second solenoidal but not lamellar, are examples of vectors with constant inten sities, which have lines which are not straight lines parallel to each other. Prove that if the lines of a lamellar point function which has a constant tensor are parallel straight lines, the vector is solenoidal. Prove also that if the lines of a solenoidal vector point function which has a constant tensor are parallel straight lines, the vector is lamellar. 140. The vectors (x + 2 zy, y + 3 xz, xy), (2 zy, 3 xz, xy + 2z) have everywhere equal divergences and curls and their tensors MISCELLANEOUS PROBLEMS. 383 are equal all over the surface x 2 + y- 4 z- + 6 xyz 0. It is evident, therefore, that such vectors as these are not deter mined when their curls and divergences are given. What additional information would determine an analytic vector which does not vanish at infinity ? The scalar potential function of a certain vector has the value unity from r = to r 1, where r 2 = or + if + z- and the value 1 /r from r = 1 to r = oo . Is the vector everywhere solenoidal and lamellar ? Can you determine an everywhere lamellar and solenoidal vector which has the value 13 at infinity? /^l41. If at any surface the normal component or a tangential component of a vector is discontinuous, must we suppose that there is divergence at the surface ? Illustrate your answer by a simple numerical illustration. 142. S is a portion of an analytic surface bounded by the closed gauche curve s. S is a surface which divides space into two portions in each of which the components of a vector Q are represented by analytic functions. At S , some of the com ponents of Q parallel to the surface are discontinuous. S cuts S in the curve s which divides S into two portions, Si and S z . Two curves in S l and S 2 respectively drawn parallel to s and very close to it shall be called V ands 2 - K n shall be the con tinuous component, in the direction of the normal to S, of the curl of Q. That portion of s which with s/ embraces practi cally the whole of Si shall be called s t ; that portion of the remainder of s which with s. 2 embraces nearly the whole of S 2 is to be denoted by s 2 . Apply Stokes s Theorem to Si as bounded by s 1 and s/ and to S 2 as bounded by s 2 and s 2 , and show that the line integral of the tangential component of Q around s is not in general accounted for by the surface integral of K n over S, unless we assign to K n on s a value such that its line integral along the line is finite. What is this value ? On page 113 Stokes s Theorem is predicated only of analytic vectors. Justify the uses made of the theorem on page 219 and in Sections 82 and 88. 384 MISCELLANEOUS PROBLEMS. 143. Assuming that the surface integral of the normal out ward component of any vector taken over any closed surface S, within and on which the vector is analytic, is equal to the volume integral of the divergence of the vector taken through out the space within the surface, show that if in spherical coordinates R, 0, 3> are the components of a vector Q, taken in the directions in which r, 0, <f> increase most rapidly, the divergence of Q is given by the expression D r (r*R) / r 2 + D e (sin ) / r sin + D^ / r sin 0. 444. Assuming that, if , ^ are three analytic functions which define a system of orthogonal curvilinear coordinates, and if h^ h^ h$ are the gradients of these functions, the sur face integral, taken over any closed surface S, of 7 -cos (, ri) (where U is any function analytic within and on S, and (, ri) is the angle between the exterior normal to S at any point on S, and the direction at that point, in which increases most rapidly) is equal to the volume integral extended through the space enclosed by S, of /^ \ -h^-d \U j\ ^] /#, show that, if Q& QW Q$ are ^ ne components in the directions in which , r), and increase most rapidly, of an analytic vector Q, the normal component of U integrated all over S gives Write down an expression for the divergence of an analytic vector in terms of , y, , and, assuming that in the case of spherical coordinates h r 1, h e = !/?, 7fy = 1/r sin 0, show that this yields the result stated in the last problem. 145. Let P be a fixed point and P a movable point in the unlimited region T, without a given surface S, and let P P be denoted by r. Show that if a function G can be found which (1) on S has the value 1/r, which (2) is harmonic at P and at every other point of T, and which (3) vanishes at infinity like the Newtonian Potential Function of a finite MISCELLANEOUS PROBLEMS. 385 mass, G is unique. Show also that if G = G + I/?*, and if ic is any function harmonic in T, which vanishes at infinity like a Newtonian Potential Function and has the value tc (> at P , 47rz (w-D n GdS, where n represents an exterior normal to S. Some writers call G " Green s Function " for the given S and the given P ; others reserve this name for G . Attach a physical meaning to G. Define a Green s Function for space inside a closed surface S. Show that if S is a plane and if r is the distance of P from the image, in the plane, of the pole P , the function G is 1/r-l/r . 146. Show that the expression I I 2 p l log (r/r ) dA lt where r is any constant, might be used for the logarithmic potential function of a columnar distribution of repelling matter. 147. Show that in general the surface density of a charge distributed on a conductor is greatest at points where the convex curvature of the surface of the conductor is greatest. 148. Show that if Z, ra, n are scalar point functions which define a set of orthogonal curvilinear coordinates in an electric field in air where the potential function is V, and if L, M, N represent the force components taken at every point in the directions in which the coordinates increase most rapidly, L = - h t D f V, M= - h m D m V, N= * h n D n V, and Laplace s Equation can be written A (L /h m /g + D M (M/h t k) + D n (N/h, . h m ) = 0. 149. Prove that if a distribution of electricity over a closed surface produces a force at every point of the surface perpen dicular to it, the potential function is constant within the surface. 150. Two conducting spheres of radii 6 and 8 respectively are connected by a long fine wire, and are supposed not to be 386 MISCELLANEOUS PROBLEMS. exposed to each other s influences. If a charge of 70 units of electricity be given to the composite conductor, show that 30 units will go to charge the smaller sphere and 40 units to the larger sphere, if we neglect the capacity of the wire. Show 25 also that the tension in the case of the smaller sphere is 2oO7T per square unit of surface. 151. The first of three conducting spheres, A, B, and C, of radii 3, 2, and 1 respectively, remote from one another, is charged to potential 9. If A be connected with B for an instant, by means of a fine wire, and if then B be connected with C in the same way, C"s charge will be 3 6. [Stone.] If, in the last example, all three conductors be connected at the same time, C s charge will be 4 5. 152. A charge of M units of electricity is communicated to a composite conductor made up of two widely separated ellip soidal conductors, of semiaxes 2, 3, 4 and 4, 6, 8 respectively, connected by a fine wire. Show that the charges on the two ellipsoids will be -J M and f M respectively. Compare the values of 2 Tnr 2 at corresponding points of the two conductors. 153. Can two electrified bodies attract or repel each other when no lines of force can be drawn from one body to the other ? 154. Two conductors, A and B, connected with the earth are exposed to the inductive action of a third charged body. Do A and B act upon each other ? If so, how ? 155. A spherical conductor A, of radius a, charged with M units of electricity, is surrounded by n conducting spherical shells concentric with it. Each shell is of thickness a, and is separated from its neighbors by empty spaces of thickness a. Show that the limit approached by V A as n is made larger and larger is (M/a) log 2. 156. The superficial density has the same sign at all points of a conducting surface outside which there is no free electricity. MISCELLANEOUS PROBLEMS. 387 157. An insulated and uncharged spherical conductor of radius 4 centimetres contains an eccentric spherical cavity the radius of which is 2 centimetres. At the centre of the cavity is a point charge of 10 units. Show that the charges on the inner and outer surfaces are uniformly distributed and that the value of the potential function at all points within the cavity is 10 /r 2.5. 158. A spherical conductor of 10 centimetres radius is sur rounded by a concentric conducting spherical shell of radii 12 centimetres and 15 centimetres. The sphere is at potential zero and the shell at potential unity. Show that the charges are - 60, 60, and 15. 159. Prove that the electrical capacity of a conductor is less than that of any other conductor in which it can be geometrically enclosed. 160. Show that two exactly similar conductors symmetri cally situated on opposite sides of a plane, so that one is the optical image of the other in the plane, repel each other if raised to the same potential. 161. Prove that the following statements are true : If any conductors, some or all of which are charged, are exposed to one another s influences but are far removed from all other charged bodies, the charge on one, at least, of the conductors must have the same sign throughout. If two charged con ductors, uninfluenced except by each other, have equal and opposite charges, the surface density at every point of one has one sign and the surface density at every point of the other the opposite sign. A charge, 1, concentrated at any point P produces a distribution of one sign throughout upon a conductor C which carries a total charge of 1 + /x, /x being any positive quantity whatever. If two conductors influenced only by each other are at potentials of the same sign, the distribution has the same sign throughout upon that one of the conductors the potential of which is the greatest in abso lute value. If two conductors influenced only by each other 388 MISCELLANEOUS PROBLEMS. are at opposite potentials, the distribution in each has the same sign everywhere that the potential function has. A charged conductor is always attracted, in the absence of other charged bodies, by every other conductor, in its neigh borhood, which is put to earth. [Duheru.] If n is the number of unit Faraday tubes, per square centi metre, which pass through any small portion of an equipoten- tial surface of an electric field in air, the strength of the field on this small area is kirn. 162. If when a unit charge is placed on a conductor C in the presence of conductors C lt C 2 , kept at potential zero, the charges on these are e lt e. 2 ; then if C be discharged and insulated and G\, C 2 be raised to potentials V v F 2 , the potential of C will be <?!?! + C, F 2 . 163. A soap bubble of surface tension T has a charge Q. Show that its diameter is Q^/(2-n-T)^ 164. Prove that the capacity of n equal spherical condensers when arranged in cascade is only about -th of the capacity of one of the condensers ; but that if the inner spheres of all the condensers be connected together by fine wires, and the outer conductors be also connected together, the capacity of the complex condenser thus formed is about n times that of a single one of the condensers. 165. A conductor the equation of the surface of which is +j+*.i 25 16 9 is charged with 80 units of electricity ; what is the density at a point for which x = 3, y = 3 ? If the density at this point be a, what is the whole charge on the ellipsoid? 166. A charged insulated conductor A is so surrounded by a number of separate conductors B, (7, D, , which are put to earth, that no perfectly straight line can be drawn from MISCELLANEOUS PROBLEMS. 389 any point of A to the walls of the room without encountering one of these other conductors. Will there be any induced charge on the walls of the room? 167. Assuming that in the case of a conductor surrounded by dry air, 8007r dynes per square centimetre is the greatest pressure that a charge on the conductor can exert at any point upon the air without breaking down the insulation, show that a spherical conductor must have a diameter of at least 0.126 centimetres in order to hold, in dry air, one elec trostatic unit of electricity. 168. Prove that two pith balls each 4 millimetres in diame ter and 3 milligrammes in weight, suspended side by side by vertical silk fibres 10 centimetres long, cannot be so highly charged with electricity that the fibres shall make an angle of 60 with each other. 169. Discuss the following passage from Maxwell s Elemen tary Treatise on Electricity : " Let it be required to determine the equipotential surfaces due to the electrification of the conductor C placed on an insu lating stand. Connect the conductor with one electrode of the electroscope, the other being connected with the earth. Elec trify the exploring sphere,* and, carrying it by the insulating handle, bring its centre to a given point. Connect the elec trodes for an instant, and then move the sphere in such a path that the indication of the electroscope remains zero. This path will lie on an equipotential surface." 170. A condenser consists of a sphere A of radius 100 sur rounded by a concentric shell the inner radius of which is 101, and outer radius 150. The shell is put to earth, and the sphere has a charge of 200 units of positive electricity. A sphere B of radius 100 outside the condenser can be connected with the condenser s sphere by means of a fine insulated wire passing * A very small conducting sphere fitted with an insulating handle. 390 MISCELLANEOUS PROBLEMS. through a small hole in the shell. B is connected with A ; the connection is then broken, and B is discharged ; the con nection is then made and broken as before, and B is again discharged. After this process has been gone through with five times, what is A s potential ? What would it become if the shell were to be removed without touching A ? [2(101) 4 /(102) 5 , 2 (101) 5 /(102) 5 .] 171. If the condenser mentioned in the last problem be insulated and a charge of 100 units of positive electricity be given to the shell, what will be the potential of the sphere ? of the shell ? If we then connect the sphere with the earth by a fine insulated wire passing through the shell, what will be the charge on the outside of the shell? What will be the potential of the shell ? If next A be insulated, and the shell be put to earth, what will be A s potential ? What will be its potential if the shell be now wholly removed ? VKftf Wu *U flat M^nfcaL ir fa- carlX te *&4 tfw^ 4m e4> [2/3, 2/3, - 4040/41 ; 60/41, 2/205, - 2/205, - 202 /205.] 172. A conductor is charged by repeated contacts with a plate which after each contact is recharged with a quantity (E) of electricity from an electrophorus. Prove that if e is the charge of the conductor after the first operation, the ultimate charge is E e / (E e). 173. If one of a system of n conductors entirely surrounds all the others, 2 n 1 of the coefficients of potential have the common value p. If the outside conductor be put to earth, it loses a quantity Q of electricity. Show that the energy loss is \pQ\ 174. A conductor is formed of two infinite planes inter secting at right angles and is connected with the earth. A long straight wire, parallel to the intersection of the planes, at distances b and a from them, has an electric charge e per unit length. Show that the electrification of the first MISCELLANEOUS PROBLEMS. 391 plane at a distance x from the line of intersection of the - 4 abex/7r [(a 2 + b- + x 2 ) 2 - 4 aV]. 175. The energy, per unit of surface, of a plane parallel plate condenser in which the superficial charge density is <T O is 2 7ro- 2 a when the distance between the plates is a. Show that if the distance be decreased to a Aa the energy is 27ro- 2 (>- Aa) if the charge remains constant, and 2 mr *a* / (a - Aa) if the potential remains constant. Hence prove that the rates of change of the energy are equal in value but opposite in sign in the two cases. 1 76. The foot of the perpendicular dropped from any point P upon the line A^A Z shall be marked J/. At A^ is a point charge m l and at A 2 a point charge ra 2 , m being greater than ^P = r v A 2 P = r v AJf = x, MP = y, A,A 2 = a, rn^/m^ Show that the surface integral of normal force parallel to the x axis over an infinite plane through M perpendicular to A^A 2 is 2 TT (m l m. 2 ) if x >a ; 2 TT (?n l + m 2 ) if < x < a ; and 2 TT (m 2 m^ if a; < 0. The induction outward through an infinite spherical surface with centre at any finite point is 4 TT (m l ra 2 ). Show that the value at any point on a spherical surface of radius r^ with centre at A l} of the normal outward component of the force is rn^/r-f ra 2 cos (r 15 r 2 )/r 2 2 , and this is positive for every point of the surface if r l >a,fji/(p 1). It follows from this that no line of force can come from infinity to the charge on A\ but 47r^ 2 of the 4 irm v lines which start from A^ reach A 2 . Show that all the lines of force which cross the two planes drawn perpendicular to A^ through A t and A 2 cross them from left to right. The inductions across these planes are 2irm z and 2-irm^ Through M } any point of A^A^ imagine a plane drawn perpendicular to 392 MISCELLANEOUS PROBLEMS. A-iAt and let a circumference be drawn on this plane with If as centre and MR as radius. Let the angles which A^R and A 2 R make with the line from A l to A 2 be o^ and o> 2 , then the induc tion through the circle is 27^?% (1 cos o^) -f m 2 (cos o> 2 1)] or 2ir[m l (L cos o^) + in a (1 + cos o> 2 )] according as A^M is greater than a, or positive, and less than a. If in the last case the radius be so chosen that the circle shall include all the FIG. 125. lines which converge to A 2 , we must equate the induction to 4 Trm 2 . This yields m L cos o^ m 2 cos o> 2 = m l m 2 , which may be regarded as the equation of the surface of separation between the lines which go from A l to A 2 and those which go to infinity. 2 TT [m : (1 cos wj) + wi 2 (1 + cos o> 2 )] = (7 is the equation of a surface of revolution which includes everywhere C lines MISCELLANEOUS PROBLEMS. 393 of force. Since every meridian curve of this surface is itself a line of force, the equation just written may be regarded as the general equation of the lines of force. If m v = m 2 , the lines are sometimes called "magnetic lines." In this case the equation becomes cos o^ cos w 2 = const., and the lines have the forms of the curves which pass through the points N, S in Fig. 125. Show that if p. = 1, if R is the resultant force at any point P, and if Q is the point where the line of action of R cuts A^A^ jK/[sin (r lt r 2 )] = ??i/[r 1 2 sin (It, >- 2 )] = w/[r 2 2 sin (R, r^~\ or, since sin (A,PQ) = (sin PQA and sin ( A 2 PQ) = (sin PQAJ (QA 2 ) / r 2 , If Q is fixed, P must move so that i\/r. 2 is constant : its locus is, therefore, a circle. [See Mascart et Joubert, 168 and 169, and also Nipher s Electricity and Magnetism, Ch. III.] 177. Two condensers A and B have capacities Ci and C 2 . A is charged by a certain battery and then discharged ; it is then charged and its charge is shared with B ; finally A and B are both discharged. Show that the energies of the different discharges are to each other as [Clare College.] 178. An earth-connected circular disc 5 centimetres in radius is suspended horizontally from one arm of a balance, and an insulated plate is placed parallel to it and 1 centimetre below it. When the lower plate has been electrified there is found to be an attraction equal to the weight of 1.274 grammes between the two. Show that, assuming the electricity to be uniformly distributed, the potential of the lower plate is about 6000 volts. 394 MISCELLANEOUS PROBLEMS. 179. S is an equipotential surface due to a distribution of matter of which it encloses a portion M v and excludes a por tion M 2 . Let M l be distributed on S according to the law 4 TTCT = D n V: then superpose on the system thus formed the negative of the original system, so as to have the surface S at zero potential due to the distribution on it and to the negative of MI within it. What will now be the value of the poten tial function without S? At a distance 8 2 from the centre of a spherical cavity of radius r, in a conductor which is at potential zero, is a point charge of m 2 units. Find by aid of the formulas given in Section 65 the density of the charge on the wall of the cavity. 180. If a conductor C, which entirely surrounds a system of charged and insulated conductors, be at first insulated and at potential V, and then put to earth, the potentials of all the interior conductors will be diminished by V. If this system be now discharged, the loss of energy is the same as if C had not been put to earth but had had the interior conductors put into connection with its inner surface. [M. T.] 181. Show that r/Sths of the unit Faraday tubes proceeding from an electrified particle, at a distance 8 from the centre of a conducting sphere of radius r, which is put to earth, meet the sphere, if there are no other conductors in the neighbor hood, and that the rest go off to infinity. 182. If a charge m l is placed at a point A l distant 8 X from the centre of a conducting sphere of radius r (Section 65) kept at potential zero, the charge induced on the surface has v the density cr = m 1 (8^ r 2 )/4 7 rrr 1 3 at a point distant r } from AV and the total amount of the induced charge is ra^/Sp The attraction between the point charge and the induced charge is m^rS l /(8 ] 2 r 2 ) 2 . If now a charge M be distributed uniformly over the sphere so as to raise its potential to M/r or F , the new density will be - m, (V - r 2 )] /4 TT rrf, MISCELLANEOUS PROBLEMS. 395 the new charge E = V Q r ?i> /Si, and the attraction F = [<> 8i/ (8f - r) 2 - m, F r/&fi, or m*r ^/(Sf - f 2 ) 2 - m.E/S 2 - < r/Sf. This attraction is zero when 8 L satisfies the equation E B, (8f - r 2 ) 2 = m, r 3 (2 8^ - r 5 ). If M=+m l r/8 l , the total charge on the sphere will be zero. In this case F = ^1/81, and the force of attraction is w 1 2 ^(28 1 2 *)/(8i 2 - r 2 ) 2 ^ 3 : this expression is always posi tive. The density on the sphere is zero, if anywhere, on a circumference determined by the equation E + ?H! T/S! = i r (^ ~ ^)/r?. and A 2 are inverse points with respect to a spherical surface S of radius VSf r 2 , the centre of which is ^. If, therefore, T is any point on S, A. 2 T-B l = OT- VS L 2 r 2 and, if Jf = m x r j VSf -r, the potential function has the same uniform value on *S and on the conductor. The intersection of the two surfaces is a line of no force and no density. The potential function due to m l alone is the same as that due to ???j and the charged sphere, at all points on the spherical surface OP j A. 2 P = M 8 X /m l r : if E= 0, this is the plane which bisects A 2 at right angles. The mutual potential energy of the point charge m^ and the distribution on the sphere is - i w 1 2 r s /8 1 2 (8 1 2 - >~). Show that if a charged conducting sphere of radius 10 centi metres is at potential F in the presence of a point charge of 12 units at a distance of 20 centimetres from the centre of the sphere, the whole charge on the sphere is 10 (FJ, f). 396 MISCELLANEOUS PROBLEMS. Show also that F is 2/15, 6/5V3, or 3.6 according as the density of the surface charge is zero, at the point of the sur face farthest from AI, at a point just visible from A^ or at the point nearest AI. Show that if the whole charge 011 the spherical surface is 14/3 there is no attraction between the point charge and the surface charge ; and that if the sphere was originally uncharged and insulated, its potential was constantly equal to 12 /8 X as the point charge gradually approached its present position from infinity. Show that the integral of (Sf r 2 )/^ 3 taken over the sur face of the sphere is 4 Trr 2 /^. How much of the charge on the sphere is visible from A v ? Find the surface density on a spherical conductor at poten tial zero under the action of two equal external point charges situated at equal distances on opposite sides of the centre. Consider separately the case where the point charges have opposite signs. 183. An insulated conducting sphere of radius r charged with m units of positive electricity is influenced by m units of positive electricity concentrated at a point 2r distant from the centre of the sphere. Give approximately the general shape of the equipotential surfaces in the neighborhood of the sphere. Give an instance of a positively electrified body the poten tial of which is negative. 184. Prove that if the spherical surfaces of radii a and b, which form a spherical condenser, are made slightly eccentric, c being the distance between their centres, the change of elec- 3 abc - cos trification at any point of either surface is - -77, r-> 47r(& a)(b 3 a s ) where is the angular distance of the point from the line of centres, and where the difference between the values of the potential function on the two surfaces is unity. 185. Show that if an insulated conducting sphere of radius a be placed in a region of uniform force (^o)? acting parallel to MISCELLANEOUS PKOBLEMS. 397 the axis of x, the function X Q x (1 a 3 /?- 3 ) + C satisfies all the conditions which the potential function outside the sphere must satisfy, and is therefore itself the potential function. Show that the surface density of the charge on the sphere is j and prove that this result might have been obtained by 4 ird making ^ infinite in the formulas near the top of page 206. 186. If <7 n , </22 are the coefficients of capacity of two of a set of conductors, and if ^ 12 is their coefficient of mutual induction, the capacity of the compound conductor formed by joining these two conductors by a fine wire is q n + 2 q lz + q w if all the other conductors be put to earth. If p n , p^, p lz are the coefficients of potential of the two conductors, and if all the other conductors of the series are uncharged and insulated, the capacity of the compound conductor is If the distance b between the centres of two conducting spheres of radii a v a 2 is large compared with the diameter of either, p u = 1/a lf p. 22 = l/ 2 , and _p 12 is approximately 1/6, so that if e lt e 2 are the charges of the spheres and V lf V z their potentials, F x = e l /a l + e.,/b, V 2 = e l /b + e. 2 /a. 2 . Show that, approximately, q. 2 . 2 = ajr / (Ir a^a^). 187. If on the radius vector OP drawn from a fixed point to another point P a new point P be taken, such that OP OP = a 2 , where a is a constant chosen at pleasure, P and P are said to be inverse points, is the centre of inversion, a sphere of radius a with centre at is the sphere of inversion and a the radius of inversion. One of a pair of inverse points is without the sphere of inversion and the other within, unless both coincide, The straight line which 398 MISCELLANEOUS PROBLEMS. joins the points of contact of tangents to the sphere drawn from an outside point P passes through the inverse point P. If P, P and Q, Q are pairs of inverse points, the triangles OPQ and OQ P are similar. If one (P) of a pair of inverse points moves along a curve, or over a surface, or through a space, the other (P 1 ) will generate the inverse curve, surface, or space. A plane at a perpendicular distance b from inverts into a spherical surface of radius a 2 / 2 b, passing through 0. A spherical surface of radius c with centre at a distance b from inverts into another spherical surface of radius a 2 c/(b 2 c 2 ) with centre at a distance a 2 b/(b 2 c 2 ) from 0. If a 2 = b 2 c 2 , the spherical surface inverts into itself, though the inverse of the old centre is not the new centre. The centre of inversion inverts into the region at infinity. Prove that if the origin be the centre of inversion, a point P or (x, y, TV), distant r from the origin, inverts into a point P or (a; , y j z ), distant r from the origin, where rr = a?, x/r = x /r j y /r = y /r , z/r = z /r , x = a 2 x /r 12 , y = a 2 y /r 2 , z = a 2 z /r 2 , x = a 2 x/r*, y = a?y/r>, z = a?z/r>. An element of arc ds at P inverts into an element of arc ds at P , such that ds = r 2 -ds /a 2 = a 2 -ds /r 12 . An element of area dS at P inverts into an element of area dS 1 at P , such that dS=i*-dS /a 4 = a 4 >dS /r *. An element of volume dr at P inverts into an element of volume dr at P , such that dr = r Q -dr /a fi = aP-dr /r 6 . The angle between two curves which intersect at P is equal to the angle between the inverse curves which intersect at P . If P and P be drawn in different diagrams, in which the rectangular Cartesian coordi nates are x, y, z and x , y , z respectively, = a 2 x /r 2 , y = a 2 y /r 2 , = a?z /r 2 , define a set of orthogonal curvi linear coordinates in the second diagram, and the Cartesian coordinates of P in the first diagram are equal to the curvi linear coordinates of P in the second diagram. Any func tion F(x, y, z) has the same numerical value at P that the MISCELLANEOUS PROBLEMS. 399 function F(a*x /r *, a*y /r *, aV/V 2 ) = f (x f , y , z ) has at P . Prove that (D? + #/ + A 2 ) ^ (*, y, ) at P = (r YO (A- 2 + D/ + A- 2 ) KA ) atp-. If P is zero on any surface or throughout any space in the first diagram, aif//r is zero on the corresponding surface or throughout the corresponding space. If F has the constant value c on the surface S, cnf//r has the value ac/r , which is not constant on the corresponding surface S . If F is the potential function due to a volume distribution of density p in a region T, together with a superficial distri bution of density a- on a surface S and a point charge e at a point (), ( ai A/ 7 ) i s th potential function due to a volume distribution of density p = a 5 p/r 5 in the region T f , corre sponding to T, together with a superficial distribution of density o- = <z 3 cr/r 3 on the surface S , which corresponds to S, and a point charge e = r e/a at the point Q , which is the inverse of Q. The inverse of a point charge e at the centre of inversion is a charge at infinity, which raises all finite points to potential e/a. If F is the potential function of a distribution o-, p which keeps a certain surface S at potential zero, (a\f//r ) will be the potential function of a distribution a- , p which keeps the corresponding surface S at potential zero. If F is the potential function of a distribution o-, p which keeps the surface S at potential c, (a^//r ) will be the potential function of a distribution or , p which keeps S at the potential ac/r : if, however, we add to the distribution o- , p a point charge ac at the origin, the new potential function will keep S at potential zero. 188. Show that if a point charge e be anywhere between two infinite planes which form a diedral angle of 60, these planes would form a surface of potential zero due to the original charge and five images in the planes. Find the den sity of the charge on two planes which form an angle TT/W, 400 MISCELLANEOUS PROBLEMS. if they are kept at potential zero in presence of a point charge between them, invert the system with respect to the charged point. 189. A homogeneous sphere of density p and radius c has its centre at a point C distant d from an outside point 0. The value of the potential function at a point P outside the sphere is $wp&/CP. Show that if the distribution be inverted, using as centre, the new distribution is a heterogeneous, cen- trobaric sphere of mass %irp<?a,/d, the baric centre of which is the inverse point of C. [Routh.] 190. A point charge -f- e lies on the x axis at a distance + b from the origin between two conducting plates, x = 0, x = 2 c, both of which are kept at zero. Show that the images of the point charge in the planes are an infinite series of point charges numerically equal to e but alternately positive and negative at points on the x axis. The coordinates of the nega tive images are I, (4c + ft), (8c + ft), -,(4c b), (8c b), (12 c ft), . . ., and those of the positive images are (4 c -f- ft), (8 c + ft), (12 c + ft), , - (4 c - ft), - (8 c - b) , - (12 c - ft), - . - . Show that the force at any point between the planes might be computed from these images and the original point charge. Indicate a method for determining the density of the induced charges on the plates. State clearly the result of inverting- the system, using the original charged point as centre of inver sion, and each of several different values for a. If in this problem the charge e is at a point midway between the plates, and if this point be chosen as origin, ft = c, and there are positive images at points the x coordinates of which are 0, 4 c, 8 c, 12 c, , 4 c, 8 c, 12 c, , and nega tive images at points the x coordinates of which are 2 c, 6 c, 10 c, , 2 c, 6 c, 10 c, . Show that if the system be inverted, using as centre of inversion and c as radius of inversion, and if the inverse of the charge at be omitted, the result is a conductor formed of two spherical surfaces of radius r = $c, in contact at potential F" = e/c under MISCELLANEOUS PROBLEMS. 401 positive charges equal respectively to e, | e, -fa e, , at points the x coordinates of which are J- c, | c, T T 5 c, ; and negative charges \ e, ^ e, T V e > " > a ^ Pi n ^s the x coordi nates of which are i c, c, y 1 ^ c, . The total charge in each of the spheres is - i (1 - I + 4 - i ) = - le log 2 = [> log 2, and their mutual repulsion, F 2 (log 2 i). 191. If two spherical conductors each of radius a have charges e^ e 2 and are at a great distance apart, the energy of the system is (e? -f 2 2 )/2 a. If the two are brought up into contact, the whole charge of the compound conductor thus formed is (e l + e, 2 ), it is at potential (e l + e. 2 ) / 2 a log 2, and the energy of the system is (^ + e 2 ) 2 /4a-log2. Show that the work done against the mutual repulsions of the two charges during the approach of the spheres is about [(0.722)6^ - (0.139) (ef + e.^/a, and discuss separately the special cases e l = 0, e l = e.,, e^ = 5 &,, e = | e* 192. Show that if a point charge be situated at a point 0, between two concentric spherical surfaces, it is possible to find a series of electric images which together with the origi nal charge would keep each of the surfaces at potential zero. What would be the result of inverting the system, using O as centre ? 193. A certain condenser consists of a closed conducting surface S l surrounded by another closed conducting surface $>, separated from the first by a homogeneous dielectric. When the condenser is charged, the lines of force between S l and S.> are the same as if S 2 were removed and ^ freely charged. What do you know about S l and $ 2 ? 194. The semiaxes of a conducting prolate ellipsoid of revolution are 10, 8, 8. Find, by help of the formulas of Sec tions 6 and 23, the external field when the conductor has a 402 MISCELLANEOUS PROBLEMS. free charge of 60 units, and show that the surface density at the equator is then 3 / 16 IT. 195. A prolate conducting spheroid of major axis 2 a and minor axis 2 b, has a charge of electricity E. Prove that the attraction between the two halves into which it is divided by its diametral plane is E~ log(a/b )/4(a 2 - b 2 ). [St. John s College.] 196. If a particle charged with a quantity e of electricity be placed at the middle point of the line joining the centres of two equal spherical conductors kept at zero potential, the charge induced on each sphere is - 2 em I -ra 2??t 2 where m is the ratio of the radius of either of the spheres to the distance between their centres. 197. A conducting sphere of small radius a is situated in the open air at a considerable height h above the ground. Show that its electrical capacity is increased by the neighbor hood of the ground in the ratio of 1 + - J to 1, very nearly. 198. A negative point charge, e. 2 , lies between two posi tive point charges e l and e 3 on the line joining them and at distances a and b from them respectively. Show that if where 1 < X 2 < b a a -f b there is a circumference at every point of which the force vanishes. 199. Two spherical conducting surfaces of radii a and b form a condenser. Prove that if the centres be separated by a small distance d, the capacity is approximately ab C abd 2 b-a I (b-a)(b 3 When d = 0, the capacity is } - a 3 ) J MISCELLANEOUS PROBLEMS. 403 200. A small insulated conductor, originally uncharged, is connected alternately with two insulated conductors A and B at a considerable distance apart. Prove that if e and e are the original charges of A and B, e l and e^ their charges after the carrier has touched A and then touched B, the charge of B when the carrier has touched A and B each n times is ab ej - e e <?/ ab-1 (ab - 1) a n - l b*- 1 where a = ej e^ and b = (e + e ej /e^. The charges of & B, and the carrier, are ultimately in the ratios 201. If a series of conductors were constructed which might be made to coincide with the closed level surface of a harmonic function w which vanishes at infinity like a Newtonian Potential Function, the capacities of any two of these conductors would be to each other in the ratio of the reciprocals of the values of w on the corresponding surfaces. If two of the surfaces for which iv = w l and w = w 2 < u\ be constructed of metal, and if charges E l and E 2 be given them, the energy is 2\ cc 9 H l ~cT^ where Ci and C 2 are the capacities. The energy becomes \(E l + E 2 ) 2 /C 2 if the two are connected. 202. An insulated conducting sphere of radius r, bearing a charge ra, is introduced into a field of force due to a fixed distribution M of electricity. Show that if the value of the potential function due to M at the centre of the sphere is (7, the value of the potential function within the sphere is C + m/r. 203. Compute the force at the point (#, ?/, z) due to a par ticle of mass ra at the point (a, 0, 0) and a particle of mass 404 MISCELLANEOUS PROBLEMS. + rn at the point ( a, 0, 0), and show that if m and a be made to increase indefinitely in such a manner that the ratio of m to a 2 is always equal to the constant A, the field becomes ultimately a uniform field of intensity X = 2 A. 204. It is evident that the value at any point P in the xy plane, of the potential function due to two slender, infinitely long, homogeneous, straight filaments (of mass m and m respectively per unit length) which cut the plane perpendicu larly at the points A and B, is 2 m log (AP/BP), and that the equipotential surfaces are circular cylinders [one is a plane] such that A and B are inverse points with respect to every one of them. If the radius of any one of these cylindrical surfaces the axis of which cuts the xy plane at C be denoted by r (Fig. 126), and if AP = r lt BP = r 2 , AC = 8 1? BC = 3 2 , 8 l /r=r/8 2 = r l /r 2 , and the triangles BCP and ACP are similar if P lies on the cylinder. The resultant force F at P has the direction of the normal to the cylinder, the repulsion due to the 126. filament which cuts the plane at A is 2 m/r L ,and the attraction due to the filament which cuts the plane at B is 2m/r 2 . If the angle APB be denoted by a, the Principle of the Parallelogram of forces applied at P yields F/ sin a = 2 m / [r 2 sin (r, ?-,) ] = 2 m / [^ sin (>, r 2 ) ] , and the Theorem of Sines applied to the triangle APB yields AB /sin a = 7-i/sin (r, r^) = r 2 /sin (r, r 2 ), so that F= 2 mAB/r^ -r 2 = 2 m^AB/r ^ 2 = 2 m^AB/r- r* ? = 2m(r 2 - S 2 2 )/r-r 2 . MISCELLANEOUS PROBLEMS. 405 The value of the potential function on the cylinder is 2 m log 8, / / = 2 m log r / 8 2 . If, now, the mass of the filament which cuts the plane at B be spread on the cylindrical surface so that the surface density at every point is the potential function outside the cylinder will be unchanged. If, finally, a mass m per unit length be spread uniformly over the cylinder, the value of the potential function within and on the surface will be 2 m log (Sj / r) -f- 2 m log r, and, by a suitable choice of m , this may be given any value. The whole charge on the unit length of the cylindrical surface is m m = M, the value of the potential function on the sur face is V s =2 m log-^/r) -+- 2(M + m) log r, and the surface density at a point distant r^ from the straight line which cuts the paper perpendicularly at A is (M + m) / 2 TT>- - m (V - r 2 ) / 2 TT>- r*. At any point Q without the cylinder the value of the potential function is 2m log (AQ/BQ) + 2(J/+ m) log CQ. Show that the force of attraction between the charge on the cylinder and the unit length of the filament through A is 2 m [m8 L / (V - r 2 ) - (M + m) / 8J. This force vanishes if V/ " 2 = ( M + w)/^- Show also that if the cylinder is at potential zero in the presence of the fila ment through A, 3f= m log (S^r). If we superpose upon this distribution a second consisting of a homogeneous filament of mass m per unit of length and cutting the paper perpendicularly at A and a similar filament of mass + m per unit length cutting the paper at B, and notice 406 MISCELLANEOUS PKO13LEMS. that the potential function due to the new distribution has the value 2ralog(S 1 /r) at every point of the cylindrical surface, we shall see that the potential function due to the two distributions has at any point Q outside the cylinder the value 2m log (CQ) = 2 (M + m) log CQ and at any point within the value 2 m log r + 2 m log 8 X / r 2m log (r^ / r 2 ) = 2 (M + w) log r + 2 m log B^/rr^ On the cylindrical surface the potential function has the constant value 2 (Jf -f- m) log ^ and the surface density at any part of it is, as before, (M+ m) /2 TIT m (8 X 2 r 2 )/2 Tmy 2 , or (^+ra)/27rr + w(8 2 2 -r 2 )/27r?T 2 2 . What is the physical meaning of the special case where M + m = ? 205. Let < (a?, ?/) be a logarithmic potential function due to a body distribution of density p through an infinite cylinder the right section of which made by the xy plane is the region T, together with a superficial distribution of density o- on an infinite cylindrical surface the right section of which is the curve s in the xy plane. Let a=/i (x, y) and /3=/ 2 (x, y) be any two conjugate functions analytic in the region considered, and form arbitrarily the new function 3> (x, y) = 4> [fi(x, y),f 2 (x, y)~\ by substituting for x and y in <, a and ft respectively. To avoid confusion call the rectangular Cartesian coordinates in thfe plane in which T and s are drawn a and ft, instead of x and y, and draw a new xy plane in which to study the new function <l>. In this second figure the curves in which a.=fi(x,y), P=fz(x,y) are constant form a set of orthogonal curvilinear coordinates. A point P which in the first diagram has Cartesian coordinates (oo, A,) is said to be transformed into a point P in the new diagram, the curvilinear coordinates of which are (OQ, /? ). The Cartesian coordinates of P are (aj , y )j where /i (x , y ~) = a 0) /2 ( x o> 2/o) = Po- It is evident that 4> (x, y) has the same MISCELLANEOUS PROBLEMS. 407 numerical value at P that <(a, ft) has at P. The points which lie on the curve s in the old diagram are transformed into points which lie on a curve s in the new diagram, so that the curve s is transformed into the curve s and, similarly, the region T into the region T . It is evident from the proper ties of conjugate functions that two curves which cut at an angle 6 at a point P in the old diagram transform into two curves which cut each other, in general, at the same angle at the point P . Show that 3> is the logarithmic potential function due to a body distribution through the infinite cylinder of which T is the cross-section, together with a sur face distribution on the cylindrical surface of which s is the trace. Show also that if k 2 represents either of the two equal quantities (D x a) 2 + (D v a) 2 , (D^y + (D y ft)*, the numerical relations, at corresponding points in the two diagrams, of the corresponding elements of arc and area, of the corresponding values of the volume and surface density, etc., etc., are truly given by the equations ds = /ids -, dA = h*dA ; p h? = p ah = o- ; < = <; A< = A< ; h D& = ZK<, 7r A 2 < = A 2 3> ; h D n $ = D a & ; p dA = p dA ; a- ds = cr ds . 206. Given in a plane two circles of radii a and b respec tively, which have no points in p omnion, it is possible to find two points (ft, Q 2 ) on the line which joins their centres (A, fi), such that if ^ and r. 2 represent the distances from ft and ft of any moving point, both circles belong to the family of curves represented by the equation ?- 1 /;- 2 = c. Show that if AB = d, and if the circles are mutually exclusive, ft and ft are between A and B, and AQ l= (a 2 + d* -b*- K)/2d, BQ, = (# + d> ->* where R* = (a 2 - b* - d*y - 4 b*d*. If one of the circles lies within the other and if a > b, ft and ft lie beyond B on the 408 MISCELLANEOUS PROBLEMS. line ABj Q l is within both circles, and Q 2 outside both. In this case, AQ l = (a* -b 2 + d 2 - R )/2 d, BQ,= (a 2 -b 2 -d 2 + K) /2 d t where R 2 = (b 2 a 2 d 2 ) 2 4 a 2 d 2 . In the second case the four points of contact of tangents drawn from Q 2 to the two circles lie on a straight line through Q l . What is the corre sponding fact in the case first treated ? Consider the special case where a = b = i- d. Prove that the values of AQ l} BQ 2 given in the subjoined table are correct and draw to scale a diagram for each of the four examples. a b d AQ, BQ* 1 2 4 0.35 1.09 4 2 1 1.37 10.62 3 1 1 1.14 6.85 5 3 1 1.62 14.38 Given a circle of radius a, with centre at A, and a straight line in its plane at a distance d from A greater than a ; the line and the circumference belong to the family of curves r l /r 2 = c, where r t and r z represent the distances from two points (Q lt $2) equidistant from the line 011 opposite sides of it and lying on the perpendicular to the line drawn through A. Show that if Q 1 Q 2 = 2 m, m 2 ^ d 2 - a 2 . 207. If r 2 = (x + a) 2 + y\ r 2 = (x- a) 2 + y\ tan 0, = y/(x + a) , tan 2 = y I (x a) ; (j> = A log^/rg), and \l/ = A(O l 2 ) are conjugate functions, and if, moreover, . , , a(c 2 4- 1) 2 a 2 ac c 2 = e 2 ^^a= ^ 2 _i ; a-^-ni andr=^ -, where the upper or lower sign is to be used according as c 2 is greater or less than unity ; (x a) 2 -f (y O) 2 = ?- 2 , and the curves of constant < are the circles surrounding the points (a, 0) and ( a, 0) represented in Fig. 59. Values of c MISCELLANEOUS PROBLEMS. 409 between and 1 correspond to circles which lie to the left of the y axis, and positive values of c greater than 1 to circles on the right of the y axis. When c > 1, c = (a + a)/r, and a = + but when c < 1, c = (a + a) /r, a = -f- On two circumferences of the system, of equal radii, on oppo site sides of the y axis, r x /r 2 has reciprocal values. Using these formulas, prove that the charge per unit length on a long cylindrical wire of radius 0.5 centimetre, kept at potential unity at an axial distance of a = 600 centimetres from an infinite plane kept at potential zero, is 0.06424 units. In this case c is about 2400, a about 599.9998, and A, 0.128. Show also that if r = 0.5 and a = 10 ; a = 9.988, c = 39.975, A = 0.271, but that if r = 0.5 and a = 1 ; a = 0.866, c = 3.732, A = 0.759. It is to be noticed that 300 volts are equivalent to 1 electrostatic unit of potential difference, 1 microfarad to 900,000 electrostatic units of capacity, 1 ohm to I/ (9 X 10 11 ) electrostatic units of resistance, 1 ampere and 1 coulomb to 3 X 10 9 corresponding electrostatic units. In general, if an infinite conducting cylinder of revolution kept at potential F be placed with its axis parallel to an infinite conducting plane at a distance a from it, the charge per unit of length is ^ F" /log^-Jl L ? and the surface density is inversely proportional to the distance from the plane. 208. A condenser is formed by two long conducting circular cylinders, one of which is entirely inside the other. Prove that if r and r are the radii, d the distance between the axes, and 2 a the distance between the limiting points of the coaxial system to which the cylinders belong, the inverse of the capacity per unit length is MISCELLANEOUS PROBLEMS. 209. Electricity is distributed in equilibrium over the surface of an infinitely long right cylinder, the cross-section of which is x* + y* = a 4 . Prove that the attraction at any external point (r, ff) is inversely proportional to and that its direction makes with the axis of x an angle [Clare College.] 210. A long, right circular cylinder of radius a is placed with axis parallel to a plane at potential zero. Show that the mutual attraction per unit length of the cvlinder between it and the plane is E*/ Vr 2 a 1 , where c is the distance of the axis of the cylinder from the plane and E the quantity of electricity on the unit length of the cylinder. [M. T.] 211. A v Q, A* three points in order on a straight line, such that J, = m, QA^ = n, have charge* *! = A Vm(m + n), e^ = X V, . = X Vn (m -h n) respectively. The charges e^ % produce potential zero on a spherical surface ^ of radius a = Vwi (i + n) with centre at A v and the charges e^ e^ produce potential zero on a spherical surface Sj of radius b = Vr< ^m -f- n) with centre at A* S l and 5, cut each other orthogonally and together form the equi- potential surface X due to e< e v and e* Show, by a method analogous to that of Section 65. that the resultant force at any point P of SH due to e^ and * is directed towards A l and is numerically equal to Xfr 3 / a - A^P*, so that the whole force at P has the direction A^ and the intensit F l = X[l/a - tf/a Find a similar expression for the whole force at anv point of the surface ,. Show by the help of Section 31 that the surface taken over the larger segment of S^. of the normal MISCELLANEOUS PROBLEMS. 411 components of the forces due to e v e , and e z respectively are 2ire 1 (l + m/a), 2ve w 2^(1 - n/b). Prove that if e w e l} and e z were distributed on the surface composed of the larger segments of S 1 and S 2 according to the law a = F/4= IT, the surface woulgL be at potential A, and there would be no density at the circle of intersection of Sj_ and &. The charge under these circumstances on the larger segment of S l would be \ 1X1 + m/a)+e + e,(l - /&)], or \ X (a -f- & 4- m Vwm ri), or A6[1 + 8 +(8 S - S - I)/ Vl -f - ], where B = a/b. If & is very large compared with a, the larger segment of S l becomes nearly hemispherical ; its charge is about 3 Acr /4 6 and its mean density 3\/Sirb. The mean density on S t when the ratio of a to b is small is nearly equal to A. (4 br 3 a-} /1 6 Trb s . If a/b = 0, we have a hemispherical boss on an infinite plane ; the ratio of the average densities of the charges on the boss and the plane is 3/2. 212. A point charge e at (4 b, 0, 0) and a point charge e at ( 4 b, 0, 0) keep the plane x = at potential zero. Show that if the system be inverted, using the point ( 2 b, 0, 0) as centre of inversion and 2 b for radius of inversion, we obtain a spherical surface of radius b, with centre at ( b, 0, 0), kept at potential zero by the charge e at the point ( 46, 0, 0), and the charge -J e at ( | b, 0, 0) : this is the problem of Section 65. If the centre of inversion were ( 4 b, 0, 0) and if a were 4 b, we should obtain by inversion a spherical sur face of radius 2 b, with centre at ( 2 b, 0, 0) at potential zero, under a charge 1 e at its centre, and an infinite charge at infinity which lowers the potential function at all finite points by e/4 b. If this last were omitted, the value of the potential function on the spherical surface would be e/4 b, as is otherwise evident. Invert a spherical surface uniformly charged with density o-, using any point not its centre as centre of inversion. 412 MISCELLANEOUS PROBLEMS. 213. If z = w -f- e w ; x = <f> + 6$ cos \J/j y = ^ + e^ sin ^, the slopes of curves of constant < are given by the equation dy I dx = (1 + e* cos \j/) /e* sin ^, and the slopes of curves of constant ^ by the equation dy /dx = e^ sin ^/(l + e* cos i/^). Show that : (1) The curve $ = is the axis of a:, and on it x = </> -f e$ ; large positive values of x and <f> correspond, and large negative values correspond. (2) If if/ = ?r, y = TT and x = (f> e^ : the maximum value of x is 1, so that the curve \j/ = TT is so much of the line y = TT as lies to the left of x = 1. (3) The curve if/ = TT is so much of the line y = TT as lies to the left of x = 1. (4) The curve </> = has the slope ctn 1 \j/ and passes through the points ( 1, TT), (- 1, - TT), (1, 0), (0, i TT + 1), (^ J IT + i V3). (5) The curve < = m , where m is any positive quantity, lies to the left of < = 0, between the lines y = TT, y = + TT: it has the slope (e m + cos \j/)/sin \l/, which is infinite when y = Oori/= ?ror y = TT, and has the minimum value Ve 2m 1, so that for values of w greater than 3 the line is hardly distinguishable from a straight line parallel to the axis of y, at a distance of (e~ m m) to the left. (6) The curve < = + m, where m is a positive quantity, lies to the right of the curve </> 0, it cuts the axis of y perpendicularly and meets (but does not cross) the lines y = ir, y = TT from without. The curves for which < has the values 1, 2, 3, 4, 5, 6, 7 meet y = TT at points the abscis sas of which are - 1.72, - 5.39, - 17.08, - 50.60, - 143.4, - 397.5, - 1090 respectively. (7) If so much of the planes y = 4- TT, y TT as lie to the left of x = 1 be considered conducting and be charged to potential + TT and TT respec tively, \j/ represents the potential function in the air near them. In this case the charge on either side of a strip of either plane between the planes x = x l} x = x 2 is equal, per unit length of the strip parallel to the z axis, to the MISCELLANEOUS PROBLEMS. 413 difference on that side of the plane between the values of < for x = x l and x = x z , divided by 4 ?r. On a strip of the plane y = TT, between x = 1 and x = 50.6, there is, per unit height of the strip, a charge I/TT on the upper side and of 50.6 / 4 TT on the under side: the charge on correspond ing portions of y = TT being equal and opposite to these. [Helmholtz, CrellJs Journal, Vol. LXX.] State carefully some problem in electrostatics which might be solved by the use of the function z = A [cw + e 010 ]. A condenser consists of two very thin, large, plane, metal sheets of the same area parallel to each other at a distance of 1 millimetre. The dielectric is air and the difference of potential between the plates is 1 electrostatic unit (300 volts). Show that the density of the charge 2 millimetres from the edge is about 5/2 TT per square centimetre on the inside of the plate. Discuss at length the function sin (mr) where n is any real constant between and [Harris, Ann. of Math., 1901], and state some problems of electrostatics which can be solved by its aid. 214. Three closed surfaces 1, 2, 3 in order are equipotential surfaces of an electrostatic field in air. If an air condenser were constructed with the faces 1, 2, its capacity would be A, but if the faces were 2, 3, its capacity would be B. Show that if a condenser were constructed with faces 1, 3 while a homoge neous dielectric of inductivity /x filled the space 1, 2, and a second dielectric of inductivity /x the space 2, 3, the capacity of this condenser would be (7, where 1/C= 1/pA -f 1 / ^B. 215. A condenser is formed of two concentric spherical con ducting surfaces of radii a and c, separated by two dielectric shells bounded by a spherical surface of radius b concentric with the conducting surfaces. Prove that if in one shell 414 MISCELLANEOUS PROBLEMS. ^ = m/r 2 and in the other w /r 2 , the capacity of the condenser is mm f /[m ( a) + m(c )]. 216. If the space between two closed equipotential surfaces in air be filled with a dielectric the inductivity of which is either uniform or a scalar point function the level surfaces of which coincide with the equipotential surfaces of the field, the potential function without the shell will be unchanged, but its value within will be increased by a constant. 217. An infinite dielectric is bounded by an infinite con ducting plane which is maintained at a potential Ar 2 , where r is the distance from a point in the plane. Prove that if the inductivity of the dielectric varies as the distance z from the plane, the potential at any point is X (u 2 z 2 ), where u is the distance from an axis drawn through perpendicular to the conducting plane. 218. A distribution of matter M consists of two portions M 1} in a homogeneous medium of inductivity /x 1? and M 2 , in a homogeneous medium of inductivity /x 2 surrounding the other medium and reaching to infinity. An equipotential closed surface Si surrounds MI, excludes M 2 , and lies wholly in the first medium, a second closed equipotential surface S 2 surrounds MI, excludes M 2 , and lies wholly in the second medium. Prove that if r is the distance from a fixed point 0, if normals are drawn outward on $ 2 and inward on Si, and if dri and dr z are elements of space within Si and without $ 2 respectively, and 4 TT^ V = - t* 2 dS 2 + 4 if is without S 2 , and MISCELLANEOUS PROBLEMS. 415 and 4 TT/X, ( V - r Sl ) = - if is within Si. Show from these equations that if S, the surface of separa tion of the two media, is equipotential, J J J ^p" is equal to /x 2 V if is without S, and to ^ V + (p. 2 ^ V s if is within Give physical interpretations to these last results. How is the force at any outside point affected by the sub stitution of one homogeneous dielectric for another in the whole region bounded by S? 219. The open surface S is a surface of zero potential due to a distribution J/ x in an infinite homogeneous medium of induc- tivity fj.i on the right of S, and to a distribution M 2 in an infinite homogeneous medium of inductivity /x. 2 on the left of S. S is the common boundary of the two media. Show that if r is the distance from a fixed point 0, j J J ^^- = ^ V or /x 2 Vj according as is to the right or to the left of S. 220. The function W so vanishes at infinity that i&D r W, where r is the distance from any finite point, is not infinite. The normal derivative of W is given at every point of an infinite plane. Prove that if W is harmonic everywhere in the space on one side of the plane, it is determined in that region. Prove also that if W is harmonic in the region on one side of the plane except at the given points -Pj, P 2 , P 3 , , P nJ at each of which it becomes infinite in such a manner that, if r k is the distance from P k , and if m k is a constant belonging to this point, W - is harmonic at P k , W is determined in the region in question. 221. Two homogeneous media of inductivities /^ and ^ have a plane surface of separation but are otherwise unbounded. In the first medium at a point P at a distance a from the 416 MISCELLANEOUS PROBLEMS. common surface S of the two media is a charge e = ^e. At Q, any point on S, the force due to this charge has the normal component ea/(PQ) s , or 8, pointing into the second medium. If jVj and N 2 are the normal components of the whole force at Q pointing into the two media, and if o- is the apparent density of the surface charge on the plane at ft N, = 2 7r<r - 8, N z = 2 TTO- + 3, and whence N, = S [(^ - /x 2 ) / (^ + ^ - 1], and ^ = 2^8/0^ + ,*,). Prove that ^ might be caused by an apparent charge (/A! /x 2 ) e / (/A! + /u, 2 ) at P , the image of JP in the plane, together with an apparent charge e at P and that N 2 might be due to an apparent charge 2 /x x e / (^ 4- /u, 2 ) at P. Hence show by the aid of the theorem stated in the last problem that the potential functions due to these apparent charges are identical (one in the first medium, the other in the second) with the values of the actual potential function in the case described in this problem. The charge at P is urged towards the dielectric with the force -~ - - * ^2 + ^ 222. Using the notation of Section 62, let the plate A of a spherical condenser be charged with m units of positive elec tricity and separated from the plate J5, which is put to earth, by a spherical shell of radii r and r f made up of a given dielectric. Let us first ask ourselves what the effect of the dielectric would be if it consisted of extremely thin concentric conducting spherical shells separated by extremely thin insu lating spaces. It is evident that in this case we should have a quantity m of electricity induced on the inside of the innermost shell, a quantity + m on the outside of this shell, a quantity m on the inner surface of the next shell, a MISCELLANEOUS PROBLEMS. 417 quantity + m on the outside of this shell, and so on. If there were n such shells in the dielectric layer, and n + 1 spaces, and if S were the distance from the inner surface of one shell to the inner surface of the next, and AS the thick ness of each shell, the value, at the centre of J, of the poten tial function due to the charges on these shells would be V =m l " * l 8 r - AS + 8 r + 2 S / - AS + 2 S + + 1 . r + -wS r - AS + ?*S J (H-28)(r- A8+28) + " J This quantity lies between r*r but these differ from each other by less than c = fiuv&dx so that m\j , which is easily seen to lie between G and H, differs from V A by less than e. If, then, S is very small in comparison with rand ?., V , differs from m\( j V< V by an exceedingly small fraction of its own value. This shows that the effect, at the centre of A, of such a system of conducting shells as we have imagined would be practically the same as if a charge ?n\ were given to the inner surface of the dielectric, and a charge 4- m\ to its outer surface, while the charges on the surfaces of the thin shells within the mass of the dielectric were taken away. That is, the value of the potential function in A would be m(l \)(- - J instead of m (- -Y 418 MISCELLANEOUS PROBLEMS. Such, a system of shells introduced into what we have hitherto supposed to be the electrically inert insulating matter between the two parts of a spherical condenser would increase the capacity of the condenser in the ratio of 1 to 1 A. It is to be noticed that X is a proper fraction : X = and X = 1 would correspond respectively to a perfect insulator and to a perfect conductor. If the coatings of a parallel plate air condenser be in the planes x = 0, x = a, and if the first have a uniform superficial charge of density <r and be kept at potential zero, the potential function in. the air between the plates is evidently 47TO-X. Show that if a number of plane plates of metal of small thickness AS be uniformly distributed between the coat ings parallel to the yz plane so as to be separated from each other by air spaces of thickness (1 A) 8, the capacity of the condenser will be increased in the ratio of //, to 1, where /JL = 1/(1 A). Show also that if 8 be made infinitesimal and A a function of x, we have between the coatings in the limit, pD x V = 4 TTO-, or D x (^D x F) = 0, the differential equa tion which F would satisfy in a real dielectric of inductivity varying with x. Treat again, on the assumption that A varies with r, the case of the spherical condenser considered above. 223. The potential function F due to an electrical or mag netic distribution in an inductive medium, may be computed according to the Newtonian Law by taking into account both the intrinsic and the induced charges. If p and <r are the intrinsic volume and surface densities, and if the integrations extend all over the space where p and o- are different from zero, the potential energy of the distribution is usually written 4 or MISCELLANEOUS PROBLEMS. 419 Why should not the apparent volume and surface densities be used in finding the energy by the equation E = Answer this question fully, using an illustrative numerical example to explain your assertions. Assuming that the energy of an electrostatic field would be mathematically accounted for 011 the supposition that every volume element of space at which the intensity of the field is F contributes F 2 /8 TT times its volume to the whole amount, show that if a tube of force be cut into cells by a set of equi- potential surfaces drawn at equal potential intervals, these cells contain equal amounts of energy. Show how to divide all space up into unit energy cells. Discuss the mechanical action on a charged conductor in an electric field on the assumption that there is tension along the Faraday tubes which abut on the conductor, such that the normal pull on the conductor per square centimetre of its surface is F*/8 TT. Discuss the pressure at right angles to the Faraday tubes in a dielectric. 224. The space between two concentric spherical surfaces, the radii of which are a and b and which are kept at potentials A and B, is filled with a heterogeneous dielectric, the induc- tivity of which varies as the nth power of the distance from their common centre. Show that the potential function at any point between the surfaces is (Aa n+l - Bb n+l )/(a n+l - b n+l ) - a" +l b" +l (A - B) / r" +1 (a n+l - 225. A condenser is formed of two concentric spherical con ducting surfaces separated by a dielectric. This dielectric consists of three shells bounded by spherical surfaces of radii TI, r 2 , r 3 , and r 4 , concentric with the conductors. The induc- tivities of the inner and outer shells are equal to /n 1? and that 420 , MISCELLANEOUS PROBLEMS. of the intermediate shell is /x 2 . Show that if C is the capacity of the condenser, r a 226. The plates of a condenser are two confocal prolate spheroids and the inductivity of the dielectric is A j p, where p is the distance of any point from the axis. Prove that the capacity of the condenser is IT A I [log (a, + bj - log ( + &)], where a, b and a lt ^ are the semiaxes of the generating ellipses. 227. The plates of a condenser are the closed metallic sur faces S l and S 2 . When ^ is at potential zero and S 2 at potential F 2 , the potential function in the air between them is given by the equation V = f(x,y,z)* The tube of force based on a portion ($/) of ^ abuts on a portion ($ 2 ) of S 2 . If the air in this tube were displaced by a homogeneous dielec tric of inductivity /A, and if the charges on / and S 2 were increased in the ratio /*,, while the charges on the remainder of S l and $ 2 were unchanged, would the force at every point be unchanged ? Would there be a discontinuity in the sur face density of the apparent charge on $i at the boundary of S^? 228. How many square centimetres of tin foil must be used in making a single parallel plate condenser of one microfarad capacity, if the two sheets of foil are to be separated from each other by paraffined paper the thickness of which is one- fifth of a millimetre, and the specific inductive capacity 2 ? [72,000 TT.] Would the required amount be the same if the condenser were made up of a pile of sheets of foil alternating with paper, the odd sheets forming one terminal and the even sheets the other? MISCELLANEOUS PROBLEMS. 421 229. Show that the generalized Poisson s Equation, D x (pD x V) + D,(jp.D y V) + D z (fi.D z V) = - 4,rp, is equivalent to if , 77, are any orthogonal curvilinear coordinates. In the case of spherical coordinates, where h r = 1, h = !/;, h^ = I/ > sin 0, the equation is sin 2 6 D r (JJL r-D r V) + sin D e O sin OD Q V) + D+ fa Dt V) = - 4 7rp>- 2 sin 2 0, and, in columnar coordinates, where h r = 1, h e = 1/r, h, = 1, itisr.I> r (^rAn + A(^An + ^-A(^An = - 4 ~ 7r P^ 230. Show that if the poles of a battery, made up of a given number of equal cells, are to be connected by a resistance R greater than the sum of the resistances of all the cells, the greatest current will traverse R when the cells are joined up in series ; but that if R is very small, the cells should be joined up in multiple are. If R is such that by arranging the cells in a certain number of parallel rows and joining up the num bers of each row in series, the resistance of the whole battery can be made equal to R, this arrangement will give the maxi mum current. 231. A battery is joined up in simple circuit with a resistance R and a galvanometer of resistance G. After the deflection of the galvanometer has been noted, an additional wire (or shunt) of resistance S is placed across the poles of the battery, and the resistance R is decreased (to r) until the galvanometer deflec tion is the same as before. Assuming that the electromotive 422 MISCELLANEOUS PROBLEMS. force of the battery remains constant, show that the resistance of the battery is ^ [Thomson.] 232. Using the potential function F = c log r + d, where r is the distance from a fixed axis, show that the resistance of a conductor bounded by two concentric circular cylindrical sur faces of radii a and b, and by two planes, distant h from each other, perpendicular to the axis of the cylindrical surfaces, is Apply the result to the problem of finding the resistance of the liquid in a cylindrical galvanic element. 233. Using the potential function, V= dog(?\/r 2 ), where i\ and r 2 are the distances from two parallel fixed axes, show how to find (see Fig. 59 and Problem 207) the resist ance of a conductor bounded by two parallel planes and by two somewhat eccentric circular cylindrical surfaces which cut the planes orthogonally. In the case of an element in which the zinc electrode is a cylindrical rod and the copper elec trode a cylindrical shell surrounding it, is the resistance of the liquid greater or less when the zinc is eccentric to the copper shell than when it is concentric with it ? 234. If two points, A and B, of a network of conductors which are carrying steady currents, be connected by an extra conductor W, A and B are said to be at the same potential if no current passes through W. A is said to be at a higher potential than B if a current tends to pass through W from A to B. In this case the difference of potential between A and B is defined to be the electromotive force (in volts) of a galvanic cell which introduced into IF with its positive pole towards A would just prevent any current from passing through W. MISCELLANEOUS PROBLEMS. 423 Three cells of electromotive force 2 volts, 1 volt, and 1 volt respectively, and internal resistances of 1 ohm, 2 ohms, and 4 ohms are joined up in series with a resistance of 1 ohm. Show that the potential differences between the terminals of the separate cells are + f , 0, and 1 respectively. If the external resistance were 9 ohms, the corresponding potential differences would be -+- |, + , 0. 235. The terminals of a compound condenser formed of three simple condensers, of capacity 2 microfarads, 3 micro farads, and 6 microfarads respectively, joined up in series, touch the ends of a linear conductor of 22 ohms resistance through which a current of 3 amperes is flowing. What are the charges on the single condensers ? Show that if with out loss of the charges the condensers be disconnected and joined up in parallel with their positively charged plates in connection, the difference of potential between the terminals of the new compound condenser will be 18 volts. What charge will each of the simple condensers have ? [66 ; 36, 54, 108.] 236. Prove that if a condenser of capacity k farads be charged to potential Q /k and then discharged through a large non-inductive resistance, r ohms, the charge Q of the condenser t seconds after the beginning of the discharge is t Q Q - e** ; and show that not one ten-thousandth part of the original charge remains after 10 kr seconds. Show also that the energy that has been expended up to the time t in heating the wire is Q<? - ^(1 e * ) joules. 237. The terminals of a condenser of k farads capacity are attached permanently to the poles of a constant battery of electromotive force E volts by leads of large resistance, r ohms. After the condenser has become fully charged its 424 MISCELLANEOUS PROBLEMS. terminals are suddenly connected together without removing the battery by a conductor of large resistance, R ohms. Assuming that the solution of a differential equation of the form D t y + ay = b is y = b/a + Se~ at } show that at the time t the charge on one of the condenser plates is where q = (R + r) / rRk. 238. A galvanic battery is composed of two galvanic cells, the electromotive forces of which are e l and e 2 and the inter nal resistances ^ and 6 2 , joined up in multiple arc. The poles of the battery are connected by an external resistance of r ohms. Show that if C\ and C 2 are the strengths of the currents flowing through the cells, 239. A galvanometer of 9 ohms resistance is to be furnished with two shunts, such that when the first alone is used T L of the current shall pass through the instrument, and that when both are used in parallel, 29/30 of the current shall pass through them. Prove that the resistance of the second shunt must be 9/20. 240. A storage battery is used to send a current through a cluster of incandescent lamps arranged in multiple arc. The resistance of each lamp when hot is 100 ohms. When 10 lamps are used the current through each is 1 ampere, but when 20 are used this current is only |J- of an ampere. Find the resistance of the battery and its connections and show that the electromotive force of the battery is 110 volts. 241. If a number of cells of different electromotive forces but of equal internal resistances are joined up in multiple arc, the battery thus formed is equivalent, so far as its ability MISCELLANEOUS PROBLEMS. 425 to send currents through outside resistances is concerned, to a single cell the electromotive force of which is the mean of the electromotive forces of the cells in the battery. Find the resistance of this equivalent cell and show that it would be more " effective" when doing a given amount of external work than the battery. How much work is done in the battery per second when the external circuit is broken? 242. A certain uniform cable 50 kilometres long has, when in good condition, a resistance of 450 ohms. The operator at one end finds that the resistance is 270 ohms or 350 ohms according as the other end is grounded or insulated. Suppos ing the ground connections at the two stations to be good, so that the resistance of the earth is negligible, and assuming that the.re is a single fault in the cable, show that this fault is 16.67 kilometres from the first station and that its resist ance is 200 ohms. 243. A cable 500 kilometres long with stations A and B at its extremities has a single fault, but is not so much injured that signals cannot be sent through it. With cable insulated at B, the operator at A grounds one terminal of a large bat tery and attaches the other terminal to the cable. After this has been done the operators find that the difference of poten tial between the cable and the ground is 200 volts at A and 40 volts at B. The cable at A is then insulated, and one terminal of a large battery at B is grounded while the other is attached to the cable. The difference of potential between the cable and the ground is then 300 volts at B and 40 volts at A. Show that the fault has a resistance equivalent to that of 47.62 kilometres of cable and is at 190.5 kilometres from A. Explain some way of measuring the potential differ ences in this case. 244. In a network PA, PB, PC, PD, AB, BC, CD, DA, the resistances are a, Ay, 8, y + 8, 8 + a, 426 MISCELLANEOUS PROBLEMS. respectively. Show that if AD contains a battery of elec tromotive force Ej the current in BC is X (aft + 78) E 2 XV + (8 -ay) 2 where A = a + ^ + y + S, and n = (3y + ya + a/8 + aS + /38 + 78." 245. Show that if the edges of a parallelepiped be formed of uniform wire such that the resistances of three contermi nous edges are a, b, and c respectively, and if a current enters at one angle and leaves at the opposite angle, the resistance of the network is \_(a + b + c) + abc / (ab -f- be -f- ca)]. 246. (a) A tetrahedral framework is made of uniform wire, opposite edges being equal and of lengths a, b, c. If a cur rent enters and leaves the framework at the ends of an edge of length a, the strengths of the currents in the pairs of edges of length a are in the ratio b(a + c)+c(a + b ): b(a+ c)c(a + b). [Jesus College.] (b) Show that the resistance of the whole framework is that of a length of the wire equal to [d&/(a + 0) + a0/(a + &)]. [St. John s College.] 247. Show that if n telegraph poles, each of resistance R y be joined in pairs, each to all the others, with wires of resist ance r, and if an electromotive force E be inserted in one of the wires, the current in that wire is E{R(n 2) + r} fr(nR + ?*). 248. An electric distributing conductor 6 miles long gives out continuously 50 amperes of current per mile of its length. The end of the conductor remote from the generator is insu lated, while the nearer end is kept at 1000 volts potential. Show that if the resistance per mile of the conductor is MISCELLANEOUS PROBLEMS. 427 1 ohm, the voltage at a point on the line x miles from the generator isv = 5Qx(- 6) + 1000. Find the rate at which a given portion of the line is delivering power. 249. A Wheatstone s bridge in proper adjustment consists of four conductors, AB, BC, CD, DA, which have respec tively the resistances p, q, s, and r. The galvanometer is connected with A and C and the battery with B and D. The electromotive force of the battery is E, and the resistance of the battery with its connecting wires is b. Prove that the heat developed per unit time in the conductor AB is the equivalent of the energy 1 [6 (s + >) + > (? + s)] 2 250. A generator of constant electromotive force E and of constant internal resistance B is used to charge a storage battery which now has an electromotive force e and an inter nal resistance b. Show that if the poles of the storage bat tery be connected by a conductor of resistance r, a current C = (Be + bE) * l(B + b) r + Bb] will go through this conductor. 251. The conductors AB, BC, CD, and DA have the resist ances p, q, r, and s respectively. A is connected with C by a battery of internal resistance b and electromotive force e. B is connected with D by a battery of internal resistance b and electromotive force e . Prove that if the current in AC is zero, e\b (p + q + r + s) + (p + s)(q + r)\+e (pr - qs) = 0. 252. A conductor of given dimensions made of given material has two given portions Si and S 2 of its surface kept at constant potentials while the rest of its surface is a current surface. Show that if V is the potential function within the conductor, when Si is kept at potential Ci and S 2 at potential C 2 , and 428 MISCELLANEOUS PROBLEMS. if V is the potential function, when S^ is kept at C\ and O 2 at t/2 , ~ f f n \ n n n _ U l ~ U 2 rr , 2 ^1 ~ U 2^1 " r r r r L/i ^ 2 <^i O 2 253. One end (0) of a straight wire of radius a and length I is kept at potential F , and the other end ($) at potential FI- The specific conductivity of the wire is K and its resist ance per unit length is w, so that the reciprocal of w is equal to ?ra 2 K. The wire is surrounded by an insulating sheath, the outside of which is in contact with sea water at potential zero. The rate of leakage per unit length of the wire or cable through the sheath at a place where the potential of the wire is V is 2 TraX V. The reciprocal of 2 TraX is denoted by W and is called the " insulation resistance " of the cable per unit length. The rate of flow of electricity into a portion of the cable of length A#, included between two right sections, the nearer of which is distant x from and is at potential F, is Kira?D x V. The rate of flow of electricity out of this element through the sheath and from the farther end is - KTra 2 (D X V + & X D X V) 4- 2 TraXFA^r. When the current is steady the element neither gains nor loses electricity and KTTO?& X D X F 2 TraX FAcc = 0, so that at every point D X 2 7 (3 2 V=0, where (? = w/W. The general solution of this equation is of the form F= AeP x + Be~P x , and if we determine A and B so that F= F when x = 0, and V= V when x = I, we get F = [ V l sinh (fix) + F sinh (3(1- x)~\ /sinh (ftl). Show that if the current which enters the cable at is / and that which leaves it at Q is I lf and if / denote the current in the core at a point at a distance x from 0, /= [ F cosh ft (I - x) - F! cosh (/&)] /[ Fsinh = 7 [ FO cosh ft(l-x)- F! cosh 0&;) ] / [ r cosh (01) - FJ, /i = [F - F! cosh (/?/)]/ [V^TF sinh (/?/)] = J [ F - F! cosh (#)]/[ F n cosh (#) - FJ. MISCELLANEOUS PROBLEMS. 429 Show also that if the end of the cable at Q be insulated and left to itself, V L = , " . > but if it be put to earth, cosh (/?/) V= F sinh/?(Z-a-)/sinh(0/). If in this latter case the cable were infinitely long, we should have V = V Q e~^ x and The whole core resistance of a certain cable 1000 miles long is 2000 ohms. When one terminal of a battery (the other terminal of which is put to earth) is attached to one end of the cable and the other end of the cable is grounded, the current at the sending end is to the current at the receiving end as 1.1276 to 1. Show that the insulation resistance of the cable per mile is 8 megohms. In the Atlantic cable of 1889, iv = 1.54 ohms per kilometre, and W= 9,085,000.000 ohms per kilometre. 254. The conduction resistance of a certain cable 1000 miles long is 10 ohms per mile, whilst the insulation resistance is 10 megohms : if the sending end be at a given potential and the receiving end to earth, find the whole charge of the cable when a steady current passes through it. Show that if the cable have a leakage fault at the middle point the resist ance of which is equal to that of a length of a miles of the cable, the strength of a steady current at the receiving end will be lowered in the ratio 1 : 1 H -- - r [~^I- T.I a +! 255. Prove that if any finite set of algebraic operations be performed upon the complex variable z = x + yi taken as a whole, and if the result \io =f(z)~] be written in the form <f>(x, y) + i-i(/(x, ?/), where < and \f/, which are said to be con jugate to each other, are real functions of x and y : (a) Both < and if/ satisfy Laplace s Equation. (ft) D x $ = D$ and Drf = - Dj. (c) At any point P, the derivative of < taken in any direc tion PQ in the plane xy is equal to the derivative of \f/ taken in a direction PR at right angles to PQ, and such that the angle QPR corresponds to a counter-clockwise rotation. 430 MISCELLANEOUS PROBLEMS. (d) The equations <(#, y) = c, \f/(x } y) = c represent two families of curves which cut each other orthogonally. 256. Prove that : (a) If <f> and \j/ are any two conjugate functions of x and ?/, that is, if <j> + i\j/ is a function of the complex variable x + yi t taken as a whole, then, conversely, x and y are two conjugate functions of <f> and \j/. (b) If < and if/ are any two conjugate functions of x and y, and if a and (3 are any two other conjugate functions of x and y, and if for x and y in the expressions for </> and ty we substitute the expressions for a and (3, we shall get two new conjugate functions of x and y. (c) If <i and fa, </> 2 and \f/ 2 are any two pairs of conjugate func tions, <^>! < 2 and \p l ^ 2 are conjugate functions of x and y. 257. Prove that in any case of steady uniplanar flow of electricity that is, flow which at every point is parallel to a given plane, and of such a character that its intensity and direction are the same at all the points of any line drawn per pendicular to the given plane there exists a function con jugate to the potential function. This function is called the " flow function." 258. Show by the ordinary rules for treating imaginary quantities that, if z = x -+- yi, z 2 , V, log z will yield respec tively the following pairs of conjugate functions : A(x* ?/ 2 ), n a 2Axy ; Art cos -, Art sin - ; A log r, Ad ; where r 2 = x 2 + ?/ 2 J .4 and = tan" 1 -. State some problems of steady flow within x conductors which these conjugate functions will help to solve. 259. Show that, with certain broad limitations, either one (say <) of any pair (<, ^) of conjugate functions of x and y may be taken as the potential function in empty space due to an electrostatic distribution the density of which is a function of x and y only, and which, therefore, must be constant at all points on any indefinitely extended line drawn perpendicular to the MISCELLANEOUS PROBLEMS. 431 plane of xy. Show also that in the case of the same distribution the other function \f/ will be constant along any line of force. 260. Show that either one (say <) of any pair (<, \j/) of con jugate functions of x and y may be taken as the potential func tion inside a conductor which carries a steady current flowing at every point in a direction parallel to the plane of xy, and the same in intensity and direction at all points of any line drawn perpendicular to this plane. Show that in this case the other function ^ will be constant along any line of flow, and that the two equations < = c, \f/ = c represent respectively, if c and c are parameters, cylindrical equipotential surfaces and cylin drical surfaces of flow. If ds is the element of any curve AB in the plane xy, and if D^ is the derivative of < taken in the direction of the normal to ds which points towards the right as C B one goes along the curve from A to B, the integral k I D n <f> ds gives the amount of positive electricity which crosses per unit of time from left to right so much of a right cylindrical surface erected on AB as is enclosed by two planes parallel to the plane of xy and at the unit distance from each other. Since D n <f> = D s /f, the integral just considered is equal to k({j/ B \f/ A ), and k times the difference between the values of ^ on two right cylindrical surfaces of flow gives the amount of flow across the unit height of so much of any cylindrical surface which cuts the plane of xy at right angles as is included between the given surfaces of flow. 261. Prove that: (a) If ?-!, ?- 2 , ?- 3 , , r n are the lengths of the radii vectores drawn from any point P to any n parallel axes, and if 1? 2 , Q 3 , , 6 H are the angles which these radii vectores make with a fixed line in the plane of xy which is perpendicular to the axes, 4> = A l log?-! + A 2 log 7-3 + A 3 log r 3 + h A n log r n , f = A& + A 2 2 + A S 3 + + A n O n are conjugate solutions of Laplace s Equation. 432 MISCELLANEOUS PROBLEMS. (b) The equation \j/ = c represents for each value of c a cylindrical surface which passes through all the axes. (c) For very large values of c, the equation < = c represents as many closed cylindrical surfaces, each surrounding one of the axes, as there are positive terms in the expression for <. (d) For very large negative values of c, the equation < = c represents as many closed cylindrical surfaces, each surround ing one of the axes, as there are negative terms in <. (e) If %A = 0, no one of the cylindrical surfaces \f/ = c ends at infinity. (/) The value of ( D n <f> ds taken around any closed curve in the plane xy which surrounds the jth axis and no other is equal to the change made in \f/ by going around the curve, and this is 2irAj. (g) However the axes may be distributed and whatever values may be assigned to the A s, <f> represents the potential function corresponding to a uniplanar flow of electricity* within the substance of an infinite conducting lamina, either thick or thin, when cylindrical holes, on the curved surface of each one of which <f> is constant, are cut through the lamina so as to remove all the axes, and if the curved surfaces of these holes are kept at potentials equal to the values of < on them. This is practically the case of a very large thin sheet of metal touched at certain points by the ends of wires con nected with the poles of batteries. (A) If in the value of < there is an even number (2m) of terms, half of which are positive and half negative, and if, moreover, all the A s are numerically equal, we have the case in which m similar pieces of wire connected with the positive pole of a battery touch a thin sheet of metal in in places, and m similar pieces of wire connected with the negative pole of * See papers by Foster and Lodge in the Philosophical Magazine for 1875 and 1876. MISCELLANEOUS PROBLEMS. 433 the battery touch the metallic sheet in m other places. In this case, if PI and P 2 are any two points in the nietal, the resistance of so much of the sheet as lies between the equipo- T*/* - T*P tential surfaces on which P l and P 2 lie is .. 2 . 7 * when 8 is the thickness of the lamina, and k its specific conductivity. (i) If <f> consists of two terms the coefficients of which are numerically equal but opposite in sign, we have the case of a thin sheet of metal touched at two points by the two poles of a battery. Here the curves in the plane xy, for which i// is constant, are circles (Fig. 59) the centres of which are on the line which bisects at right angles the line which gives the points where the battery electrodes touch the sheet. Show that this value of < enables us to find the resistance of a thin circular disc touched at two points on its circumfer ence by the poles of a battery, and hence, by superposition, the resistance of such a disc touched by any number of pairs of battery poles at different places on the circumference. State other problems which an inspection of Fig. 59 shows can be solved by the aid of the value of c. (j) If <f> is made up of an infinite number of terms with coefficients all numerically equal, but alternately positive and negative, and if the corresponding axes cut the plane of yy in a straight line so that the distance between any axis and the next is , certain of the lines of force in the plane of xy will be straight lines which cut at right angles the line on which the traces of the axes lie. Show that by aid of this <f> we can find the resistance of a lamina of breadth b, and of infinite length when touched at two points opposite each other, one on one edge, and the other on the other. Draw from general knowledge a diagram which shall give the shape of the lines of flow and the equipotential lines in such a lamina. 262. (a) Show that if in a thin conducting plate of indefi nite extent there are two sources and a sink, each of strength 434 MISCELLANEOUS PROBLEMS. numerically equal to m, situated respectively at points A, B, C, which lie in order upon a straight line, one of the lines of flow consists in part of a circumference of radius V CA CB drawn around C as a centre, so that the flow inside the circumference would be unchanged if the part of the plate outside it were cut away. In other words, if a circumference be drawn in a thin conducting plane plate of indefinite extent, the " image " in this circumference of a source, of strength m, situated at a point P in the plane, is made up of a sink, of strength w, at the centre of the circle, and a source of the same strength at Q, the inverse point of P with respect to the circumference. Show that if a sink be regarded as a negative source, and if, inside a circumference drawn in a thin plane conducting plate of indefinite extent, there are sources at the points A 1} A 2 , A 3 , , A k , of strengths algebraically equal to m lf ra 2 , m s , -,m k respectively, and sources of strengths algebraically equal to m lt m 2 , m s , , m k , at the corresponding inverse points, then, if m x 4- ra 2 + m s + + m k = 0, there is no flow of electricity across the circumference. If at a fixed point P in a thin plane plate (Fig. 127) there is a sink of strength numerically equal to m, and at another p A point P in the plate an equal source, and if P ^^ be made to approach P as a limit and the prod- * * / uct m PP be kept always equal to a given con- Ofa X stant fjL, we have as a limit a "plane doublet"* of strength /x, the axis of which is PX, the limit ing position of the straight line drawn from P to P . We shall find it convenient to represent sources and sinks respectively by black and unshaded circles, and doublets by circles half black and half unshaded. The black portion * Kirchhoff, Pogg. Ann., 1845, p. 497. W. R. Smith, Proc. Ed. Eoy. Soc., 1869-70. Foster and Lodge, Phil. Mag., 1875. Minchin s Uniplanar Kinematics, p. 213. Peirce, Proc. Am. Acad. of Arts and Sciences, 1891. MISCELLANEOUS PROBLEMS. 435 of a doublet circle indicates the directions in which there is a flow away from the point where the doublet is situated; the unshaded portion indicates the directions from which there is a flow towards this point. The axis of a doublet bisects both the black and unshaded portions of the doublet circle. Show that if P be used as origin and PX as axis of abscissas, the velocity potential function due to the doublet s < = and the function is = If * 2 + r * 2 + r x + yi = z, these are respectively the real part and the real factor of the imaginary part of the function : The equipotential lines and the lines of flow are circles (see Fig. 128) touching the axes of y and x respectively at the origin. A "plane quadruplet" is formed of two equal and oppo site plane doublets in the same manner that a doublet is formed out of a source and an equal sink. An " octuplet " is formed in a similar way of two equal and opposite quadruplets, and so on. We may use the word " motor " to denote in general a source, a sink, a doublet, a quad ruplet, or any other combination of sources or sinks at a single point. (b) The upper circle in Fig. 130 shows the plane quadru plet formed by combining the two plane doublets indicated in the lower part of this diagram. Show that the flow function due to a quadruplet of this kind at the origin is 2 kxy / (x* + ?/ 2 ) 2 , while the flow function due to such a quadruplet as that shown in Fig. 131 will be k(x 2 - if}/(y? + y 2 ) 2 . One of these quadruplets is evidently equivalent to the other turned FIG. 128. 436 MISCELLANEOUS PROBLEMS. through 45. Find the flow function clue to an octuplet of the kind shown in Fig. 132 at the origin. (c) Show that the lines of flow due to a plane doublet may be regarded as the lines of force due to a columnar magnet of infinitely small cross-section. (d) Show that the functions 112 6 te^->-yy--y- each of which is the derivative with respect to z of the one which precedes it, yield a series of pairs of conjugate func tions which represent in order the velocity potential functions and the flow functions due to a source at the origin, to a plane e > < FIG. 129. FIG. 130. FIG. 131. FIG. 132. doublet at the origin, to a plane quadruplet at the origin, to a plane octuplet at the origin, and so on. (e) Show that if two plane doublets L and M exist together at a point 0, and if the directions of the two straight lines OA, OB show the directions of the axes of L and M respectively, and the lengths of OA and OB the strengths of L and M on some convenient scale, then the direction of the axis of the resultant of L and M will be given by the direction, and the strength of the resultant by the length, of the diagonal of the parallelogram of which OA and OB are adjacent sides. Plane doublets, then, can be compounded and resolved by com pounding and resolving their axes like forces or velocities. 263. If a charge -f m concentrated at a point Q be made to approach on any analytic curve a point charge m at a fixed point P 011 the curve, and if as Q approaches P, m is made to increase in such a manner that the product of m and PQ is MISCELLANEOUS PROBLEMS. 437 always equal to the constant p, the limiting value of the poten tial function of the system is said to be due to a space doublet of strength /A at the point P, and the axis of the doublet is said to be the limiting position of the secant PQ. Show that if r is the distance of any point P from P and if is the angle between the axis of the doublet and PP t the value at P of the potential function due to the doublet is /*cos Q /i*. The force components along and perpendicular to r are 2/A cos O/ 1* and /t sin / r 3 . The potential function (Section 69) due to a doublet at the origin with axis coincident with the x axis is px/ r 3 . The potential function due to a mass m at the point (b, 0, 0), a mass -f m at the point (b + 8, 0, 0), a mass ma/ (b + 8) at the point (a? / (b -f 8), 0, 0), and a mass ma/b at the point (a 2 / b, 0, 0), where b and 8 are smaller than , has the value zero on the spherical surface x- + if -f- z- = a 2 . Prove that if, while a and b are constant, 8 be made to decrease indefinitely and m to increase in such a manner that their product shall always be equal to the given constant p., the limiting value of the potential function will be (x - b)/[(x - b) + if + + ap [l(x~ + if + 2 2 ) - 2 ) 2 + 6 2 (/ -f If 6 = 0, this expression becomes p x ( a3 "~ y 3 ) /a 3 " 3 ? where r 2 x 2 -f- ^ H- 2 2 . What problem in electrostatics can be solved by the aid of this last function ? Is the image of a doublet in a spherical surface another doublet ? 264. A straight wire of radius a which forms the core of a cable of length I lies in the axis of x with one end at the origin and the other at the point (7, 0, 0). The whole of the outside of the insulating covering of the cable and the core at the point (I, 0, 0) are kept at potential zero, while the core at the origin is at the potential F . Show that if c is the capacity per unit length of the cable considered as a 438 MISCELLANEOUS PROBLEMS. condenser, k the ratio of the conductivity per unit length of the core to c, and h the rate of loss of electricity by leak age through the insulation per unit length of the cable when the difference of potential of the core and the outside of the cable is unity, the value of the potential function V in the core satisfies the equation D t F= k D* V- ~ V, and if V = we- ht/c , D{w = k- D x 2 w . c Show also that in the final state, when V satisfies the equa tion D*V = hV/kc and is equal to F when x = 0, and to zero when x = I, the value of V is given by the expression [ F r sinh /3a; + F e -smh/3(Z e)]/sinh/3Z, where (3 2 = h/kc. Prove that any quantity of the form A s - e~ xt cos(nx 8), where X = kn 2 , satisfies the equation D t w = k D x 2 w, and that if 8 = J-TT, n = sir /I, where s is an integer, and A s = 2 ckirs - COS Sir / (hi 2 + cksW) ; the expression w l = N A 8 e~ Kt sin TIX vanishes when jc = or sc = Z, s=l and, when = 0, is equal to sinh ft (I x) / sinli pi. Hence prove that the expression V= F [sinh ft (I - a) /sinh ftl - w^~ u ^ gives the value of the potential function in the cable, if, when the whole core is at potential zero and the farther end permanently grounded, the point x = is suddenly raised to potential F at the time t = 0, and kept there. The current (C) at any point is given by the negative of the derivative of the potential function with respect to x, divided by the resist ance p of the core per unit length, so that " cos nx]. MISCELLANEOUS PROBLEMS. 439 If the insulation is so good that h may be neglected, and the current is (Fo/p/X 1 + 2-COSS7T. V"e~ A COSTIX). 265. The terminals of a battery of electromotive force E Q volts and internal resistance b ohms are suddenly connected, through a non-inductive conductor of resistance r b ohms, with the coatings of a condenser of k farads capacity. Show that after t seconds the condenser is charged to potential difference E volts, where E = E ( ,(l - e~ ( kr ) = E T, and that the charge on the positive plate is Ek units. If t = ^ kr, T = 0.095 ; if t = J- kr, T = 0.181 ; if t = J kr, T = 0.393 ; if t = kr, T= 0.632 ; if t = 2kr, T- 0.865 ; if t = 3kr, T= 0.950 ; if t = 5 kr, T = 0.993, and if t = 7 kr, T = 0.999. Show that if the condenser just mentioned had been leaky, its dielectric having a resistance of only R ohms, the charge on the positive coating after t seconds would have been ^TT5(1 - e- < r + *>/**), and the final charge E Q kR / (r + K). 266. The coatings of a perfect condenser of 2 microfarads capacity which are connected together by a non-inductive resistance R of 2500 ohms are attached to the terminals of a constant battery. After the condenser has become fully charged, a bullet moving at a velocity of v metres per second cuts first one of the battery leads at a point A and, 2 metres farther on in its course, the resistance R at a point B. While the bullet is moving from A to B the condenser loses 1 1/e of its charge through R. Show that, e being the base of the natural system of logarithms, v = 400. 267. If Si and S z , the plates of a condenser separated by a poorly conducting medium of inductivity /x and of conductivity X, are at potentials V l and V% respectively, and if V denotes 440 MISCELLANEOUS PROBLEMS. the potential function in the dielectric, the capacity of the con denser and the strength of the current that flows through the dielectric, when the difference of potential of the plates is unity, are Show that if the condenser be charged to such a potential that each plate requires Q units of (positive or negative) electricity and then, left to itself, the charge on one of the plates after t seconds is given numerically by the expression Q**Q0-+*. 268. A Leyden jar loses 0.000001 of its charge per second by conduction through the glass. The specific inductive capacity of the glass is 8. Show that the resistance of a cubic centi metre of the glass is roughly 14 X 10 17 ohms, having given that one electrostatic unit of resistance is equivalent to 9 X 10 11 ohms. [M. T.] 269. A submarine telegraph cable 1885 miles long is formed of a copper conductor inches in diameter surrounded by a gutta-percha coating inch in diameter. The specific inductive capacity of gutta-percha being 4.2, show that the capacity of the cable is equal to that of a sphere of the same size as the earth. [St. John s College.] 270. The outer coatings of two condensers A and B are put to earth and their inner coatings are connected through a galvanometer the resistance of which is 4000 ohms. The capacity of A is 3 microfarads, that of B is 1 microfarad, and the two condensers are charged to potential 1 volt. The inner coatings of A and B are then put to earth simultane ously through resistances of 1000 and 2000 ohms respectively. Show that the whole amount of electricity which will flow through the galvanometer is one-seventh of the charge of the smaller condenser. [St. John s College.] MISCELLANEOUS PROBLEMS. 441 271. The outer coatings of two condensers A and B are put to earth and their inner coatings are connected together through a galvanometer of g ohms resistance. The capaci ties of the condensers are C and c respectively. Both are charged initially to potential F" and then have charges Q Q and q . Show that if the inner coatings of the condensers are put to earth simultaneously through non-inductive resistances R and r, and if X=rRC, X =rR C) p. = C r(g + E), p = CR(g + r\ m = CcrRg, k 2 = 4 AA + (/i-/) 2 ; ^ - XX = CcrRff (y + r + fi), and the charge on A after t seconds will be Q e-^+ / 2[(A; + p + ,j, - 2 m/ CR) <F / 2 " + (k-n-iL + 2m/ CR) e- kt/ 2m ] /2 k. Show also that the whole quantity of electricity which passes through the galvanometer during the discharge is 272. Prove that the potential and stream line functions due to electrodes placed at certain points of a spherical current sheet can be deduced directly from the solutions for the plane current sheet which is its stereographic projection. If E l and E 2 be two electrodes on a complete spherical sheet, show that the stream lines are small circles through E l and E 2 and the equipotential curves small circles the planes of which pass through the line of intersection of the tangent planes at E and E 2 . 273. Verify the statement that the value of the potential function at any point P of a solid homogeneous sphere of specific resistance K, when a current of intensity C flows between two electrodes A and B at opposite ends of a diameter, is BP AB 442 MISCELLANEOUS PROBLEMS. where N is the foot of the perpendicular from P on the diameter AB. [M. T.] 274. The two concentric spherical surfaces which bound a shell are kept at different constant potentials. Prove that if the conductivity of the shell is a function of the distance from its centre, the potential function within it satisfies the equation* D r (r*k-D r V) = 0. Show that if w = l/r, this is equivalent to the equation given on page 250. 275. Prove that if a quantity of electricity equivalent to Q absolute electromagnetic units be discharged through a ballistic galvanometer which has a suspended system the magnetic moment of which is M, the moment of inertia /, and the reduced complete time of swing T , 47T/ where GM is the couple exerted upon the suspended system in its position of equilibrium when a steady current of 1 unit passes through the galvanometer coil. 276. When a bar magnet of magnetic length 2 1 and moment M is placed in Gauss s A position with its centre at a dis tance d from the centre of a magnetic needle of length 2 A, the needle is deflected through an angle a, such that 2Hi2M.ad-l dl d + l d + l M where r-f = (d I A sin a) 2 + A 2 cos 2 a, r 2 2 = (d I + A sin a) 2 + A 2 cos 2 a, r, a = (d + I - A sin a) 2 + A 2 cos 2 a, r = (d + I + A sin a) 2 + A 2 cos 2 a. Show that if 1 = 4: centimetres, d = 40 centimetres, A = 0.5 centimetre, a = 20, and H= 0.2, this formula makes m = 285.43, whereas the approximate formula, M t,n /2\a m_ = i^.^J_ tan a) yields m = 285.40. // +j d MISCELLANEOUS PROBLEMS. 443 277. A magnetometer is set up with the centre of its needle vertically above a point in the axis of a horizontal metre rod n centimeters from the centre. The rod is perpendicular to the meridian. A homogeneous, short bar magnet is placed in Gauss s A position with its centre first 50 d centimetres from one end of the rod and then 50 d centimetres from the other end, d being greater than n. If the deflections of the magnetometer needle in the two cases are 8j and 8 2 respec tively, the relative error made by computing M/H by means of the formula d 3 (tan 3 L + tan 8 2 ) /4 is [(1 + 3 ^ 2 ) / (1 - e 2 ) 3 ] - 1, where e n/d. 278. The track upon which the carriage of the short deflect ing magnet slides in an apparatus for determining M/H in Gauss s A position makes an angle with the east and west line instead of being exactly perpendicular to the meridian. Show that if the centre of the deflecting magnet is at a dis tance d from the centre of the needle, and if the deflection changes from S l to 8 2 when the deflector is turned end for end, H 2 cos (Si + (9) 2 cos (8 2 - 0) . ctn 8 l ctn 8 2 where tan 6 = ^ 2i 279. In order to obtain the temperature coefficient of a cer tain magnet, of moment M lt it is placed in a water bath at a short distance from a magnetometer needle, its axis being perpendicular to the magnetic meridian at the centre of the needle. The needle is brought back to its zero position by a compensating magnet placed on the opposite side of the magnetometer at a distance d from it, its axis being also perpendicular to the meridian at the centre of the needle. The moment of the compensating magnet is M ot its magnetic 444 MISCELLANEOUS PROBLEMS. length 2 1 . When the magnet M l is heated a given number of degrees, its moment decreases to Jf/, and the magnet ometer needle is deflected over n divisions of the scale. The scale distance being a, prove that where the deflection n is small. Show that if a x is the angle through which M would deflect the needle if M l were absent, M-, M, f tan a n - = - j where tan a = 7 M l tan c^ 2 a 280. Two magnets, m^ and ra 2 , are placed, with their axes parallel to each other but opposite in direction, in Gauss s B position with respect to a magnetometer. The centre of m l is north of the magnetometer and the centre of m z south of it. The distances (d l and d 2 ) of the centres of w&i and ra 2 from the centre of the magnetometer needle are such that the needle is undeflected. Show that if fa and p 2 are the strengths of the "poles" of m l and m 2J and if 21 1} 21 2 , and 2 A. are the "lengths" of m^ m 2 , and the needle respec tively, /AJ is to fji 2 as J _ r , 2 I [4 2 + W - *) 2 ] [^ + ( 281. A fixed bar magnet of magnetic length SN= 2 L and of pole strength M, and a magnetic needle of magnetic length sn = 21, of pole strength w, are in the same plane, with their centres (C, c} at a distance r from each other. The angles NCc and scC are equal to $ and < respectively. The lines MISCELLANEOUS PROBLEMS. 445 ns, A r meet when produced in F. The perpendicular dis tances of Nj C, and S from ns are r sin < L sin (< -f <), r sin <, and ? sin <f> + L sin (< + <J>), so that the length of the perpendicular dropped from c upon Nn is I sin (FnN) or Z [> sin < Z sin (< + <)] / Nn. The lengths of the perpendiculars dropped from c upon Sn, Ns, and /Ss are I [> sin < + L sin (< + <)] / Sn, I \r sin < L sin (< + <)] / JVs, and I [> sin < + sin (< Show that the sum of the moments, taken about c, of the forces which tend to decrease <, is D = Mm J =rr^ + =z > ^ r sin <f> L sin (<f> 4- Ll-ar* jvvj L in ^ jj or J r rin 4- 1 =i + = r- = ~ = [ |_3?i ^j? Sn bs J Show also that Nn 2 = r + L 2 + f 2 + 2 >/ cos ^> 2 ?-Z cos - 2 Zi cos ^ 4- *, or = - + 2/cos ^ ~ 2L cos r 2 "|~ g , and that, if both I and L are small compared with r so that only the first powers of l/r and L/r need be kept, the approximate value 1 f-i j_ 3 (^ cos ^ - 7 cos $)~| - ^ 1 -h - may be used for ^ n~ 3 . Treat ing JVs, Sn, and /Ss in the same way, prove that if J/ and 440 MISCELLANEOUS PROBLEMS. m are the magnetic moments of the magnet and the needle respectively, we may use for D the approximate value MQ m [3 cos < sin </> sin (< + <fr)] /r\ A better approximation can be obtained by keeping higher powers of the ratios //rand L/r. It is to be noticed that < = and <fc = 90 correspond to Gauss s " Principal Positions." 282. Prove that the magnetic force at a large distance in the prolongation of its axis, due to a bar magnet of moment M y lies between 2 M/rf and 2 M/rf, where r lt r z are the distances from the two ends of the magnet. 283. If I, m, n are the direction cosines of the axis of a small magnetic needle free to turn about its centre in a mag netic field, and if //, M, N are the components of the couple which acts on the needle, D t L -f D m M+ D n N= 0. 284. The accurately flat north end of one of two exactly similar, uniformly polarized, perfectly hard bar magnets is placed in close contact with the south end of the other, so that the two form a long, uniformly polarized, straight bar. What force is necessary to separate the magnets lengthwise ? Compute the work necessary to separate into short elements a long, uniformly polarized, magnetic filament. 285. Has a polarized rigid distribution an axis in the sense that a straight bar magnet has a magnetic axis ? Consider first a bent, solenoidally polarized, magnetic filament. 286. Show that if a polarized electrical distribution were enclosed in a thin " metallic skin connected with the earth," there would be induced upon the inner surface of the skin a charge, E, of total amount zero. Show also that the effect of the given distribution together with the charge on the inner surface of the skin would be nothing at outside points, and the effect at outside points of the given distribution the same as that of a charge on the skin equal to the negative of E. This charge is sometimes called "Green s Distribution" and sometimes "Poisson s Surface Distribution." MISCELLANEOUS PROBLEMS. 447 287. A solid soft-iron sphere is placed in a uniform magnetic field. Show that about three times as many lines of force pass through any closed curve within the sphere as through an equal and parallel curve at an infinite distance. 288. In the case of a certain sphere of radius a polarized parallel to the axis of x, I = A - r"- 3 , where r is the distance from the centre, and the function, /, mentioned on page 192, is r"-*. Show that the values of the potential function within and without the sphere are 4ir^ xr"~ 3 /;i and 4 TrA^x/nr* respectively. Show that at the surface of the sphere the normal component of the induction is continuous, and the tangential components in general discontinuous. The tangential components of the force are continuous, and the normal component in general discontinuous, by the amount 47ro-. 289. An uncharged conducting sphere of radius a is in a uniform field of force F, and consists of two hemispheres in contact with the plane of division perpendicular to the field. Show that if the hemispheres are separated, each will have a charge 3a*F/ir. 290. The field inside a shell bounded by. two concentric spherical surfaces of radii a and b and uniformly polarized in the direction of the x axis, has the potential function zero. Out side the shell the potential function is 4 irxl(l* a^/Sr 8 . 291. Show that - 3Xx/(p + 2) + <7, for values of r less than a, and - Xx + a?Xx(p - l)/[r 3 (/x + 2)] + C, for values of r greater than o, represent the potential function within and without a sphere, of radius a, with centre at the origin, composed of a homogeneous dielectric of inductivity /tx, placed in a uniform field in air of intensity X. 292. If a cylindrical surface which circumscribes an oval body P touches it in a curve which is the perimeter of a right section of the cylinder of area Q, and if P be uniformly polarized in the direction of the axis of the cylinder to 448 MISCELLANEOUS PROBLEMS. intensity /, the amount of matter in the positive distribution on P s surface is Q T. If an ellipsoid the semiaxes of which are a, b, c, be uniformly polarized to intensity / in the direction of the axis a, the moment, M, of the distribution is ^irdbcl and the amount of matter, m, on either the positive or negative half is irbc I. The distance of the centre of gravity of either the positive or negative part of the distribution from the centre of the ellipsoid is M/2 m or f a. In what sense is the " magnetic length" of a uniformly polarized sphere -J a? 293. In the case of any purely polarized distribution bounded by a surface S, the volume and superficial densities are accounted for by a vector /, of components A, B, C, such that within S, p = Divergence /,and on $, <r = I- cos (n, 7). Show that the polarization might be equally well represented by any vector which differs from / by a solenoidal vector O every line of which, if it meets $ at all, lies wholly on S. Is the induction within a hard magnet definite ? 294. Matter is distributed on the ends of a cylinder of revolution of length I and radius a. The density within the cylinder and the superficial density on its curved surface are everywhere equal to zero. On one end a quantity 2 TT<I* of matter is distributed with density o- = a 2 /r, where r is the distance from the axis; on the other end a quantity 2-n-a s is distributed with density a- = 3 r. Can you affirm that the cylinder is not polarized solenoidally ? 295. Show that if, in the case of a polarization symmetrical about the axis of z so that the lines of the vector / lie in planes which pass through this -axis, Z be the component of I parallel to the axis of z and Mthe component perpendicular to the axis, p = [D r R -f R/r + D Z Z~\. Consider the volume den sity in, and the superficial density on, a cylinder of revolution of length I and radius a, the axis of which coincides with the axis of z, when R = (r a)f(z), and + (2 r - a)f(z) /r\ dz. f MISCELLANEOUS PROBLEMS. 449 Assuming both / and \j/ at pleasure, draw the lines of polari zation for the simple case which you have chosen. 296. A solenoidal vector, the components of which par allel to the columnar coordinates r, 0, x are (a r)f (x), 0, (2 r a)f(x)/r, represents the polarization within a magnet bounded by the cylindrical surface r = a and the planes x = 0, x b. Determine the surface density a- and draw two of the lines of polarization when f(x) = x. Show that a- is zero when/(z) = sin(Trx/b). 297. Show that a vector the components of which in the directions of the columnar coordinates r, 6, x are [/ (x) . J?(r)], 0, -/(x) [I"(r) +F(r)/r-\, is solenoidal, and use this form to determine two or three different polarizations within a bar magnet for which both p and <r shall be everywhere zero. 298. Prove that an infinitely long cylinder of revolution of radius a, the axis of which coincides with the z axis, when polarized uniformly in the direc tion of the x axis, gives rise to the potential function 2 irla^x/r 2 at outside points. This is identical with the potential function due to a plane doublet of strength 2 irla 2 at the origin. Within the cylinder the resultant force has the intensity 2 7r/and the direction of the negative x axis, while the induction has the intensity 2irl and the direction of the positive x axis. The lines of induction and the lines of force have the same direction without the cylinder and opposite directions within. The lines of force are shown in Fig. 133. Show that the normal component of the induction is continuous at the surface of the cylinder. FIG. 133. 450 MISCELLANEOUS PROBLEMS. 299. A solenoidally polarized distribution inside which the lines of polarization are straight and parallel need not be uniformly polarized. 300. Prove that the mutual potential energy of any two small magnets at a distance apart large compared with their linear dimensions is M-i - M 2 (cos < 3 cos O l cos 2 ) / ^ where M ly M 2 are the moments of the magnets, < the angle between their directions, and O ly 2 the angles which these directions make with a line drawn from the centre of the first to the centre of the second. 301. Show that for a simple magnetic shell in the form of a circle, the direction of the vector potential at any point is per pendicular to a plane through the point and a normal to the plane of the shell through the centre. [St. Peter s College.] 302. Prove that if m is the pole strength of a slender, straight, uniformly magnetized magnet AB, a vector poten tial may be found which has at any point P the value (cos PAB 4- cos PBA), where p is the length of the per pendicular dropped from P on AH, produced if necessary. Show that the direction of this vector potential is perpen dicular to the plane PAB. [M. T.] 303. Show that if V is the value of the potential function, and F that of the vertical component of the magnetic force at the earth s surface, the earth s field in outside space may be considered as due to a surface distribution of density F/2 TT F/4 ?ra, where a is the earth s radius. 304. A magnetic needle is placed near an infinite plane face of a mass of soft iron. Show that the reaction of the iron on the needle may be represented as due to a negative image of the needle in the plane face, reduced in intensity in the ratio of (/x 1) / (fi + 1), where /x is the permeability of the iron. [St. John s College.] MISCELLANEOUS PROBLEMS. 451 305. A magnetic element /SaVof pole strength a and moment b lies in a magnetic field which has the potential function V. Show that if I, m, n are the direction cosines of the axis of the element, the mutual potential energy of the element and the field is \E=a(V y -V^ = b(l-D x V+m.D y V+n-D s V). If the element is a rectangular parallelepiped, dx dy dz, taken from a magnetized body in which the polarization is /, b = Idx dy dz, &E= (A-D x V+B-D y V+C.D z V)dxdy dz, and the mutual energy of the field and the magnet is the integral of this last expression. . If the magnet is a simple shell of strength 4>, E = *(l -DtV+m- D fl V+ n D z V) dS, or cos (* n ) + Y> cos 0, n) + Z- cos (z, n) ] dS, where the integration is to be extended over one face of the shell. 306. A simple plane circular magnetic shell of radius r lies in the yz plane, with its centre at the origin, in a magnetic field symmetrical about the x axis. The intensity of the x compo nent of the field is F (x), where F is a continuous function, such that F (oo) = 0. Show that the force which urges the shell is equal to ira 2 & - D X F. The centre of the rigid shell is to move along the x axis to infinity while the plane of the shell is parallel to the yz plane. Compute the work done on the shell by the field during the motion. Has the field any com ponent perpendicular to the x axis ? Compute the work done on the shell by the field, with the help of the method discussed at the top of page 218. Show that a vector which has the com- ponents F(x), y C/(^+z 2 ) - \y F (x), zC/(y> + z 2 ) - \z . F (x), is solenoidal and is symmetrical about the x axis. 307. Show that if A 1 , B\ C are the components of magneti zation at the point (x , y\ z ) in any magnet, M , and if p. denotes the reciprocal of the distance between (x , y , z ) and 452 MISCELLANEOUS PROBLEMS. (x, y, z), the components at (a*, y, z) of the ordinary vector potential of the magnetic induction are fff (C .D x .p-A .D s ,p)dr , - D,p - & D X . P ) dr . The scalar potential function of magnetic force is in the same notation + B-. !)!> + C D,p) dr . If M is a simple shell of strength 4> , the x component of the vector potential function can be written in the form <!> C C[Djp cos (y, n) D y ,p cos (z, ri)~\dS . [Maxwell.] 308. Show that if r is the distance from a fixed point, the line integral around any closed curve s of the tangential component of the vector (1/r, 0, 0) is equal to the surface integral, taken over any cap S bounded by s, of D z (1 IT) cos (y, n) -D y (l/r). cos (z, n), where n is a positive normal to the cap. Obtain two similar equations with the help of the vectors (0, 1/r, 0), (0, 0, 1/r), and prove that the components of the vector potential function of the force due to a magnetic shell of strength <I> in air are 4> f[cos(x, s) /r]d8, 4> J[cos (y, s) /r]ds, *J [cog (a, *)/*"] d*j taken around the perimeter of the shell. If (L, M, N) are the curl components of a vector (F x , F y , F z ), the latter is a vector potential function of the magnetic field. MISCELLANEOUS PROBLEMS. 453 Show that the mutual potential energy of a magnetic shell of strength <!> and the field is - 4> C(F M dx + F y dy + F z dz ), taken around the shell in positive direction. If the external field is caused by another shell of strength <1>, we have where the integrals are to be taken around the perimeter s of the second shell, and the mutual potential energy of the two shells is Sx& J J [cos (x, s) cos (x, s ) + cos (y, *) - cos (y, 5 ) + COS (S, A ) COS (z, )] [flfe (/* / /] or - <M> f f[cos (C/A-, ds ) / r] rf ds . The integral by which <M> is multiplied has been called the " geometric potential " of the two curves. 309. Prove in two different ways that the energy of the surface distribution a- = /-cos(?i, /), on a sphere of radius a uniformly polarized to intensity /, is 87r 2 / 2 a 8 /9. In what sense is this the energy of the distribution? Give a sum mary of the reasoning of Lord Kelvin in his paper " On the Mechanical Values of Magnets." 310. If a polarized distribution is placed in a field of force which has a potential function I , the mutual potential energy of the field and the distribution as a whole is VI - cos (n, /) dS - J J J F(/V1 + /> y # + D z C) dr, where the first integral is to be extended over the surface of the distribution and the second through its volume. Show that this energy is equivalent to (A-D X V+B-D V V+ C-l 454 MISCELLANEOUS PROBLEMS. 311. A sphere of radius a uniformly polarized to intensity / is placed in a uniform field of force of intensity X. Show that if the directions of the field and the polarization coincide, the mutual potential energy of the sphere and the field is 47ra 3 /X/3. What would be the energy if the direction of the field and polarization were opposed ? It would be zero if these directions were perpendicular to each other. 312. If V is the potential function due to a volume distri bution of density p l in a region T l9 and a surface distribution of density <TI on a surface Si, and if U is a continuous function + 4 CC Ucr, where the volume and surface integrations in the second member are to be extended respectively through and over a spherical surface of radius r so large as to include 2\ and S^ If U vanishes at infinity, the last surface integral vanishes when r is infinite. Use this equation to compute the mutual potential energy ( ira*IX) of a sphere of radius a, uni formly polarized to intensity / in the direction of the x axis, and a uniform field (X, 0, 0), in which it lies. In this case the value of the last term in the second member is 32 ?r 2 a 8 /X/9. 313. At a distance of 10 centimetres from the middle point of a wire 140 centimetres long, the magnetic force due to a current in the wire would be within one per cent of that which would be produced if the wire were infinite. 314. Show that the magnetic force within a square circuit (of side = 2a) at a point midway between two sides, at a dis tance x from the centre of the square, is a \ a x a + x MISCELLANEOUS PROBLEMS. 455 Draw a curve which shall represent this force as a function of x. What if x is greater than a ? Show that the force at a point distant y from the plane of the circuit in the axis of the circuit is SCa 2 1 } j Show that if a current of A amperes be sent through a tangent galvanometer which has a square coil consisting of n turns of wire, 5 all tan 8 A - -p -- 2V2-7* 315. If a circuit carrying a steady current C is a regular polygon of 2 n sides, and if a is the radius of the inscribed circle, the magnetic force at the centre is (4 n C/a) - sin(7r/2 n). 316. The plane of the ring of a tangent galvanometer which consists of a single turn of fine wire is the vertical plane of the magnetic meridian. Show that if a current of A amperes be sent through the ring, the strength of the field at a point P in its axis at a distance z from the centre is (* - where r is the radius of the ring. Hence prove that if the centre of the galvanometer needle is at P, the deflection will be given by the equation A = [ 5 (z 2 -f r 2 ) 3 IT tan a] /in*. 317. Show that at a point on the axis, at a short distance (z) from the centre of a tangent galvanometer coil of radius a, the intensity of the electromagnetic field due to a steady current passing through the coil is to the intensity of the same field at the centre asfl- ^ ) to 1, nearly. 318. Show that if around a ring formed of a piece 2 b centi metres long of a thin metal tube of inside radius a and of outside radius a + 8, a steady current of strength 2b8C uni formly distributed through the conductor could be sent, the 456 MISCELLANEOUS PROBLEMS. strength of the resulting electromagnetic field at the centre A S~il of the axis of the coil would be = =. What would this intensity become if the tube were to shrink indefinitely in length while the whole current around it remained unchanged ? Assuming that when x is small (1 + x)-* = 1 - 1 x + l& - ^ .r> -f . , deduce from your results the usual correction, , for the breadth of the coil of a tangent galvanometer. 319. The vertical coil of a tangent galvanometer makes a small angle 8 with the east and west line through the centre of its needle. If a steady current, of such strength that it would cause a deflection of 45 if the plane of the coil were in the meridian, be now sent through the coil, it will cause a deflection of 8. 320. The centres of the rings of a two-coil tangent galva nometer are 20 centimetres apart and the mean radius of each of the coils is 20 centimetres. The centre of the needle is on the common axis of the coils halfway between their centres. When the instrument is properly set up in a cer tain place a steady current of half an ampere sent through both coils in series causes a deflection of 45. Show that if there are 20 turns in each coil, ff= 167T/10 (5) 3 . 321. A tangent galvanometer has two equal vertical coils, each of mean radius r, placed at a distance apart of 2 r ( VS 1)*. The short compass needle is placed midway between the coils on their common axis. Show that the needle deflection caused by any current which passes in the same direction through both coils in series will be the same as if the same current passed through only one coil, while the centre of the needle was at the centre of this coil. 322. Show that if the vertical coil of a tangent galva nometer makes an angle 6 with the meridian, and if a current MISCELLANEOUS PROBLEMS. 457 of C amperes be sent through it first in one direction and then iii the other, causing deflections of ^ and S 2 respectively, then n-nC _ sinSi sin8 2 5777 ~~ cos (0 - ^ ~ cos (0 + 8 2 ) and tan = (ctn &, ctn 8^, where r is the mean radius of the coil and n the number of turns of wire on it. 323. From a thin, flat sheet of copper of thickness 8 is cut a ring of inside radius a d and outside radius a d. If a steady current of strength 2C&d could be made to circulate around this ring, what would be the strength of the electro magnetic field at the centre of the ring? What would this strength become if the ring were to shrink to a fine wire ring of radius a concentric with the original ring without change of the current strength ? Assuming that when x is small deduce from your results the usual correction (one-twelfth of the square of the ratio of the depth of the coil to its mean radius) for the depth of the ring of a tangent galvanometer. 324. A certain galvanometer coil is wound upon a large square frame. When the vertical plane of the coil makes an angle with the meridian a certain current C sent through the coil deflects the short needle through an angle towards the coil. Show that C would cause the same deflection if the coil were in the meridian. 325. On the axis of a fixed circular ring of wire which car ries a steady current C is a molecular magnet of moment m. Show that if the axis of the magnet makes an angle with the axis of the ring the moment of the couple which tends to diminish is (2 irmC sin 8 < sin 6) /a, where a is the length of a radius of the ring and < the angle subtended at the molecule by the radius. 458 MISCELLANEOUS PROBLEMS. 326. A perfectly flexible wire fastened at two fixed points carries a current of given strength. Prove that in a uniform field of magnetic force it will tend to assume the shape of a helix. [M. T.] 327. A very long straight wire which carries a steady cur rent C is at right angles to the plane of a circular ring of radius a which carries a current C . The ring is free to turn about the diameter which intersects the straight wire. Prove that the couple tending to turn the ring is 2TrCC a 2 /r or 27r(7C"r, according as a is less or greater than ?, the distance of the wire from the centre of the ring. [Trinity College.] 328. A plane ring can move about a diameter parallel to an infinite straight wire, the distance of which from the centre of the ring is equal to the radius of the latter. Show that when currents CC are sent through the two circuits the couple tending to turn the ring is 4 7rCC a(cos </> - cosi </ V2 cos 0), when a is the radius of the ring and <j> the acute angle which the normal to its plane makes with the perpendicular to the straight wire drawn from the centre. [M. T.] 329. If a layer of n turns of wire carrying a steady current of unit strength and forming a coil k be wound uniformly on such a ring coil, k, as that shown in Fig. 77, the induction due to the current in k has at every point within the coil the value 2/xw /r. The integral of n times this quantity taken over a cross-section of the ring R on which k is wound gives the mutual inductance of the two coils. Show that if R may be regarded as formed by revolving a circle of radius a about a line in its plane, distant b from its centre, the value of the integral is 4 TTfjinn (b V& 2 a 2 ). If R were formed by the revolution of a rectangle with sides of length b parallel to the axis and a perpendicular to it, the value of the integral would be 7 , c 4- a /2 2 nn fib log MISCELLANEOUS PROBLEMS. 459 where c is the distance of the centre of the section of the ring from the axis. Show that the self-inductance of k might be found for the two cases just mentioned by putting n equal to ri in the expressions for mutual inductance, and imagining k to move into coincidence with k. 330. A thin tubular conductor of circular section has a radius a and carries a steady current (7; prove that the mechanical action between the different portions of the current produces a transverse tension in the tube, of intensity C*/7ra. [St. John s College.] 331. The ponderomotive forces which act upon a portion A^A 2 of a circuit which carries a steady current of strength C in the field of a magnetic pole of strength m at the point 0, have a resultant moment M about any straight line OZ drawn through 0. Let PP represent an element As of the circuit ; let OP = r, OP = r + Ar, ZOP = 0, ZOP = + A0, (/,*) = 8, and denote the angle between the planes ZOP and POP by <. The fundamental equation of spherical trigonometry yields cos (0 + A0) = cos cos POP + sin sin POP cos <f>, and it is evident, since A0 is not greater than POP , that the limit of the ratio of (cos A0 - cos POP 1 ) /sin POP is zero, so that cos< is approximately equal to sin A^/sin POP . The Theorem of Sines applied to the plane triangle POP yields the equation PP / OP = sin POP /sin OPP . Prove that the moment about OZ of the elementary force exerted by the pole upon As may be written, AJf = mC- sin 8 cos < sin As/r, and that for purposes of integration this is equivalent to mC sin dO, so that M = mC(cos 2 cos ^), where 0! = ZOA lt 6 2 = ZOA 2 . If O l = 2 , as in the case of a closed circuit, M is zero. Consider the possibility of rotation about a straight line, of a closed circuit bearing a steady current C under the action of any number of magnetic poles on the line. 460 MISCELLANEOUS PROBLEMS. 332. In the case of a solitary linear circuit s carrying a current C in its own magnetic field, all the lines of force are closed curves threading the current, and the line integral of the force taken around any one of these curves is 4?rC. The equipotential surfaces fill all space ; each of them is a cap bounded by the circuit, and the surface integrals of the induc tion taken over these caps are all equal. Use the reasoning of page 270 to show that since = Cff-ds, and^= C C - dBdr, where dp is the increment of the induction flux through the circuit, due to a small increase in the current. Show from the equation E dp/dt = rC that, besides the energy dissipated in heat, the generator in a solitary circuit must furnish an amount of energy C dp while the current in the circuit is changed from C to C + dC, and that the difference dW between this quantity and the increment dT of the electro- kinetic energy shows the amount of energy which is used in some other way than in increasing this energy. Prove that and use this expression to compute (see page 291) the energy loss due to hysteresis during a cycle of magnetization. 333. The distance between the axes of two infinitely long, straight, round, non-magnetic wires (A lf A 2 ) of radius a and parallel to each other is b. One wire carries a steady current (7, uniformly distributed, in one direction, and the other wire an equal current, uniformly distributed, in the other direction. MISCELLANEOUS PROBLEMS. 461 If the cross-section of each wire be divided into n elements of equal area, every element, dS, is the section of a filament which carries a current CdS/wa 2 . Imagine a circuit made up of a certain filament F l in A^ distant r, , r^ from the axes of the wires, and a filament F. 2 in A., distant r 2 , r 2 " from these axes. The flow of induction through this circuit due to A^ is / 9 Cr C r i 2 C I -dr + I -dr or 2 C[(a 2 - r/ 2 )/2 a 2 + log (r 2 /a)] */r CL \s a T and that due to A s is 2C[(a 2 - r 2 " 2 )/2a 2 + log /)] If the filaments are symmetrically situated, >- 2 = r/ , r/ = r 2 " and the induction through the circuit is The electrokinetic energy of a set of circuits is equal to one-half the sum of the products formed by multiplying the induction through any circuit by the current in that circuit. The contribution which the elementary circuit just mentioned would make to the electrokinetic energy T is, therefore, one- half the product of the induction through it and the current which it carries, so that T= . ; f fjl - r/ /Sa _ r> / 2 a - + log(r, /a) TTO"*/ / where the integration is to be extended over all such elementary circuits, that is, over the cross-section of either wire. We may write for dS, either r/ drj dO l or r z " dr 2 " dO at pleasure, and we may use the first of these for the first, second, and fifth terms of the integrand, and the second form for the other terms. The limits of will be and 2 TT, and those of >/ and r/ , and a. Assuming that, if m>n, \ log (m -f n cos 0) dd = TT log \ J (i 462 MISCELLANEOUS PKOBLEMS. show that T= <7 2 [i + log(6 2 /a 2 )] and that it makes no differ ence how the separate filaments of A l and A 2 are combined into elementary circuits. Show also that the inductance, per unit of its length, of the circuit made up of the wires, is 2 log (b 2 / a 2 ) -f 1. Prove that if the wires had inductivities /A L and /u, 2 and radii a ly a 2 , we should have L = 2p log/ (&Xa 2 ) +(/*! + /x 2 ), where /u, is the inductivity of the surrounding medium. [For a discussion of the inductance of the circuit when the cross- sections of the long parallel wires are of any form, the reader is referred to A. Gray s Absolute Measurements in Electricity and Magnetism, Vol. II, p. 288, and to Drude s Physik des Aethers, p. 207.] 334. Obtain Heaviside s expressions (Electrical Papers, p. 101) for the coefficients L 19 L 2 , M, of self and mutual induction for two parallel wires of length I, radii a ly 2 , and inductivities /x 1? ^ suspended parallel to each other and to the earth at heights h l} h 2 and at a horizontal distance d apart, if the current is supposed to return through the earth in a thin sheet, and if h and A 2 are small compared with L These expressions are L,/l = J^ + 2 log (2 /h/fli), L 2 /l =1/^ + 2 log (2 h 2 /a 2 ), n i <& + (^1 + hzY M/l = log Yf 7^ 335. Show that if K and //, represent the dimensions of electric and magnetic inductivities respectively, the dimen sions of /ut in terms of L, M, T, K are L~ 2 T 2 K- 1 , while those of K in terms of L, M, T, /x are L~ 2 T 2 ^- 1 . Show that the dimensions of electric quantity in the two systems are f*, Zi*Jf*fi *j those of magnetic quantity .* ; those of electric field strength L * ; those of magnetic field strength MISCELLANEOUS PROBLEMS. 463 i those of electric potential LrM*T~ l K~ h , those of magnetic potential L*M*T~ 2 K*, tnose of conductivity LT~ 1 K, Z" 1 !/*" 1 ; those of electric current L*M-T~ 2 K , L-M*T~ V~" 5 those of capacity LK, L~ 1 T 2 P~ 1 ; those of inductance L~ 1 T K~\ Z/x; those of magnetic moment L*M*K~*, llf*2 r "V*; those of electric surface density L~*M*T~*K*, L~*M*n~*- 336. A rigid plane wire of any shape is free to turn about a point in its plane distant a and b from the nearer and farther ends of the wire. The plane of rotation is perpen dicular to the lines of a uniform field of induction of intensity B. Show that if the wire forms part of a circuit which carries a current (7, the moment about of the forces which, acting on the wire tend to set it in motion, is ^BC(b 2 a-). If the wire rotates with angular velocity to, it cuts the lines of the field at the rate ^ o>B (lr a 2 ). 337. A copper disc perpendicular to the lines of a uni form magnetic field is spun in its own plane about a fixed point and is continuously touched at two points by the fixed electrodes of a galvanometer. Show that the current in the galvanometer is proportional to the difference of areas swept out by the radii vectores from O to the points touched. [M. T.] 338. A magneto-electric machine, driven at a constant rate, sends current through the coil of another magneto-electric machine used as a motor. When the second machine is held still, a power JFis used in the circuit. Prove that the maximum power obtainable from the second machine is J JF, and that then the first machine absorbs ^ W from the engine which drives it. [M. T.] 339. Show that (1) if a conductor be moved along a line of magnetic induction parallel to itself, it will experience no electromotive force ; (2) if a conductor carrying a current be free to move along a line of magnetic induction, it will experi ence no tendency to do so ; (3) if a linear conductor coincide 464 MISCELLANEOUS PROBLEMS. in direction with a line of magnetic induction and be moved parallel to itself in any direction, it will experience no electro motive force in the direction of its length ; (4) if a linear conductor carrying an electric current coincide in direction with a line of magnetic, induction, it will not experience any mechanical force. [Maxwell.] 340. Discuss the following statements of different writers : "When an electromotive force E is suddenly applied to an inductive circuit of resistance J?, the counter-electromotive force of self-induction is initially equal to E and the current caused by E is initially zero." When the current is rising a portion of E, viz., .#(?, is employed in maintaining according to Ohm s Law the current C already established ; the other portion of E, viz., L D t C, is employed in increasing the electro magnetic momentum LC." "At the beginning the whole of the electromotive force acts to increase the current." " If a current is established in a coil and the coil left to itself, short circuited without any electromotive force to maintain the current, then as the decaying current reaches a value C the electromotive force RC is equal to L D t C" "The reactance does not represent the expenditure of power, as does the effective resistance, r, but merely the surging to and fro of energy. While the effective resistance, r, refers to the energy component of the applied electromotive force or the electromotive force in phase with the current, the reactance, x, refers to the wattless component of the electro motive force or the electromotive force in quadrature with the current." 341. Compare the differential equation of motion of a body of mass L, moving with velocity C, under the action of an impressed force E, which tends to increase the velocity, and a resistance rC, proportional to the velocity, with the equa tion which the current in an inductive circuit must satisfy. Why should LC in the electrical case be called the "electro magnetic momentum " ? MISCELLANEOUS PROBLEMS. 465 342. Given that j sinpt sin(^ -f- a) dt = [(2pt sin 2 pt) cos a cos 2 pt - sin a] /4j9, show that the average value for any number of whole periods of the product of two simple harmonic functions of the same period is half the cosine of their phase difference. Show that the activity, or " power/ of a harmonic alternating current is equal to the product of the effective current, the effective electromotive force applied to the circuit, and the cosine of their difference of phase. 343. Two coils, the resistances of which are r ly r 2 and the inductances L u L are in series in a simple circuit carrying a harmonic current. Is the impedance of the two taken together equal to the sum of their impedances ? 344. The ends of the coil of an electromagnet are subject to a rapidly alternating electromotive force. Show that the energy expended in the battery when a given amount of heat is produced in the wire will be greater than would be the case if the electromotive force were constant in direction and magnitude. 345. Show that if a number of linear circuits s lt s 2 , s 3 , , carrying currents C\, C 2 , C 3 , , exist together, and if the total flow of induction through s t be denoted by j^., the electro- kinetic energy, T, may be written $2C k p k and p k = %Dc k T. The quantity p k is sometimes called the electrokinetic momen tum of s k . Compare this result with the equation, p k =Dc k T, given on page 296. 346. Show that the total flux of electric current induced in a thin circular coil ofc radius a and resistance R, made up of n turns of wire, when the coil is turned through two right angles in the earth s uniform magnetic field H, is 2 ira?nH / R. Show also that if a sphere of soft iron of the same radius a be pushed completely into the opening in the coil, the flux is increased in the ratio of 3 /*/(//, + 2). [St. John s College.] 470 MISCELLANEOUS PROBLEMS. the currents (I l9 / 2 ), similarly measured, in tlie two branches. If one of the branches (TI) of the di\ ided circuit is non-inductive, the instantaneous difference of potential between its ends is r l C l and the instantaneous rate of expenditure of power in the other branch is r^C^ the average activity in this latter branch is i\ times the average value of C 1 C 2 . Show that the power expended in r. 2 is ^^(J 2 / x 2 / 2 2 ). What is the " Three Ammeter Method " of measuring power ? 363. Show that if u -f vi is a solution of the equation L-D t C ) v is a solution of the equation L - D t C + rC = E^ sin (pt + a). Prove that the complete solution of this last equation is Ae~ rt/L + E sin (pt + fl) / Vr 2 + where tan b = (r sin a Lp cos a)/(r- cos a -f- Lp - sin a). 364. Show that if u and v are solutions of the equations L.D t C + rC=E l -sin (pj + a^, ^ __ L DC + rC =fi 2 > sin (p 2 t + 2 ) respectively, u + v is a solution of the equation L - DC + rC = E- sin l t + % + ^2 sin and write down the complete solution of this last equation. Write down also an expression for the current in an inductive circuit to which n simple harmonic electromotive forces of given periods are applied. 365. Show that if u + vi is a solution of the equation u is a solution of the equation r D t C + C/k =pE cos (pt + a). MISCELLANEOUS PROBLEMS. 471 Prove that the complete solution of this last equation is Ae- t/kr + kpE* sin (pt + P) / VI + where tan ft = (rpk sin a + cos a) / (rpk cos a sin a). 366. Show that if u and v are solutions of the equations r D t C + C/k = A l cos (pj + i), r.D t C + C/k = A cos (pj + a 2 ), u + -y will be a solution of the equation =Ai> cos (^ + a,) + A, cos (^ + a 2 ), and write down the complete solution of this last equation. Write down also an expression for the current in a non- inductive circuit of capacity k to which n simple harmonic electromotive forces of given periods are applied. 367. If < stands for the operation D t , < 2 for the operation Dfj and so on, the result of applying the operation (po to e** is equal to the product of e kt and Show that if we denote the result of applying the operation to e** by u, so that u satisfies the equation feo + Vi<t> 4- fi a^ 8 + ) a special value of u is the product of e kt and the fraction 2 4- Show how the complete value of u might be found and why the special solution alone is needed in many practical problems. 470 MISCELLANEOUS PROBLEMS. the currents (I 19 / 2 ), similarly measured, in the two branches. If one of the branches (r^ of the divided circuit is non-inductive, the instantaneous difference of potential between its ends is i\C l and the instantaneous rate of expenditure of power in the other branch is r C^ C 2 ; the average activity in this latter branch is i\ times the average value of CiC 2 . Show that the power expended in r 2 is % r (P If / 2 2 ). What is the " Three Ammeter Method " of measuring power ? 363. Show that if u + vi is a solution of the equation v is a solution of the equation L D t C + rC = E - sin (pt + a). Prove that the complete solution of this last equation is Ae~ rt/L + E sin (pt + ft) / Vr 2 + where tan b = (r - sin a Lp cos a)/(r- cos a + Lp- sin a). 364. Show that if w and v are solutions of the equations sin (pj + a^, sin (^ + 2 ) respectively, M + v is a solution of the equation L - DC + rC = E- sin j + a, + ^ sin and write down the complete solution of this last equation. Write down also an expression for the current in an inductive circuit to which n simple harmonic electromotive forces of given periods are applied. 365. Show that if u + vi is a solution of the equation u is a solution of the equation r - D t C + C/lc =pE cos (pt + a). MISCELLANEOUS PROBLEMS. 471 Prove that the complete solution of this last equation is Ae-"* + kpE sin (pt + /?)/ VI + where tan /? = (rpk sin a -f- cos a) / (r/>& cos a sin a). 366. Show that if u and v are solutions of the equations r D t C + C/k = AI cos (p^t + t ), r D ( C + C/k = A.,- cos QA/ + 2 ), ?* + v will be a solution of the equation r D t C 4- C/& = A l cos (^ + e^) + .4 2 cos (pj + 2 )> and write down the complete solution of this last equation. Write down also an expression for the current in a non- inductive circuit of capacity k to which n simple harmonic electromotive forces of given periods are applied. 367. If < stands for the operation D t , <f> 2 for the operation D t 2 , and so on, the result of applying the operation to e** is equal to the product of e kt and Show that if we denote the result of applying the operation ^ = (Po +Pi<t> +P*tf + -)/fe + !/i* + <l*f + ) to ef* by u, so that u satisfies the equation Go + ?i* + <!<& + ) = (^o +^i* -f ^ 2 * 2 + )**> a special value of M is the product of e kt and the fraction Show how the complete value of u might be found and why the special solution alone is needed in many practical problems. 472 MISCELLANEOUS FKOBLEMS. Compare the results of applying the operation Z and the operation (A + B<f>) / (C + Zty), where A = ( PQ -p 2 k* + ptk* ---- ), B=( Pl -p s k 2 C=(q - q,k* + qtf ---- ), D = (q, - q 3 k 2 to M -sin(A; -f e). Show that (a + b<f>) [M> sm(kt + a)] = M Va 2 + b 2 k 2 - sin [kt + a + tan- 1 (bk/a)~\ or M -\la? + b 2 k 2 - sin (A: + 8), where tan 8 = (bk cos a + a sin a) / (a cos a bk sin a), and that a special value of Va 2 + We 2 . is M sin Vc 2 + <W This symbolic notation is treated at length in Forsyth s Treatise on Differential Equations and in Perry s Calculus for Engineers. 368. Prove that if < stands for the operation D t) !> 2 )[Jf.sin(fo where tan X = bk/ (a ck 2 ), and that a special value of m(kt -f a)] is Jf sin (kt + a. p)/ V(7 + rJr) 2 + w 2 /c 2 , where tan ^ = mk/(l nk 2 ). Hence show that a special value of . . is J/ .1 = sm/^ + a + A - MISCELLANEOUS PROBLEMS. 473 369. If <f> were an algebraic quantity, the expression 1 would be equivalent to 1 J_/ - \ Are these two operations equivalent when < = D t and when a special value suffices? 370. If an electromotive force E = E m sinpt be applied to a circuit consisting of a coil of resistance r and inductance L, in series with a condenser of capacity A, we have the equation E-L-D t C-Q/K=rC, where C = D t Q. Show that this equation can be written in the form and write down its solution in the form needed for practical use. Treat in the same manner several of the equations of Section 86. 371. Two circuits, s l and s 2 , have resistances r 1? r 2 , induc tances LI, L 2 , capacities A 1? K, and a mutual inductance M. They contain variable electromotive forces E lt E^ and carry currents C l5 C z . Show that if R, = r, -f L<& + where <^> represents the operation D t , If the capacities of the circuits are negligible, we are to put R l = r x -}- Lrf, J? 2 = ?* 2 4- L. 2 <^ in these results. If E z = 0, Kt = oo, A", = oo ; and C 2 = - M+E! 1-^2 + r^ + r 474 MISCELLANEOUS PROBLEMS. Write down the solutions of these equations and compare them with the results given in Section 87. 372. Show that if C v C 2 are the currents in the primary and secondary of a transformer, and if the secondary circuit has no capacity and contains no internal applied electromotive force, M<l>C l +(r 2 +L 2 <t>)C 2 = or C 2 = -M^C,/^ +Lrf), and if C l = C m sin (pt a), C 2 = ~ P m sin (pt - a + | - tan-lL^/r,). V 2 y + r* If, as is often the case in practice, i\ is small compared with L 2 p, we have, approximately, C 2 = MC m sin (pt - a) - /L 2 = M C m - sin (pt a ?r) /L 2 . In the general case C 2 and C are not zero at the same instant. 373. An electrodynamometer consists essentially of two coils, one fixed and the other movable. The movable coil is furnished with an index which moves on a fixed scale, and the readings are to be considered equal to the product of the strengths of the steady currents C v C 2 in the two coils. If, however, these currents alternate rapidly, the readings are proportional to the average value of C r C 2 . Assuming that J sin pt- sin qt dt = Sin O -?)*/ 2 (p -q)- Sin (p + q)t /2 (p + q), \ siii 2 pt dt (pt sin^tf cospt)/2p, and that C l = m - sin pt, C 2 = n- sin qt, show that the reading is zero when p and q are not equal. What is it when p = q ? Plot a curve which shall show the readings for different values of a, when C l = m- sin pt } C 2 = n sin (pt a), assuming that sin pt cospt dt = cos 2 pt /4 p. If a = % ir, the reading is zero. MISCELLANEOUS PROBLEMS. 475 374. A function z=f(t) may be represented in polar coordinates at any instant by a point P, the distance of which from the origin is equal to the numerical value of , while the vectorial angle XOP is equal to pt, where p is any con venient constant. The plane path traced out by P during any time interval shows the march of z during the interval. If z is either a cos (pt 8) or a sin (pt 8), the path of P is a circumference of diameter a passing through the origin : the vectorial angle of the centre of the circumference is 8 in the first case and 8 -f- $ TT in the second. If z is known to be a simple sine or a simple cosine func tion of frequency p/Zir, it is completely determined when the vector OP , which represents the diameter of the circumfer ence, is given. If the plane of the diagram were the ordinary complex plane, P would represent the complex quantity 2 = x + jy m where x and y Q are the horizontal and vertical projections of the diameter of the circumference, and j the imaginary unit ; and it is often convenient, as Steininetz has shown in a series of remarkable papers, to represent the har monic function z by the quantity Z Q . With this understand ing of the meaning of the sign of equality, we may write in general z = x + yj: the modulus of z is given by the equa tion |z| = Vz 2 + 2/ 2 . Show that if C +j C" represents the current C=C m .sm(pt-$), where tan 8 = Lp / ;, in a simple circuit of resistance ?, induc tance L, reactance x = Lp = 2 -n-nL, and impedance Z\ the " electromotive force consumed by the resistance " is rC=r(C +j-C"), the " electromotive force produced by the reactance " is jxC=jxC -xC", 476 MISCELLANEOUS PROBLEMS. and the electromotive force required to overcome the reactance is xC" jxC , so that the applied electromotive force E is (rC + xC") +j(rC" - xC>) = e r + e x -j = (r - jx) C, and the impedance is (r jx). Write down an expression in complex form for the impe dance of a simple circuit made up of a number of coils of resistances r, r 2 , r 3 , , and inductances L l} L z , L 3 in series. Show that if an electromotive force, E m -cospt, be applied to a simple circuit of resistance r and inductance L which con tains a condenser of capacity k, the impedance Z has the form rj(x x ), where x = Lp and x = \/kp. 375. A long straight wire parallel to the x axis, of resist ance r and self-inductance L per unit length, is covered by a thin layer of insulation the outside of which is kept at poten tial zero. The capacity of the cable per unit length is k, and the rate of leakage through the insulation of a point where the wire is at potential V is A. V per unit of length. Show that if C is the current in the wire, or, - DJC - Lk-D?C- (L\ + rk) D t C - r\C = 0. [Heaviside, Electrical Papers, Vol. I, XX ; Poincare, Comptes jRendus, 1893 ; Picard, Comptes Rendus, 1894 ; Boussinesq, Comptes Rendus, 1894 ; Bedell and Crehore, Alternating Currents ; Webster, Electricity and Magnetism ; Pupin, Transactions of the American Mathematical Society, 1900; The Electrical World and Engineer, October, 1901, and February, 1902.] 376. Two circuits s lt s 2 , which have self-inductances L ly L z , and a mutual inductance M, carry currents C ly C 2 in a magnetic field due to these currents only. The first circuit, which is rigid, contains a generator of constant electromotive force E l ; the second, which is deformable, contains no generator, so that MISCELLANEOUS PROBLEMS. 477 , + MC^/dt = rjCi, - d(3IC\ + L,C*)/dt = r,C,. Show that if one (.r) of the generalized coordinates which define the conformation of the second circuit receives the increment dx during the time dt, so that L. 2 , M, C^ C 2 are changed while Z x remains constant, the work d W done by the electromagnetic forces is % C dL. 2 + C^- dJf t and the change dT in the electrokinetic energy is C, 2 dL, + L&- dC, + MC, dC, + MC, dC, + dC, dM + L 2 C, dC 2 . Show that the equation ( dp 2 = r 2 (7 2 dt) yields r,Cr - dt + C,L, dC, + C 2 2 dL, -h MC 2 - rfC L + C.C, dM = 0, and that the energy (Cfrdt + C^dp^ furnished during the inter val dt by the generator in the first circuit is equal to d W + dT plus the energy dissipated in heat in the two circuits. If C 2 is originally zero, the expressions for d W and c/Tare much sim plified. In any given case dx /dt is virtually determined by the mechanical equation of motion of the moving parts of s 2 ; its value will evidently be greater or smaller, other things being equal, according as the electromagnetic forces are assisted or opposed by external forces. L. 2 . J/ are to be regarded as given functions of x and other variables which do not here enter, and dL 2 /dt, dM/dt can be written D^L^dx/dt) and D^I^dx/dt). The mechanical equation and the first two equations of this problem form a set which completely deter mine x, C 1? C. 2 as functions of the time. If a circuit s is threaded by J/4> lines from a magnetic shell of strength 3>,E.dt- d(3f3> 4- LC) = rC dt and, in the general case, JIT, 3>. L, and C are all functions of the time. 377. If fji is the mass of the slider AB in Fig. 69, and if a constant force A" be applied to AB towards the right, if DG = I, GB = x, r = Dp, we have H . D t i- = i C--DJ. + CHI + X, E-dt-d(LC+ Hlx) = rC- dt. 478 MISCELLANEOUS PROBLEMS. Show that if we can neglect the effect of the field due to the current in DAJ3G, and if the change in r during the motion is inappreciable, C = \_E/r + X/Hl~\&-K- XI HI, v = [rX/HH* + E/HZ] (1 - *-"), where A = H*l*/nr. If X is positive, the current changes sign when HHH = fir log \_(EHl + rX)/rX~\. 378. A condenser made of two circular pieces of tin foil, each 28.58 centimetres in diameter, separated by a plate of plane glass of inductivity 6, J of a centimetre thick, is dis charged by means of a piece of non-magnetic wire 1 metre long, bent into the form of a nearly complete circle. The resistance of the circuit is 0.001 ohm, and its self-induction 1000 electromagnetic absolute units. Assuming that the farad is equivalent to 9 x 10 11 electrostatic absolute units of capacity, show that the discharge will be oscillatory with a period of about 2.3 x 10 ~ 7 seconds. The time constant, 2L / R, is 0.002 second. The amplitude would be reduced to YoVu of its initial value in about 0.014 second, and to T _ TJ ^_ Tr of this value in about 0.028 second. 379. A spherical shell of copper of small uniform thickness and of radius a is in a magnetic field of uniform intensity H. Show that the work required to withdraw it instantaneously from the field is J H*a*. [M. T.] 380. An alternating electric current C cos pt is made to flow along a straight wire of uniform circular section. Prove that the current strength at a distance T from the axis of the wire is given by the real part of Cka- J (kr) e pti / [2 ira? Ji (ka)], where a is the radius of the wire, p its specific resistance, and k = (1 - i) V2~^/ Vp. [M. T.] MISCELLANEOUS PROBLEMS. 479 381. If x is a function continuous within the extremely short time interval T, and if T represents any instant during the interval. ( x dt is small and I dt \ x- dt much smaller. JQ Jo */o A wire circuit of conductivity C and self-inductance L is situated in the field of a magnetic system which undergoes a disturbance of impulsive character such that at the end it has returned to its initial state. Prove that if M denote the change in the induction through the circuit due to the field at any instant during the disturbance, and if N is the time inte gral of M taken throughout the whole time of disturbance, then the induced current at any subsequent time tisNe~ t/CL -/ CL 2 . [M. T.] 382. Show that if the branches p, q, r, and 5 of the Wheat- stone net have self-inductances L p , L qJ L r , L s , and contain con densers which have capacities k p , k q , k r , k s , and if the current C in the main circuit is a given function of the time, the currents in the other branches are to be found from the equations + C r /k r - L s DfC. - s D t C, - C s /k s -g.D t C g = 0. (1) Prove that if C x is the current in any branch, x, when C = F(t) t and C x " the current in the same branch when C =/(*), C x + C x " will be the current when C = F(t) +/(*) (2) Show that if C x + i C x " is the value of C x obtained from these equations when C = F(t) -f i-f(t), C x would cor respond to C = F(t) and C x " to C =/(*). (3) Show that if C = A e xt , the equations are satisfied, when the coefficients are properly determined, by the values 480 MISCELLANEOUS PROBLEMS. and that if b x = 1 4 Xk x (x + X L x ), (b p k q 4 *A) 4. + 0**A4r = kM (& A + &A) A - ( * A + &A + ?AJfc A) A = M Find 4,. (4) Show that if the condensers are all removed, if C=A- e A <, and if 4 a q ) (a r 4 a,) t sin mt), A g has the form 4 Op 4 a q + a r + a s)~ (5) If X = mi, C A (cos ra M + JVi , and C g the form (Jf + Ni) (cos m^ H- i sin mt) = (M> cos mt N- sin m^) 4- *(-3f sin m^ 4- -^- cos mf). If the condensers are all removed, and if C = A cos mt, what is the condition that no current shall pass through g ? 383. Show that if (1) the branches p, q, r, and s of the Wheatstone net have self- inductances L J} , L q , L r , L s , and are in parallel with branches of negligible resistance having capacities k p , k q , k r , k,, if (2) the current C in the main circuit (Fig. 134) is a given function of the time, and if (3) the cur rents in the branch x of the net and in the condenser circuit parallel to it are denoted by C x and CJ respectively, the currents are to be determined with the help of the equations FIG. 134. C r 4 C r + C, 4- C. 1 = C, C p C p = C r 4 MISCELLANEOUS PKOBLEMS. 481 and ! C r -s.C, -g. C g = -L r -D t C r If the results of applying the operations [1 + A- x (x . D t + L x - D, 2 )], [x + L x D J to C x be denoted by <f>(C x ), ^(C>) respectively, these equations yield the five equations which follow : Show that if C = A e**, these equations are satisfied, when the coefficients are properly determined, by the values C p = A p - e", C q = A q . <P, C r = A,. e A , C s = A s e", C g = A v - e", and that if a x = x +L x -\, b x = 1 + k,\(x +,- A), If A is real, and if X = mi, where m is real, C = A (cos mt + / sin mt), A g is of the form M -f- Ni, and C g of the form (M+ Ni) (cos mt + i sin mt) or (JIf cos mt N- sin ?w#) -f- i (M- sin ??i# + jVcos mt). The real part of C g is of the form ^/ M~ -\- N 2 cos (mt - 8), where tan 8 = N/M. Prove that this would be the value of C g if the value of C were A cos /#. Show that if C = A- cos mt, if the condensers are all removed, and if L q = L^ = 0, C g will be zero for all values of m if X^/Z r = fj/s =p/r. Prove that if C = A - cos mt, if the inductances are neg ligible, and if A*, = A*. = 0, C g will be zero for all values of m 482 MISCELLANEOUS PROBLEMS. Prove that if all the L s and & s except k p and L s are zero, C g will be zero if, and only if, L s = qrk p = psk p . Show how to find the value of C g if C = A sin mt. 384. An infinite mass of metal has one plane face, which is the yz plane. At the time zero, uniform currents parallel to the z axis are induced in the plane by a sudden change in the magnetic field, which after the change remains constant. It is evident that u and v will remain zero and that w is a function of x only, so that (Section 88) 4 ?r/xX D t w = D^w and w = A e u /Fj where u = Tr/xAx 2 / t. Show that w will have its maximum value at a distance x from the plane face at a time t = 2 ir^Xx^ and that this value is A /(x V2 Tr/uAe). When t is large, w has nearly the same value for all moderate values of x. Assuming that for copper /xA = 1/1600 and for iron 1/10, show that the maximum current will be attained at a depth of 16 centimetres in copper and 1.26 centimetres in iron after 1 second. 385. An infinite conductor has one plane face (the yz plane), but is otherwise unlimited. Periodic currents parallel to the z axis and independent in intensity of y and z are induced in this conductor by some cause on the negative side of the yz plane. Since u and v are zero, w must (Section 88) satisfy the equation 4 TT\(JL D t w =D x z w, and of this equation Ce lt e~ nx , where I = kH/^icX^ n 2 = kH, n = k(l + i)/^/2, is a special solution. The real part of a complex solution of this equation is itself a special solution ; and if we assume that w = C m cos pt, when x = 0, we learn that w = C m e~* V2n ^ p . cos (pt - x V2 7rA/xp), a simple harmonic expression the amplitude of which is C m e~ 2nxV ^ f , where /is the number of alternations per second. Show that this amplitude has I/ rath of its surface value at the depth log w/(2ir VXf/). Show that if the frequency is 100, the amplitudes 1 centimetre from the surface will be 0.208 C m in the case of copper, and 0.0000000024 C m if the MISCELLANEOUS PROBLEMS. 483 conductor is made of iron. Show also that if the frequency is 10,000 the amplitude in the case of copper at a point 1 centi metre from the surface would be 0.00000015 C n . 386. Show that if U y F, W are the components of a vector taken at every point in the direction in which the orthogonal curvilinear coordinates u, v, u- increase most rapidly, the components of the curl of the vector are A very interesting proof of Stokes s Theorem in curvilinear coordinates has been given by Prof. A. G. Webster in the Bulletin of the American Mathematical Society for 1898. 387. The whole induction flux through a linear circuit, s k , is equal to the line integral, 7 t , of the tangential component of a vector potential of the induction taken around the circuit, so that the electrokinetic energy, T, of a set of circuits 5 lf s. 2 , s 3 , , which carry currents (7 15 (7 2 , C 3 , -, is C k l k = C *l F * cos (x, 5,) + F v - cos (y, If the linear circuits are the filaments of a massive conductor in which the components of the current are it, r, w, and if S k is the area, at any place, of the cross-section of the filament s k ; C k cos (x, s k ) = u S k , C k cos (y, s k ) = v S k , C k cos (2, s t ) = w - S k . Show that we may write where the integration is to be extended over all space where currents exist. ANSWERS. ADDITIONAL ANSWERS TO PROBLEMS. CHAPTER I. d\ x 113Q32 Y 96501 7? o^ ~ 274625 " 274625 * + .0 = 40 29 +. (2) R = m\, cosa= , cos/3 = -, cosv = -, where crA tr\ cr\. (5) If 2 Z be the length of the wire, and if the axis of the wire be taken for the axis of abscissas with the origin at the middle point, the required equation is (6) If the radius of the earth be taken as 3960 miles, and the mass of a cubic foot of water as 62.5, one poundal is equiv alent to 952 million attraction units approximately. (8) If c be the distance of the point P from the centre of the sphere, the required attraction is m\ , if P is [c 2 8(cr) 2 J without the solid ; -^, if P is within the cavity, and 8r 2 m if P is a point in the mass of the solid. (9) The attraction of a hemisphere of radius R at a point P facing the flat side of the hemisphere and lying on the perpen dicular to this side erected at the centre is 484 ANSWERS. 485 M - 2 a 3 + (2 a 2 - ^ 2 ) V.ft 2 + a 2 ] , where a = OP. The attraction of the other hemisphere ma} be found by sub tracting this quantity from the attraction due to the whole sphere of which this hemisphere is a part. See 9. (10) (a) That the density varies inversely as the distance from the centre. W _^ir?+^ i_i ivJPLr 981itr 327?iJ* (12) Here d is supposed to be greater than a. (15) The attraction is 34.9, and its line of action makes an angle of 1 49 with the line joining the centre of the sphere with the point in question. CHAPTER II. (7) That the force is constant. (9) Fo.^ _ 3 jr 2 /-_ 14 .A T ^ 2 40 F 2 _ 3 = - 7J 7> 2/ --- c 2 ; Fi_.-p 3 \ c / o c (11) Yes; 1.46 -. ifr>5, F=^l2. r (15) No. (20) (1) About 1,830,000 tons of 2000 pounds each. INDEX. Activity, 316, 465. Alternate currents in inductive circuits, 312, 329, 464-482. Ampere, 265. Ampere, the, 233, 298. Apparent charges, 183. Apparent electromotive forces, 317. Attraction. centrobaric bodies, 362, 371. cones, 8, 25, 349. curved wires, 25, 345. cylinders, 7, 26, 337, 346, 347, 348, 358, 368, 369, 376, 377. cylindrical distributions, 60, 72. cylindrical shells, 26, 349. discrete particles, 2, 25. ellipsoidal homoeoids, 16. ellipsoidal surface distributions, 141, 160, 378. elliptic cylinders, 376, 377, 379. focaloids, 378. given mass, 19, 350. hemispheres, 13, 15, 26, 352, 353, 35r\ hollow spWes, 26, 27. paraboloids, 28, 353. similar solids, 350. solid ellipsoids, 117, 378. special laws of, 351, 359, 361. spheres, 13, 18, 26, 27, 350, 358. spherical distributions, 56, 72, 368, 375. spherical sectors, 351. spherical segments, 27, 352. Attraction, spherical shells, 10, 11, 18, 26, 27, 35, 58, 350, 352. spheroids, 28, 380, 381. straight wires, 3, 4, ,25 > _2fi r _34, 71, 344, 345, 346, 359. thin plates, 22, 28, 347, 348, 349. two rigid bodies, 24, 350, 353. two spheres, 23, 337, 338, 339. two wires, 23, 73, 344. Ballistic galvanometers, 442. Bocher, 371. Capacity, 159, 161, 307, 309, 321. (See Condensers.) Centrobaric distribution, 362, 371. Charged conductors, 146, 148, 157, 167, 171, 385, 386, 387, 388, 389, 390, 393, 394, 395, 396, 397, 402, 403, 410. Coefficients of potential, induction, and capacity, 157, 390, 397. Columnar coordinates, 63, 141, 354, 421. Condensers, 161, 164, 166, 176, 184, 307, 388, 389, 390, 393, 396, 401, 402, 409, 413, 418, 419, 420, 439, 440, 441, 466, 467, 478, 479, 480. Conditions which determine func tions, 104, 107, 133, 134, 135, 136, 137, 138, 180, 203, 245, 371, 382, 415. 486 INDEX. 487 Conductivity, 227. Cones, 8, 25, 349. Conjugate functions, 412, 429, 430, 431, 432, 433. Convergence, 111, 138. Coulomb, the, 233. Coulomb s Equation, 89, 130, 179, 183. Curl, 111, 138, 139, 143, 382,383,483. Current induction, 291. Currents in cables, 428, 429, 437. Curvilinear coordinates, 65, 136, 137, 141, 182, 384, 385, 421, 483. Cylinders, 7, 26, 346, 347, 348, 358, 369, 376, 377. Cylindrical distributions, 60, 72, 368, 369, 371, 375, 404, 408, 409, 410. Darwin, 364. Depolarizing force, 206, 379. Derivatives of the potential func tion, 30, 31, 32, 36, 40, 45, 50, 72, 73, 91, 360, 361, 366. Derivatives of scalar functions, 115, 116, 138, 382. Dielectrics, 146, 176, 199, 413, 414, 415, 416, 418. Dimensions of physical quantities, 210, 338, 462. Dirichlet, 50, 104, 125. Displacement currents, 335. Dissipation function, 240. Divergence, 111, 138, 141, 382, 383, 384. Divided circuits, 235. Double layers, 144, 214. Doublets, 196, 434, 436. Effective electromotive forces, 317. Electrical displacement, 177. Electrical images, 167, 170, 400, 401, 404, 410, 411. Electrical intensity, 177. Electrodynamic potential, 273, 275. Electrodynamics, 262, 267, 271, 273, 276, 297, 454-459. Electrodynamoineter, 474. Electrokinematic equilibrium, 222, 241, 245, 246. Electrokinematics, 222, 246. Electrokinetic energy, 271, 281, 296, 461, 465, 477, 483. Electrokinetic momentum, 297,465. Electromagnetic fields due to closed linear circuits, 259, 454, 455, 456-459. Electromagnetic fields due to straight currents, 251, 255, 454. Electromagnetic units, 233, 298. Electromagnetism, 251. Electromotive force, 230. Electromotive force, triangle of, 316, 319, 320, 321. Electrostatic potential functions within conductors which carry currents, 241, 246. Electrostatic units, 233. Electrostatics, 145. Ellipsoidal conductors, 160, 401, 402. Ellipsoidal homceoids, 16. Ellipsoidal shells, 16, 378. Ellipsoidal surface distributions, 141, 378. Ellipsoids, 117, 378, 380. Elliptic cylinders, 376, 377, 379. Energy, 43, 97, 183, 269, 280, 364, 368, 391, 401, 418, 453, 454. Energy of charged conductors, 171, 175, 183, 391, 394, 401, 418. 488 INDEX. Equations of the electromagnetic field, 332. Equilibrium of fluids, 70, 73, 74, 365, 381. Equipotential surfaces, 37, 71, 72, 73, 122, 141, 357, 369, 371, 372, 403. Ewing, 291. Farad, the, 233. Faraday s disc, 272, 298. Faraday tubes, 152. Field components, 21, 30, 177, 255, 265, 283, 332. Fleming, 291. Flow of force, 151, 365. Focaloids, 378. Galvanometers, 455, 456, 457, 458. Gauss s Theorem, 52, 66, 78, 129, 151, 177. Gibbs, 221. Gradients, 115, 137, 138, 384, 385, 421, 483. Gravitation, 1. Gravitation constant, 2, 370. Gravity, 15, 342, 343, 344, 347, 370. Gray, 269. Green s distribution, 109, 446. Green s Function, 384. Green s Theorem, 91, 129, 384. Hard and soft media, 202, 204, 208. Harmonic Functions, 45, 100, 103, 104, 105, 135, 137, 143, 382, 415. Heat developed in circuits which carry currents, 238, 272, 297. Heaviside, 221, 282, 402. Helmholtz, 221. Hemispheres, 13,15, 26, 352, 353, 358. Hilbert, 105. Hollow conductors, 152. Hysteresis, 289, 460. Impedance, 315, 318, 465, 476. Induced charges, 146, 153, 156, 167, 184, 201, 394, 395. Induced currents, 292, 298, 299, 302, 326, 463, 464. Induced electromotive force, 292, 295, 297, 302. Induced polarization, 203, 204, 207. Inductance, 278, 458, 461, 462, 476. Induction, 146, 152. Induction coils, 326. Induction flux, 151, 260, 292. Induction vector, 178, 201, 202. Inductivity, 176, 200. Intrinsic charges, 183. Intrinsic energy of a distribution, the, 43, 97, 183. Inversion, 397, 399, 400, 401, 411. Joule, the, 239. Kelvin, 105, 206. Kirchhoff, 234, 280. Kirchhoff s laws, 234. Laplace s Equation, 44, 65, 71, 245, 357, 359, 360. Laplace s Law, 262. Law of gravitation, 1. Law of nature, 75, 145. Linear conductors, 226, 230, 421. Linear differential equations, 302 304, 307, 309, 311, 314, 470, 471, 472, 473, 474. Lines and level surfaces of vectors, 112, 123, 140, 382. Lines and surfaces of flow, 243, 246, 247, 249. INDEX. 489 Lines and tubes of force, 55, 71, 72, 73, 122, 150, 187, 188, 251, 260, 288, 359, 367, 372, 373, 391, 394. Logarithmic potential functions, 126, 385, 406, 407. Magnetic energy, 269. Magnetic induction, 260, 287. Magnetic lines, 391. Magnetomotive force. 287. Magnets, 199, 442, 443, 444, 445, 446, 450, 451, 453. Maximum and minimum theorems, 103, 135, 136, 240. Maxwell, 334. Maxwell s Current Equations, 281. Mechanical action on a conductor which carries a current in a magnetic field, 262, 264, 267. Motion under gravitation, 71, 338, 339, 340, 341-344. Mutual energy, 42, 269, 273, 276, 368, 395, 401. Mutual energy of distribution and field. 451, 452, 453, 454. Mutual inductance of two circuits, 276, 278. Neumann, 273. Newton, 349. Non-homogeneous conductors, 244, 245. Normal force, 89. Ohm, the, 233, 298. Ohm s Law, 227. Paraboloids, 28, 353. Pendulums, 26, 342. Permeability. 176. Perry, 469, 472. Planetary motion, 341. Poisson, 334, 446. Poisson s Equation, 61, 66, 79, 129, 147,178,182,201,202,360,421. Poisson s Integrals, 102, 132. Polarization, 185, 192, 198. Polarization moments, 186, 193. Polarization vector, 186, 194. Polarized cylinders, 448, 449. Polarized ellipsoid, 189, 207. Polarized shells, 214, 450. Polarized spheres, 187, 447. Potential difference, 231, 318, 422, 466. Potential function as measure of work and energy, 41, 78. average value on spherical sur face, 67. definition, 29, 354. derivatives of, 30, 31, 32, 36, 40, 44, 45, 50, 61, 72, 73, 89, 91, 130, 179, 183,360, 361,365, 366. properties and characteristics of, 32, 40, 44, 67, 68, 78, 80, 86, 107, 179. special cases, 34, 35, 36, 58, 60, 71, 72, 74, 80, 82, 125, 197, 355-365, 375. Poynting, 371. Pupin, 467, 476. Rayleigh, 307. Eeactance, 315. Real charges, 183. Reluctance, 287. Repelling matter, 75, 70. Resistance, 227, 247, 249, 422, 426, 433. Resonance, 323, 467. Ring magnets, 286. 490 INDEX. Self-inductance, 278, 296, 300-331. Solenoidal and lamellar vectors, 111, 138, 139, 140, 143, 144, 221, 382, 383. Solenoidal polarization, 198, 203, 204, 449, 450. Solenoids, 284. Solid angles, 11, 49, 53, 215, 261, 349. Space derivatives of scalar func tions, 115, 138, 382. Specific inductive capacity, 176. Spheres, 13, 18, 23, 26, 27, 350, 358. Spherical condensers, 161, 184, 389, 390, 413, 419. Spherical conductors, 159, 161, 167, 394, 396, 401-403. Spherical coordinates, 63, 384, 421. Spherical distributions, 56, 72, 184, 210, 368, 372, 375. Spherical segments, 27, 352. Spherical shells, 10, 11, 18, 27, 35, 58, 350, 360, 375. Spheroids, 28, 144, 370, 379, 380. Steady currents in linear circuits, 222-241, 421-429, 441, 454-459. Steinmetz, 475. Stokes s flux function, 367. Stokes s Theorem, 113, 219, 252, 282, 295, 332, 383, 483. Strength of field, 2, 147. Superficial induced currents, 299, 479, 482. Surface distributions, 83, 85, 88, 109, 146-176, 385-420. Surface pressure, 90. Susceptibility, 200, 281. Theorems involving surface and volume integrals, 47, 54, 66, 93, 94, 95, 97, 98, 100, 101, 102, 103, 104, 113, 132, 135, 136, 137, 144, 220, 356, 357, 384, 414, 415, 452, 453, 454, 460. Thomson, J. J., 299. Thomson s Theorem, 104. Tide-generating forces, 363, 364. Transformers, 331, 474. Triangle of resistances, 316. Two-fluid theory, 145. Uniform polarization, 186, 188. Uniformly polarized distributions, 186, 189, 205, 207. Units, 233, 298, 462. Units of force, 2, 25, 31, 80, 210, 337, 338, 462. Variable currents, 423, 437, 439, 440, 441. Variable currents in inductive cir cuits, 301, 326. Vector lines and surfaces, 112, 123. 140, 382. Vector potential functions, 112, 139, 140, 218, 294, 452, 453. Vector product, 293. Vectors, 14, 111, 139. Volta s Law of Tensions, 229. Volt, the, 233, 298. Webster, 206, 265, 483. Wheatstone s net, 236, 241, 305, 306, 311, 427, 479, 480. Wires or rods, 3, 4, 23, 25, 26, 34, 71, 73, 344, 345, 346, 359. Woodward, 370. Work, 41, 78, 354, 355, 401. Zonal harmonics, 261, 373, 374, 375. ADVERTISEMENTS Text-BooKs on Mathematics FOR HIGHER SCHOOLS AND COLLEGES ^ M>n?tig price price Anderegg and Roe s Trigonometry 3-75 $0.80 Bailey and Woods Plane and Solid Analytic Geometry 2.00 2.15 Baker s Elements of Solid Geometry 80 .90 Beman and Smith s Higher Arithmetic 80 .90 Beman and Smith s Elements of Algebra 1.12 1.22 Beman and Smith s Academic Algebra 1.12 1.25 Beman and Smith s New Plane and Solid Geometry 1.25 1.35 Beman and Smith s New Plane Geometry 75 .85 Beman and Smith s New Solid Geometry 75 .80 Beman and Smith s Famous Problems of Elementary Geometry 50 .5^ Byerly s Differential Calculus 2.00 2.15 Byerly s Integral Calculus 2.00 2.15 Byerly s Fourier s Series 3.00 3.15 Byerly s Problems in Differential Calculus .75 .80 Carhart s Field- Book retail, $2.50 Carhart s Plane Surveying 1.80 2.00 Faunce s Descriptive Geometry 1.25 1.35 Hall s Mensuration 50 .55 Hanus Determinants 1.80 1.90 Hardy s Elements of Quaternions 2.00 2.15 Hardy s Analytic Geometry 1.50 1.60 Hardy s Elements of the Calculus 1.50 1.60 Hill s Geometry for Beginners i.oo i.io Hill s Lessons in Geometry 70 .75 Hyde s Directional Calculus 2.00 2.15 Manning s Non-Euclidean Geometry 75 .80 Osborne s Differential Equations 50 .60 Peirce s (B. 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(7 Tables) .50 .55 Wentworth and Hill s Five-Place Log. and Trig. Tables. (Complete) i.oo i.io Wheeler s Plane and Spherical Trigonometry with Peirce s Tables., i.oo i.io GINN 6 COMPANY Publishers Boston New Yorh Chicago San Francisco Atlanta Dallas Columbus London TEXT-BOOKS ON HIGHER MATHEMATICS BY ARTHUR SHERBURNE HARDY, Recently Professor of Mathematics in Dartmouth College. Hardy s Analytic Geometry. 8vo. Cloth. 239 pages. For intro duction, $1.50. THIS work is designed for the student, not for the teacher. Particular attention has been given to those fundamental concep tions and processes which, in the author s experience, have been found to be sources of difficulty to the student in acquiring a grasp of the subject as a method of research. The limits of the work are fixed by the time usually devoted to Analytic Geometry in our college courses by those who are not to make a special study in mathematics. Hardy s Elements of Quaternions. Crown 8vo. Cloth. 234 pages. For introduction, $2.00. THE chief aim in this book has been to meet the wants of beginners in the class-room, and it is believed that this work will be found superior in fitness for beginners in practical compass, in explanations and applications, and in adaptation to the methods of instruction common in this country. Hardy s Elements of the Calculus. 8vo. Cloth. 239 pages. For introduction, $1.50. Part I., Differential Calculus, occupies 166 pages. Part II., Integral Calculus, 73 pages. THIS text-book is based upon the method of rates. The object of the Differential Calculus is the measurement and comparison of rates of change when the change is not uniform. Whether a quantity is or is not changing uniformly, however, its rate at any instant is determined essentially in the same manner, viz.: by letting it change at the rate it had at the instant in question and observing what this change is. It is this change which the Cal culus enables us to determine, however complicated the law of variation may be. From the author s experience in presenting the Calculus to beginners, the method of rates gives the student a more intelligent, that is, a less mechanical, grasp of the problems within its scope than any other. GINN & COMPANY, Publishers, Boston. New York. Chicago. Atlanta. Dallas. ELEMENTS OF THE Differential and Integral Calculus WITH EXAMPLES A1H) APPLICATIONS. REVISED EDITION, ENLARGED AND ENTIRELY REWRITTEN By JAMES M. TAYLOR, Professor of Mathematics in Colgate University. 8vo. Cloth. 269 pages. For introduction, $2.00. H IS text-book aims to present clearly, scientifically, and in their true relations the three common methods in the Calculus. The concept of Rates gives a clear idea and statement of the problems of the Calculus ; the principles of Limits afford general solutions of these problems ; and the laws of Infinitesimals greatly abridge these solutions. The Method of Rates is so generalized and simplified that it does not involve "the foreign element of time," and affords the simplest and briefest proofs of first principles. By proving // (Ay / A^-) = dy / dx, the problem of rates is reduced to one of limits or infinitesimals. The Theory of Infinitesimals is viewed simply as that part of the theory of limits which deals with variables having zero as their common limit. The important distinction between infinitesimals and zero and that between infinites and a/o are emphasized. Taylor s Theorem is accurately stated and proved. The method of finding Asymptotes illustrates the meaning of impossible and defective systems of equations. The applications of Double and Triple Integration clearly set forth the meanings of these operations. A Chapter on Differential Equations is added to meet an increasing demand for a short course in this subject. A Table of Integrals, for convenience of reference, is appended. Throughout the work the numerous problems so set forth and illustrate the highly practical nature of the Calculus as to awaken in the reader a lively interest. Those who prefer to study the Calculus by the Method of Limits alone can omit the few demonstrations which involve rates and take in their stead those by limits or infinitesimals. QlNN & COMPANY, Publishers, Boston. New York. Chicago. Atlanta. Dallas. PLANE AND SOLID Analytic Geometry By FREDERICK H. BAILEY, A.M. (Harvard), and FREDERICK S. WOODS, Ph.D. (Gottingen), Assistant Professors of Mathematics in Massachusetts Institute of Technology. 8vo. Cloth. 371 pages. For introduction, $2.00. TT HIS book is intended for students beginning the study ^ of analytic geometry, primarily for students in colleges and technical schools. While the subject-matter has been confined to that properly belonging to a first course, the treatment of all subjects discussed has been complete and rigorous. More space than is usual in text-books has been devoted to the more general forms of the equations of the first and the second degrees. The equations of the conic sections have been derived from a single definition, and after the simplest types of these equations have been deduced, the student is taught by the method of translation of the origin to handle any equation of the second degree in which the x y term does not appear. In particular, the equations of the tangent, the normal, and the polar have been determined for such an equation. Only later is the general equation of the second degree fully discussed. In the solid geometry, besides the plane and the straight line, the cylinders and the surfaces of revolution have been noticed, and all the quadric surfaces have been studied from their simplest equations. This study includes the treatment of tangent, polar, and diametral planes, conjugate diameters, circular sections, and rectilinear generators. Throughout the work no use is made of determinants or calculus. __^^__ GINN & COMPANY, PUBLISHERS, BOSTON. NEW YORK. CHICAGO. WENTWORTH S New Plane and Spherical Trigonometry, Surveying, and Navigation By GEORGE A. WENTWORTH Half morocco. 412 pages. For introduction, $1.20 IN this book the principles have been unfolded with the utmost brevity consistent with simplicity and clearness, and interesting problems have been introduced with a view to awaken a real love for the study. Much time and labor have been spent in devising the simplest proofs for the propo sitions, and in exhibiting the best methods of arranging the logarithmic work. Answers are included. The special features of the New Plane Trigonometry are sufficient practice in the use of the radian as the unit of angular measure, the solution of simple trigonometric equations, the solution of right triangles without logarithms, a brief treatment of anti-trigonometric functions, and a chapter on the development of functions of angles in infinite series. It also contains the latest entrance examination papers of some of the leading colleges and scientific schools, and a large number of miscellaneous problems in trigonom etry and goniometry. Teachers can omit, at their discretion, the chapter on construction of tables, and many of the miscellaneous problems in trigonometry and goniometry. The Spherical Trigonometry, Surveying, and Navigation has been entirely rewritten, and such changes made as the most recent data and methods required. GINN 8< COMPANY, Publishers Boston New York Chicago San Francisco Atlanta Dallas Columbus London PLANE AND SOLID Analytic Geometry By FREDERICK H. BAILEY, A.M. (Harvard), and FREDERICK S. WOODS, Ph.D. (Gottingen), Assistant Professors of Mathematics in Massachusetts Institute of Technology. Svo. Cloth. 371 pages. For introduction, $2.00. JT HIS book is intended for students beginning the study ^ of analytic geometry, primarily for students in colleges and technical schools. While the subject-matter has been confined to that properly belonging to a first course, the treatment of all subjects discussed has been complete and rigorous. More space than is usual in text-books has been devoted to the more general forms of the equations of the first and the second degrees. The equations of the conic sections have been derived from a single definition, and after the simplest types of these equations have been deduced, the student is taught by the method of translation of the origin to handle any equation of the second degree in which the x y term does not appear. In particular, the equations of the tangent, the normal, and the polar have been determined for such an equation. Only later is the general equation of the second degree fully discussed. In the solid geometry, besides the plane and the straight line, the cylinders and the surfaces of revolution have been noticed, and all the quadric surfaces have been studied from their simplest equations. This study includes the treatment of tangent, polar, and diametral planes, conjugate diameters, circular sections, and rectilinear generators. Throughout the work no use is made of determinants or calculus. GINN & COMPANY, PUBLISHERS, BOSTON. NEW YORK. CHICAGO. WENTWORTH S New Plane and Spherical Trigonometry, Surveying, and Navigation By GEORGE A. WENTWORTH Half morocco. 412 pages. For introduction, $1.20 IN this book the principles have been unfolded with the utmost brevity consistent with simplicity and clearness, and interesting problems have been introduced with a view to awaken a real love for the study. Much time and labor have been spent in devising the simplest proofs for the propo sitions, and in exhibiting the best methods of arranging the logarithmic work. Answers are included. The special features of the New Plane Trigonometry are sufficient practice in the use of the radian as the unit of angular measure, the solution of simple trigonometric equations, the solution of right triangles without logarithms, a brief treatment of anti-trigonometric functions, and a chapter on the development of functions of angles in infinite series. It also contains the latest entrance examination papers of some of the leading colleges and scientific schools, and a large number of miscellaneous problems in trigonom etry and goniometry. Teachers can omit, at their discretion, the chapter on construction of tables, and many of the miscellaneous problems in trigonometry and goniometry. The Spherical Trigonometry, Surveying, and Navigation has been entirely rewritten, and such changes made as the most recent data and methods required. GINN Sc COMPANY, Publishers Boston New York Chicago San Francisco Atlanta Dallas Columbus London WENTWORTH S GEOMETRY REVISED. BY GEORGE A. WENTWORTH. Wentworth s Plane and Solid Geometry. Revised. 473 pages. Illustrated. For introduction, $1.25. Wentworth s Plane Geometry. Revised. 256 pages. Illustrated. For introduction, 75 cents. Wentworth s Solid Geometry. Revised. 229 pages. Illustrated. For introduction, 75 cents. THE history of Wentworth s Geometry is a study in evolution. It was the corner stone on which was built a now famous mathematical series. Its arrangement and plan have always appealed to the eager student as well as to the careful teacher. It was the first to advocate the doing of original exercises by the pupils to give them inde pendence and clear thinking. As occasion has offered, Professor Wentworth has revised the book in the constant endeavor to improve it and move it a little nearer the ideal. The present edition is a close approach to this end. It represents the consensus of opinion of the leading mathe matical teachers of the country. It stands for exact schol arship, great thoroughness, and the highest utility to both the student and the teacher. In this new edition different kinds of lines are used in the figures, to indicate given, resulting, and auxiliary lines. These render the figures much clearer. In the Solid Geometry finely engraved woodcuts of actual solids have been inserted, for the purpose of aiding the pupil in visu alization. They give just the necessary assistance. The treatment of the Theory of Limits is believed to be the best presentation of the subject in any elementary geometry. GINN & COMPANY, Publishers, Boston. New York. Chicago. San Francisco. Atlanta. Dallas. Columbus. London. 7 DAY USE RETURN TO ASTRON-MATH-STAT. LIBRARY Tel. No. 642-3381 This publication is due before Library closes on the LAST DATE and HOUR stamped below. 31 inyr: ^* rtJTtr iED 14. vjrr Uue end of CPRIN(yqterter Subject to recall a$r ^28 1983 RB17-5m-2 75 (S4013slO)4187 A-32 General Library University of California Berkeley MATH- STAf. UBtAflf