GIFT OF MICHAEL REESE THE THEORY OF ELECTRICITY AND MAGNETISM BEING LECTURES ON MATHEMATICAL PHYSICS BY ARTHUR GORDON WEBSTER, A.B. (HARV.), PH.D. (BEROL.), 'i ASSISTANT-PROFESSOR OF PHYSICS, DIRECTOR OF THE PHYSICAL LABORATORY, CLARK UNIVERSITY, WORCESTER, MASSACHUSETTS. ILonfcon MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY 1897 [All rights reserved.] PBINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE. SOME justification is perhaps necessary for the appearance of another treatise on Electricity and Magnetism, in view of the numerous ones already existing in English. This book is the result of a demand encountered in my own experience in teaching, and is based upon various courses of lectures that I have delivered at Clark University during the last six years. The classical treatise of Maxwell, which must always remain as a point of departure for the modern treatment of the subject, is ill adapted to the purpose of a text-book. To ask a student to attempt to assimilate the contents of the two volumes of Maxwell in a year, or even in two years, is only to expose him to the severest pangs of mental indigestion. Again, Maxwell's own views are there presented by him with not the greatest clearness, while severe demands are made upon the student's mathematical attainments. The excel- lent treatises of Mascart and Joubert and of Watson and Burbury follow Maxwell with considerable closeness. Professor Gray's admirable treatise, though containing much recent matter, suffers under the disadvantage of being in three volumes, while the very convenient little book of Mr Emtage is somewhat restricted in scope. Professor J. J. Thomson's altogether delightful Elements of the Mathematical Theory, which appeared when the present w. E. b VI PEE FACE. book was nearly ready for the press, while extremely modern as well as clear, is addressed to a somewhat different class of students from that contemplated in writing the present book. The theoretical writings of Hertz, Heaviside, Cohn and others have resulted in the systematization of Maxwell's theory and have made possible improvements in the mode of its presentation and nomenclature not contemplated by him. The extremely important and original contributions of Mr Oliver Heaviside are unfortunately but little adapted to the use of the student on account of their very voluminous character as a whole, as well as of an extreme conciseness of expression in individual parts. The few brilliant chapters on theoretical matters left by Hertz are hardly by way of exposition, but rather of a summing up of the conclusions of the theory. It has been my aim in the preparation of this volume to present to the student the results of the theory as it stands to- day after the labors of Faraday, Maxwell, Helmholtz, Hertz and Heaviside. Here it may be convenient to state what I consider to be the essentials of Maxwell's theory as distinguished from the old theories. To this question may very well be made the answer of Hertz : " Maxwell's theory is Maxwell's system of equa- tions." But to specify more fully the points of difference, they are in the opinion of the writer : 1°. The localization of the energy in the medium. 2°. The magnetic action of displacement currents. While starting from the standpoint of Energy, I have not thought it advisable to abolish the usual terms repugnant to so many writers, who assuming the attitude of Maxwellians par excellence, deny the existence of Electricity. Maxwell himself was not one of these. Feeling that the consideration of the Newtonian Potential Function is indispensable, not only for the old theory of action at a distance, but for the modern theory, and in addition that it introduces the student to many of the methods that he will need in various branches of mathematical physics, I PREFACE. vii have prefixed to the treatment of Electricity a rather complete treatment of the Potential by itself, including the properties of polarized distributions. It has been the custom of English writers to include chapters on the Potential in works on Analytical Statics, as in the cases of the admirable treatises of Routh and Minchin. It will probably be admitted, however, that the inclusion of this subject in a treatise mainly devoted to the consideration of rods, strings, and billiard balls is no more appropriate than in one devoted to Electricity and is less likely to attract the student of the latter. It is unfortunately the case that graduates of our American colleges are as a rule insufficiently prepared in the departments of mathematics necessary in approaching the subject of mathe- matical physics. In fact, I know of but three text-books on the Calculus in English, those of Greenhill, Williamson and Byerly, that give a treatment of Green's Theorem. I have therefore considered it expedient to prefix a mathematical introduction giving a short treatment of the important subjects of Definite Integrals and of the Theory of Functions of a Complex Variable, indispensable to a study of the Potential Function. For the same reason, I have included a treatment of the fundamental principles of Mechanics ab initio, including the deduction of the Principle of Energy, Hamilton's Principle, and Lagrange's Equations of Motion. I have followed the example of Boltzmann in making the deduction of the equations of the Electromagnetic Field depend on Hamilton's Principle by means of the properties of Helmholtz's Cyclic Systems, the treatment of which is here added. These chapters are ex- tracted from my lectures on Dynamics. In this manner it has come about that the book is nearly half finished before the word electricity is mentioned. This may be objectionable to some persons, but I consider it of great importance that the student should be well supplied with tools and practised in their use before he is called upon to use them on a new and unfamiliar subject. The physical difficulties connected with electricity are Vlll PREFACE. great enough without being mixed up with mathematical ones. It is also a pity to have the student get the idea that certain theorems pertain to electricity, when they really are simply matters of geometry or analysis. I have whenever possible at- tempted to bring out the geometrical or physical nature of the processes involved before coming to the electrical application. Thus these introductory chapters may serve as a sort of general introduction to Mathematical Physics. After a treatment of the problem of Electrostatics in a single medium by means of the Principle of Virtual Work the usual methods of attacking electrostatic problems are treated. These chapters pertain to either the new or the old theory. I have then inserted the chapter on Electrokinetics, somewhat out of its natural order, in order to bring out the geometrical ideas involved in the so-called Law of Ohm. Of these application is made in the treatment of Dielectrics and Magnetizable Bodies, which is carried out in such a manner as to show the close parallelism between the two classes of phenomena there treated, a point not always insisted on by Maxwell, but clearly brought out by Hertz and Heaviside. On account of this the symmetrical notation of Hertz is adopted in preference to that of Maxwell. I have however kept the term induction used by Maxwell for magnetism alone, instead of the term polarization used by Hertz, which I have used in the more usual sense of moment of unit volume. I regret not having been able to respond to the appeal made by Boltzmann to future writers to follow Maxwell's notation. I feel that it is more important to have a good notation than a familiar one, and that it is a first essential of a good notation that it should be symmetrical. The indiscriminate mixture of Greek, Roman and German letters used by Maxwell is as u fortunate as the dissymmetry with respect to electrical and magnetic phenomena. It is hardly necessary to say that vector methods have been used throughout, although the abbreviated notation of Hamilton PREFACE. IX and Heaviside has been little used. It is easy to lay so much stress on symbolism that the student loses sight of the real sim- plicity of the method. In order, however, to show its extreme utility, particularly in connection with the operator V, the essen- tials of the quaternion notation have been explained in the first chapter. As the aim of the book has been to present an introduction to the mathematical theory of electricity, little or no reference has been made to experimental methods — in fact it seems that such subjects as standard cells, dynamos, or galvanometers should be treated in a separate work, and I have no desire to add to the large number of such already existing. At the same time it is hoped that the principles involved in the various modes of mea- surement are all herein contained. The figures with which the book is illustrated, while but a few of them are new, have in no case been copied from existing figures, but have been, if necessary, recalculated, and in every case redrawn on a large scale and photographed down to the required size. For the amount of labor here involved I am under great obligations to Messrs W. P. Boynton and T. W. Edmondson, fellows of Clark University, who have undertaken the whole matter. As the proof has been read only by the author, it is probable that a certain number of errors have crept in, which it is hoped may be excused. In conclusion my aim has been to present a brief, connected treatise embodying the essential points of the theory and suitable for assimilation by the student in a period of time not exceeding a year. To this end I have considered only the usual methods of treating the various subjects, and included enough examples to illustrate their working, and no more. If it be considered that unnecessary matter has been included it may be replied that this may easily be omitted, and that it is safer to include too much than to make unwarrantable assumptions regarding the know- ledge possessed by the student. If the book shall succeed in X PREFACE. clearing up some of the difficulties generally encountered by the student and in inducing him to read the classical writings of Maxwell, Helmholtz, Hertz and Heaviside the object of its author will have been achieved. A. G. WEBSTER. WORCESTER, MASS., Dec. 23, 1896. UNIVERSITY LIST OF WORKS CONSULTED BY THE AUTHOR Maxwell, Treatise on Electricity and Magnetism. Green, Mathematical Papers. Heaviside, Electrical Papers. Thomson, Mathematical and Physical Papers and Reprint of Papers on Electrostatics and Magnetism. Thomson, J. J., Recent Researches in Electricity and Magnetism. Thomson, J. J., Elements of the Mathematical Theory of Electricity and Magnetism. Gray, Absolute Measurements in Electricity and Magnetism. Faraday, Experimental Researches in Electricity. Kirchhoff, Vorlesungen iiber Elektricitdt und Magnetismus. Helmholtz, Wissenschaftliche Abhandlungen. Kirchhoff, Gesammelte Abhandlungen. Hertz, Gesammelte Abhandlungen. Gauss, Gesammelte Werke. Riemann, Schwere, Elektricitdt und Magnetismus. Boltzmann, Vorlesungen iiber Maxwell's Theorie der Elektricitdt und des Lichtes. Drude, Physik des Aethers auf elektromagnetischer Grundlage. Winkelmann, Handbuch der Physik. Mascart et Joubert, Lemons sur V filectricite et le Magnetisme. Mathieu, Theorie du Potentiel et ses Applications a V filectrostatique et au Magnetisme. Duhem, Lemons sur V JZlectricite et le Magnetisme. Poincare, Mectricite et Optique. Poincare, Les Oscillations electriques. Betti, Teorica delle Forze Newtoniane. Ewing, Magnetic Induction in Iron and other Metals. du Bois, Magnetische Kreise, deren Theorie und Anwendungen. Thomson and Tait, Treatise on Natural Philosophy. Routh, Analytical Statics. Xll LIST OF WORKS CONSULTED BY THE AUTHOR. Minchin, Treatise on Statics. Peirce, The Newtonian Potential Function. Kirchhoff, Vorlesungen uber Mathematische Physik — Mechanik. Somoff, Theoretische Mechanik. F. Neumann, Vorlesungen uber die Theorie des Potentials und der Kugelfunctionen. C, Neumann, Untersuchungen uber das logarithmische und das Newton'sche Potential. Harnack, Die Grundlagen der Theorie des logarithmischen Potentiates. Bacharach, Abriss der Geschichte der Potentialtheorie. Jacobi, Vorlesungen uber Dynamik. Appell, Traite de Mecanique Rationnelle. Kronecker, Vorlesungen iiber die Theorie der einfachen und vielfachen Integrate. Riemann, Partielle Differentialgleichungen. Lame, Lemons sur les Coordonnees Curvilignes et leurs diverses Appli- cations. Darboux, Theorie des Surfaces. Picard, Traite d1 Analyse. Jordan, Cours d 'Analyse. Laurent, Traite $ Analyse. ERRATUM. Page 140, lines 12 and 13, for revolution read translation. MATHEMATICAL INTRODUCTION. CHAPTER I. NUMBER. 1. Rationals and Irrationals. The primary objects of study in Arithmetic are the natural numbers or integers, 1, 2, 3,... forming an unlimited sequence. Of these any two a and b may be added together, and we find the fundamental law that a + b = b + a. This is known as the commutative law. For more than two numbers, we find that (a + b) + c = a + (b + c). This is known as the associative law. Any two numbers may be multiplied together, and we find that multiplication is subject to the commutative law, ab — ba, to the associative law, (ab) c = a (be), and in addition to the distributive law, a (b + c) = ab + ac. Defining the operation of subtraction as the inverse of addition, so that c is defined as the result of subtracting b from a if b added to c will give a, we find that the operation of subtraction is pos- sible only if a is greater than 6. We are thus led to extend our definition of numbers in such a way as to call that to which 6 must be added in order to give a, a number, in the case where a is less than b. We are thus led to the conception of the negative ^S W. E. 1 2 . , NUMBER. [INT. I. integers,, which we can v add, multiply, and subtract according to the same laws asv the natural numbers. Defining the operation of division as the inverse of multiplica- tion, so that a divided by b is c, if c multiplied by b will give a, we find that the operation can be performed only when b is a factor of a. We are thus again led to extend our definition of numbers so as to call that which, being multiplied by b, will give a, even when b is not a factor of a, a number. We are thus led to the con- ception of fractions, which may be operated upon like the positive and negative integers. Every fraction is of the form m/n where m and n are positive or negative integers. This system of numbers suffices for all the ordinary operations of arithmetic, including the solution of equations of the first degree. Any number may be raised to any power, the process being known as involution. If we define evolution, or the operation of taking a root, as the inverse of involution, so that the 6th root of a is c when cb = a, we find that the operation can be performed only when a is one of the series of numbers, c, c2, c3, ....;. If we further extend our definition of number, so that that which raised to the 6th power will give a, even in the contrary case, we are led to the conception of irrational numbers. No irrational number can be expressed as the quotient of two integers, though for any given irrational a we can always find two integers such that their quotient differs from a by an amount that is as small as we please. In symbols, if e is any given positive number as small as we please, we can always find m and n so that < e. By | a | is meant the absolute, or arithmetical value of a, irre- spective of sign. E.g., the square root of 2 is an irrational, but the rules for the extraction of square roots enable us to find a value that differs from it by as little as we please. The ordinary theory of enumeration shows that we can express any rational number in terms of any integer, 6, called the base, as the sum of a definite number of terms, each of which is the product of some integer less than b by some power of b with positive or negative exponent, or else as the limit of a sum of such terms, where as the 1, 2] NUMBER. 3 exponents of the negative powers increase regularly in absolute value, from a certain term on the coefficients are all obtained by the repetition of a certain definite group : a = anbn + an_! bn~l + ... + a 16 e.g. ~ = 3-10° + 2-10-1, o ~ = HO1 + 010° + HO-1 + 610-2 + 610-3 + o No irrational number can be so expressed, though by taking a sufficient number of terms we may obtain a number differing from the given irrational by as little as we please. The coefficients in this case never repeat indefinitely. Since irrationals can not be expressed by means of a finite number of terms each of which is rational, they ate defined by their properties, or as the limit approached by an infinite sequence of rational numbers. 2. Limits. If we have a sequence of rational numbers, ax> a2, a3, ..., following each other according to a given law, and can find a number A, possessing the property that, corresponding to any arbitrarily given positive number e however small, we can find a number //, such that for all values of n greater than /A, \an-A fj,, 7 then A is called the limit of the sequence. E.g. the sequence =1 a -1 - a -1 l - -1 - - - has the limit 2, a rational number. The necessary and sufficient condition for the existence of a limit is that when e is arbitrarily given, we can find a number /*, such that for all integral values of n greater than //,, and for any positive integral value of p, If the latter condition is fulfilled, even though the sequence has no rational limit, the sequence has a limit, which defines an irrational number. 1—2 4 NUMBER. [INT. I. The sequence does not fulfil the above condition, for 1 1 1 p 1+- p which cannot be made as small as we please, for all values of p, no matter how great n be taken. On the other hand, the sequence 1 1 1 1.2' ' 1.2 ' 1.2.3' 1 1 1 1.2 ' 1.2.3.' 1.2.3.4' does satisfy the condition, for 1 ' ...... n+I 1 , _ L_ _1 7 "(w + 2) ...... (n+p)\ 1.2.3 n(n+l)*' which is less than e as soon as n is taken greater than 1/e. This sequence defines the irrational known as e, the natural logarithmic base, which is not the root of any algebraic equation with rational coefficients. The class of irrationals is in fact much larger than the class of algebraic irrationals, which led to their inclusion in the number-system. 3. Complex Numbers. The system composed of the rational and irrational, forming together the real numbers, is still not suffi- cient for the solution of algebraic equations. For consider the simple equation x2 + 1 = 0. 2 — 4] NUMBER. 5 Since even powers of all real numbers are positive, there is no real number that has a square equal to — 1. If we further extend the idea of numbers, so as to call that a number whose square is - 1, we have a means of satisfying the equation. If we denote the new number by i, defined by the equation i2 = — 1, we may multiply it by any real number, positive or negative, integral, fractional, or irrational, and thus get a class of new numbers, known as pure imaginary numbers. Evidently no imaginary number is equal to a real number, for the quotient of two real numbers is always real. If we consider the sum of a real and an imaginary number, we arrive at the conception of a complex number (in the narrow sense). Two complex numbers are equal when their real parts are equal and their imaginary parts also. Any equation containing complex numbers is accordingly equivalent to two equations con- taining only real numbers. In particular the equation a4-bi = 0, where a and b are real, is equivalent to the two, a = 0 and 6 = 0. A complex number vanishes only when its real and imaginary parts both vanish. 4. Complex Numbers in the Extended Sense. As we have formed numbers by multiplying the real and imaginary units 1 and i by all real numbers and forming sums therefrom, so we may still further extend the notion of numbers to include sums of terms each formed by multiplying any number n of different units by real numbers. Such numbers are complex numbers in the extended sense, a number involving n units 'es being an w-fold number. The units may have any properties by which we wish to define them. If they are all independent of one another, it is obvious that two complex numbers are equal only when composed of the same number of each unit es, so that any equation contain- ing all the units is equivalent to n equations containing only real numbers. In particular, a complex number a = a^ + «2e2 + anen, vanishes only when the coefficient a.8 of each unit es is zero. Two complex quantities satisfy the associative and commuta- tive laws with respect to addition, and accordingly the sum of a = OL& + OA + anen and 6 = 6 NUMBER. [INT. I. is defined as a + b = («!+ /3l}el + (a,, + &)e2 ...... + (<*n + 0n) en. .With respect to multiplication, the units are commutative with respect to real numbers : aberces = aerbces = abceres = eresabc, etc. The associative and distributive principles also hold, so that (aer) es — a (eres) and either a (er + es) = aer + ae8, or er (es + et) = eres + eret. We may accordingly define the multiplication of any two com- plex numbers ab = (otA + «2e2 ...... + ane») (&*! It will be convenient to consider a system of units of such a nature that instead of the commutative property with respect to multiplication we have eres — — eser where r and s are different, and for any r, e? = — 1. If we consider a set of three units, each possessing the above properties, and in addition the property that the product of any two taken in cyclic order is equal to the third, we have the system proposed by Hamilton, and denoted by him by the letters i, j, k. Accordingly by definition ij = —ji = k, jk = — kj = i, ki = — ik =j. Multiplying each equation by the first unit appearing in it, and observing the associative law, we have iij = fij = — iji = — (ij) i = — ki = —j, jjk = fk = -jkj = - (jk)j = -ij=-k, kki = k?i = — kik — — (ki) k — —jk = — i, necessitating £=ja = # = _!. Even powers of the three units are real, and equal even powers are equal, while odd powers of any unit are equal to real multiples of itself, and equal odd powers of different units are not equal. The product of two threefold complex numbers of this system, a and b, 4, 5] NUMBER. (ai + 13 j + yk) (vti+ffj + yk) = aaV + /3/8'j2 + yy'k* + affij + fa'ji + y@'kj + ya'ki + ay'ik = - (act + 00' -f yy') is accordingly equal to a real number plus a complex number of the same system, and this may be considered as a fourfold complex number compounded of the units 1, i,jt k. Such a fourfold number was called by Hamilton .a Quaternion. We shall in this book seldom need the fourfold number, but shall frequently use the threefold one. 5. Geometrical Representation of Numbers. The natural numbers may be represented by an unlimited series of points laid off at equal distances along a straight line. If we take a certain point to represent zero, the positive integers will lie on one side of it and the negative on the other. Points between the integer points will represent fractions and irrationals, and to every real number will correspond a point. For any rational number we may find others lying as near it as we please, and as we have already stated, for any irrational we may find rational numbers lying as near it as we please. It may be shown, however, that between any two rational numbers, however close together, there can always be found an irrational, consequently the rational numbers do not form a continuous series. It may be shown that every point on the line corresponds to either a rational or an irrational number, so that the whole series of real numbers is continuous. Quantities which, like the real numbers, require for their specification but a single given quantity, which may take any of an unlimited series of values, are said to have one degree of freedom. It is also said that there is a single infinity of such quantities. Complex quantities in the narrow sense, involving two different units, 1 and i, cannot be represented by points on a line. If however we lay off the real numbers on a straight line, we may lay off the pure imaginary numbers on a line at right angles with it through the point representing zero. The point i is to be taken at the same distance from zero on this line that the point 1 is on the other LLae. The two lines are called respectively the axes of reals and of pure imaginaries, or the axes of X and Y. Any complex number a = a + /3i may now be represented by a point in the plane whose rectangular x and y coordinates are respectively a 8 NUMBEK. [INT. I. and /3. Whatever the values of a and /3 we may always find a corresponding point, and to every point in the plane there corre- sponds a single complex number, including the real and pure imaginary numbers as particular cases. As each of the real numbers a and /3 may independently assume the value of any of the single infinity of real numbers, there is said to be a double infinity of complex numbers, or a complex number has two degrees of freedom. The distance of the representative point from the origin is called the modulus of the complex quantity and denoted by | a | = + Va2 + ft* since it includes as a particular case the absolute value of a real number. The angle that the radius vector from the origin makes with the X-axis is called the argument of the number. This representation of complex numbers in the plane was proposed by Argand and Gauss*. The threefold complex quantity a — cd + {3j + where n is any integer, however great, the function takes all values from 1 to — 1. 16 VARIABLES AND FUNCTIONS. [INT. II. If a function does not approach any limiting value for a certain value of the variable, it must be otherwise denned for such a point ; e.g. we may assign to the function defined at all points except x = 0 by the analytical expression sin - any arbitrary value x for the point # = 0. The function will then be completely de- fined. A quantity that approaches the limit zero is called an in- finitesimal. If y is a function of x defined in an interval a to b, where b is as large as we please, a number possessing the property that, when M is a given number as large as we please, |/(*)-4|M, for all values of x greater than M, is said to be the limit of y as x increases indefinitely or, briefly, as x approaches infinity. This is denoted as follows : = A ; e.g. \imex=l. tC=oo If in the above definition, we change M to a negative number whose absolute value is as great as we please, and consider all values of x less than M, we say that A is the limit as x approaches minus infinity. If in the neighbourhood of a point # = a, when M is any number as great as we please, we can find a corresponding number 8 such that for all values of h, whose absolute value is less than 8, \f(a + h)\>M, \h\ (y). x=a It may again approach a limit as y approaches a value b. If we consider the limit approached by u considered as a function of y, we shall have in general a function of x, lim w = ¥(«;). y=b If x then approaches a, we may have a limit, which is not neces- sarily the same as before, lim j lim u ) = lim (;?/) =.4, lim j lim u I = lim "9 (x) — B ; y=b I x=a 1 y=b x=a \ y-~b > x=a * Weierstrass, Abhandlungen aus der Functionenlehre, p. 97; Harkness and Morley, Theory of Functions, p. 58. .11, 12] VARIABLES AND FUNCTIONS. 19 m e.g. the function u = - has y lim j lim u I = lim (0) = 0, y = 0 ( 3 = 0 ) 2/=0 since the limit for x does not contain y, while lim j lim u \ = oo . #=0 ( 2/=0 ) A function of two or more variables is continuous at a point x = a, y = b, if for any positive value of e, however small, we can find Sj and 82 so that h, y + k)-f(x, y) | & as the coordinates of the point M. The level surfaces of qi, #s are the coordinate surfaces, and the intersections of pairs ((MaX (MS)* ( are tne coordinate lines. The tangents to the coordinate lines at M are called the coordinate axes at M. If at every point M the coordinate axes are mutually perpendicular, the system is said to be an orthogonal system. 16. Differential Parameter. The consideration of point- functions leads to the introduction of a particular sort of derivative. If F is a uniform point-function, continuous at a point M, and possessing there the value F, and at a point M' the value F', in 22 VARIABLES AND FUNCTIONS. [INT. ii. virtue of continuity, when the distance MM' is infinitesimal, V - V = A F is also. The ratio F'-F_AF MM' ' As is finite, and as M. M ' = As approaches 0, the direction of MM' being given, the limit ,. AF 9F lim -r— = -5- As=0 As 9s is defined as the derivative of V in the direction s. We may lay off on a line through M in the direction of s a length MQ = ^- and p as we give s successively all possible directions, we may find the surface that is the locus of Q. Let MN be the direction of the nor- mal to the level surface at M, and let v MP represent the derivative in that direction. Let M' and N be the inter- sections of the same neighbouring level surface, for which F= F', with MQ and MP. Then AF _ AF MN MNMM" MM' As MM' approaches zero, we have r AF 9F lim -r— = -=-, s=o &s vs MN MM' = COS Hence 9F 9F D,.n —- = — -cosPifQ, that is, the derivative in any direction at any point is equal to the ' {projection on that direction of the derivative in the direction of (the normal to the level surface at that point. Accordingly all | points Q lie on a sphere whose diameter is MP. The derivative in the direction of the normal to the level surface was called by Lame'* the first differential parameter of the function F, and since it has not only magnitude but direction, we shall call it the vector differential parameter, or where no ambiguity * G. Lam6. Legons sur les coordonnees curvilignes et leurs diverses applications. Paris, 1859, p. 6. 16] VARIABLES AND FUNCTIONS. 23 will result, simply the parameter, denoted by P or PF. The above theorem may then be stated by saying that the derivative in any direction is the projection of the vector parameter in that direction. The theorem shows that the parameter gives the direction of the fastest increase of the function V. If V is a function of a point-function q, V =/(#), its level surfaces are those of q, and ~dn ~ dq Sn~ dn' and if ±S^ = h,P^±f(q).h, where the sign -f is to be taken if V and q increase in the same, — if in opposite directions. Suppose now that V=f(q1, qz, q3 ...... ) = __ + _ + ds 9^ ds dq2 ds dqs ds and if hi, hz, ...... denote the parameters of qlt q2, ...... the above theorem gives P cos (Ps) = ^—h1 cos (hjs) + ^—h2 cos dV Now ± ^— hi is the parameter of F, considered as a function of qi, and we may call it the partial parameter Pi, and since Pi and hi have the same sign if ^— > 0, opposite signs if ^— < 0, we have in either case ^— hi cos (^s) = Pi cos (Pi s), and P cos (Ps) = Px cos (P^s) + P2 cos (P2s) + ....... This formula holds for any direction s and shows that the parameter P is the geometrical sum, or resultant, of the partial parameters, Hence we have the rule for finding the parameter of any function of several point-functions. If we know the parameters 24 VARIABLES AND FUNCTIONS. [iNT. II. AI, h2 ...... of the functions q1} g2 ...... and the partial derivatives «— , ~— , ...... we lay off the partial parameters in the directions /^, h2 ...... or their opposites, according as ^— >0, OQi or the opposite, and find the resultant of P1} P2, ....... If the functions qlt qz, ...... are three in number, and form an orthogonal system, the equation ?-?,+?,+?„ gives for the modulus, or numerical value of the parameter Examples. (1) in § 14. Let the distance of M in the given direction /V ffy from the plane be u. AF= AM = - -, where a is the angle between cos a the given direction and that of a perpendicular to the given plane. _ Aw cos a ' cos a ' If the given direction is perpendicular to the given plane P=l. Accordingly for q1 = x, q2 = y, q3 = z, the rectangular coordinates of a point, we have Px = Py = Pz = lj and for any f unction f(x, y, z) The projections of P on the coordinate axes are the partial parameters Consequently, if cos (sec), cos (sy), cos (52;) are the direction cosines of a direction 5, the derivative in that direction — = P1 cos (sec) + P2 cos (sy) -f P3 cos («c) ar ar . x ar = — cos (sec) + -T- cos (8y) + -T- cos («*). 003 07/ 02 P may be written in terms of the unit vectors i, j, k as 16] VARIABLES AND FUNCTIONS. 25 The operator i-r-+j-^-+k— which gives the vector differential ox oy oz parameter of a function, was denoted by Hamilton by V. (read Nabla). If f(xt y, z) is a homogeneous function of degree n, by Euler's Theorem /. y y ¥ nf= x^- + y~ + z^-. dx J ty 8z ' or nf= P {x cos (Px) + y cos (Py) + z cos (Pz)}. Now the ± parenthesis is the distance from the origin of the tangent plane to the level surface at x, y, z. Calling this 8, or the parameter of a homogeneous function is inversely proportional to the perpendicular from the origin to the tangent plane to the level surface. For example, if n= 1, V = ax + by + cz, x) = a, Pcos(Py) = b, Pcos(Pz) = c, P= The level surfaces are parallel planes, and the parameter is con- stant, V is proportional to the distance of the level surface from the origin. Pcos(Px)=—, P /5? 5J2 * / or ?/ s2 -5 +^ + \/ c 2 ' ^ 2 ' „ 2 For the surface, F=l, 1 /^++^ V oaa a,? a32 a familiar result of analytic geometry. VARIABLES AND FUNCTIONS. [INT. ii. Polar Coordinates. If we call the point-functions of Examples 2, 3, and 4, of § 14, r, 6, <£, we obtain the system of spherical, or polar coordinates. 6 and may be called the co-latitude and longitude. The level sur- faces of r being spheres, the normal coin- cides with r. Accordingly dr _ dr , _ FIG. 5. The level surface of 6 is a circular cone of angular opening 0, (Fig. 5), and ,Q W d6 I , I an = rau, ^- = — 7^ = - , /i9 = — . on rdu r r The level surfaces of are meridian planes through the axis of the above cones, (Fig. 6), and d the longitude, or angle made by the plane includ- ing the point M and the ^-axis and a fixed plane through that axis, we have the system of semi-polar, cylindrical, or columnar coordinates, for which we have immediately, 17—19] VARIABLES AND FUNCTIONS. 27 The parameter of a function f(z, p, CD) is the resultant of the partial parameters V . 8/ - 18/ ~ 19. Ellipsoidal Coordinates. The equation of a central quadric surface referred to its principal axes is where a1} a2, a3 may be positive or negative. If they are all negative, the surface is imaginary. 1°. Suppose one is negative, say while «! = a2, a2 = &2- Let a > b > c. The equation is^+^-^ = l. The surface is cut by the XY- plane in the ellipse —^ +^ = 1, whose semi-axes are a and 6, and o,J cr whose foci are at a distance from the center The section by the ^Z-plane is the hyperbola ___ — 1 a2 c2 ' with semi-axes a and c, and foci at distance Va2 + C2 = Vaj — as on the JT-axis. The section by the F^-plane is the hyperbola with semi-axes 6 and c and foci at a distance \/62 + c2 = Va2 — «3 on the F-axis. The surface is an hyperboloid of one sheet. 28 VARIABLES AND FUNCTIONS. [INT. II. 2°. Let two of the constants alt a2, as be negative, say The equation is /y?2 /i>2 ^-2 a*~P~&=1' The sections by the coordinate planes and their focal distances are XT sr£W3L Hyperbola ^•^ ^-^=1- Hyperbola Va' + c2 = Vai-a3, Ct (j ^A rg. YZ £- + -^ = — 1. Imaginary Ellipse V- (62 — c2) = Va2 - as. t? C The surface is an hyperboloid of two sheets. 3°. If al} aa, «3 are all positive, the sections are ellipses, and the surface is an ellipsoid. In all three cases, the squares of the focal distances of the principal sections are differences of the three constants a1} a2, a3. Accordingly if we add to the three the same number, we get a surface whose principal sections have the same foci as before, or a surface confocal with the original. Accordingly /T\ _i_ y V*/ ~2i,*~k2l. represents a quadric confocal with the ellipsoid for any real value of p. If a>b>c and p>— c2, the surface is an ellipsoid. If — c2 > p > — b2, the surface is an hyperboloid of one sheet, and if — 62 > p > — a2 an hyperboloid of two sheets. If — a2 > p, the surface is imaginary. Suppose we attempt to pass through a given point cc, y, z a quadric confocal with the ellipsoid 19] VARIABLES AND FUNCTIONS. 29 Its equation is a;2 2 i2 r~~ ' where p is to be determined. Clearing of fractions, the above is (2) (a2 + p) (62 + p) (c2 + p) - tf (V + p) (c2 + p) - f (c2 + p) (a2 + p) - z* (a2 + p) (62 + p) =f(p) = 0, a cubic in p. Putting successively p equal to x , — c2, — 62, — a2, and observing signs of /(/?), = + oo + - *? (62 - a2) (c2 - a2) The changes of sign of f(p) show that there are three real roots. Call these X, //,, v in order of magnitude. X lies in the interval X > — c2 necessary in order that the surface may be an ellipsoid, p in the interval — c2 > /JL > — 62 that it may be an hyperboloid of one sheet, and v in the interval — 62 > v > — a2 that it may be an hyper- boloid of two sheets. There pass therefore through every point in space one surface of each of the three kinds. If we call (3) the equation F=0 defines X as a function of x, y, z, and there- fore as a point-function. The normal to the surface X = const, has direction cosines proportional to ax ax ax a# ' dy ' dz Now since identically F = 0, ox oy oz ax and we have ax /d\\ dx ax 30 VARIABLES AND FUNCTIONS. [INT. II. Therefore 9X 2# / #2 z z* \ ax 8s ~ (c2 + X) F' (X) ' The parameter of the point-function X is accordingly given by 8x *_L_*_.+ f +^ _,./ /-v \\o I / o . -v \o I /To i ^\o • / n that is y(X))2 ((a2 + X)2 ' (62+X)2 ' (C2 + X)2J ' 4 Now the direction cosines of the normal to the surface X = const, are 1 8X (7) c _ , ^ \/v/ "" _ i O * /~2 i ^V \ ET/'X — Similarly for the normals to the surface //. = const., / \ x cos (n^x) = ± - COS WuV = + - cos (rinZ) — + - The angle between the normals to X and JJL is given by 19] VARIABLES AND FUNCTIONS. 31 (8) cos * (c2 + X) (c2 + /*)) V^7 (X) F' Now by subtracting from the equation a? y* z> tf~+\ + ¥^\ ~ ' the equation a* a we get +x ~ ^TM = Oi or f (9) (X - A*) a? (62 + X) (62 4- /O *2 1 = 0. Accordingly, unless X = yu, cos (WXWM) = 0 and the two normals are at right angles. Similarly for the other pairs of surfaces. Accord- ingly the three surfaces of the confocal system passing through any point cut each other at right angles. If we give the values of X, yit, v, we determine completely the ellipsoid and two hyperboloids, and hence the point of intersection x, y} z (and its seven symmetrical points in the other quadrants). Hence we may take X, fju, v for the coordinates of the point, and the family of surfaces forms an orthogonal system. X, /*, v are called the ellipsoidal or elliptic coordinates of the point. We shall proceed to find their parameters in a form not con- taining any coordinates but X, //,, v. We must find the rate of change of X as we go along the normal to the ellipsoid X = const. Since we have identically a? z, in terms of X, JJL, v. Observe that the function 19] VARIABLES AND FUNCTIONS. 33 has as roots X, /*, />, and being reduced to the common denominator has a numerator of the third degree in p. As this vanishes for p=\, p=p, p = v it can only be -(p-\)(p Hence we have the identity _ - (p - V) (p ~ /*) (p - " Multiplying this by p + a2 and then putting p = — a2 we get - (a' + X (a2 and in like manner (62 - c2) (62 - a2) (c2 + X)(c2 + /*)(c2+v) (c2-a2)(c2-62) If X, fj,, v are contained in the intervals above specified, these will all be positive, so that the point will be real. If we insert these values in Sx, we shall have Ti^ expressed in terms of X, ^ v. This is more easily accomplished as follows. Differentiating the above identity (14) according to p, -X p + a2 p-p p+tf p-v p + c1} If we put p = X, all the terms on the right except the first, being multiplied by p — X, vanish, and we have (a2 + \y T (62 + X)2 """ (c2 + X)2 ~ (a2 + X) (62 + X) (c2 + X) ' w. E. 3 VARIABLES AND FUNCTIONS. [INT. ii. The expression on the left is -^ • Hence In a similar manner we find , _„ /(a3 + .Q(68 +»>)(% FIG. 7. the normal distance between two consecutive level surfaces ql and q1 + dq1 is ^ = -?p , consequently if we take six surfaces qlt q! + qlf qz, q2 + q2, q3, the edges of the infinitesimal curvilinear rectangular parallelepiped whose edges are the intersections of the surfaces are dq1 dq2 dq^ hi h2 hs and since the edges are mutually perpendicular, the diagonal, or 19, 20] VARIABLES AND FUNCTIONS. 35 element of arc is ^2_^i2 ,^2 , ^32 = I7 + ~V +1?~' the elements of area of the surfaces qlt qz, qs are respectively and the element of volume is Examples. Rectangular coordinates #, y, z hx — hy = hz = 1, dSx = dy dz, dSy = dz dx, dSz = dx dy, dr = da dy dz. Polar coordinates r, 0, , dSr — r2 sin 0d0d(f> element of area of sphere, dSe = r sin 0drd(f> element of area of cone, dS — rdrdd element of area of plane, Cylindrical coordinates, zt p, co, hz = hp = I, Aw = -, dSz = pdpdco element of area of plane, dSp = pdcodz element of area of cylinder, dSa = dpdz element of area of meridian plane, dr = pdpdcodz. Elliptic coordinates, X, //,, v. ,„ _ _ d/mdv (/Lt —v)(fji- _ . ~ 4 V dvd\ »(v -\)(v- /A) (X - /A) (X - i/) , , ., /\ r'/y A"/\ / hyperboloid, __ 4 (a2 + !/) (62 + v) (c2 + v) (a2 + \) (62 + X) (c2 + X) - _ hyperboloid, a2 + X) (62 + X) (c2 -f X) (a2 + /*) (62 + ^) (c2 + ft) , _ d\dfj,dv(\ — ~ ^)(c2+ 3—2 CHAPTER III. DEFINITE INTEGRALS. 21. Definite Integral of a Function of one Variable. If we consider a continuous function of one real variable, the notion of its definite integral may be illustrated by means of a geometrical representation. If the function y =/(#) be represented as the ordinate of a curve of which x is the abscissa, and if between two points x = a, x — b, we place any number n — 1 of points #1, #2, ...... ofc, ...... #n-i> and in the intervals between them erect ordinates to the curve at points %lt |2 ...... so that the sum S = (a, - a)/(£) + (x, - represents the area of the rectangles constructed on the bases with the altitudes /(&)• The value of this sum depends on the form of the curve or of the function f(x\ on the choice of the points of division, x± ...... a?n, and of the points f& within the intervals. It can be shown, however, that if all the differences Bk are less than a certain value S, all the values that S can take are confined between certain limits, and if the number of intervals increases so that 8 decreases without limit while Sx + &2 ...... + &n remains always equal to b — a, that these extreme values of S approach a common limit. This limit will represent the area of the space bounded by the axis of X, the ordinates erected at the points x = a and x = 6, and the curve representing the function y =/(«)• 21, 22] DEFINITE INTEGRALS, 37 This conception may be extended to any function whether continuous or not, and the limit, if there be any, approached by the sum Urn S= Km *lf $*/(&), n = n = k=I as the number of intervals is made to increase without limit, is the definition of the definite integral of the function f(x) from a to b. It is denoted by "b f(x) dx, L f(x) is called the integrand, a and b the limits, and ab the field of integration. Evidently the letter x in the symbol may be replaced by any other without affecting the integral. If the sum has a limit the function / (x) is said to be integrable in the region from a to b. 22. Condition of Integrability. The oscillation of a func- tion in a given interval is the difference between the greatest and the least value that it assumes in that interval. It is evident from the definition of continuity that if e is a positive number as small as we please we may always find a number 8 such that in any interval less than 8 and lying in the region ab in which the function is continuous, the oscillation is less than e. Let fj ...... fn be a system of ordinates for a system of sub- division #! ...... xny and let f / ...... fw' be a different set of ordinates contained in the same intervals 81} 82 ...... 8n. Then I «./(£.) - 1 8./(£.') = 28. fc+« (/* —f"h+t)> where the /&"'s are arbitrarily chosen values of/ in the intervals pt, and where P =S2B [P»+1 (/. -/ V.) + /> 5=1 22, 23] DEFINITE INTEGRALS. 39 But in this sum P, for every s the greatest possible difference fs—f"n+k is in absolute value not greater than the oscillation D8. Therefore In like manner where [jPJs «/!>/. Then 2 Sr/^S «,'/,' -P—P, 1 1 n n and if lim 2 8S D8 = 0 for all systems of division, the limit 2 Ss/g tt=co 1 1 is the same for all systems of division. It is easy to show that if n the condition lirn 2 Bs Ds = 0 is satisfied for one mode of division, it is satisfied for all. This is then the necessary and sufficient condition that the function f(x) shall be integrable in the interval ab. 23. Properties of Definite Integrals. It results imme- diately from the definition b n f(x) dx — lim 2 &sfs> i n= 1 that if we interchange the limits a, b, since every Ss changes sign, the sign of the integral is changed. More generally (I) J a J b The arithmetical mean of a number of quantities is defined as their sum divided by their number. If f(x) is finite and in- tegrable in an interval ab, and 83 8n, 8/ §n> are two divisions of the interval, from the last equation of § 22, is v/;- 1; «,/.! s i s8D8+l s; A'. 1111 Consider n constant and let n' increase without limit. Then u, f(x) dx-2 Bsfs \£%88 D8, so that I Bsfs 40 DEFINITE INTEGRALS. [INT. III. is an approximate value of the integral whose error is less than We may put the Bs's all equal, so that Ss = — - . Then (2) -— f(x)dx = lim j-- j^ n That is, the definite integral of a function in a given interval divided by the magnitude of the interval represents the arith- metical mean of all the values of the function taken at equidistant values of the variable throughout the interval, when the number of values taken is increased indefinitely. From the definition it is evident that if/ (x) has the same sign rb throughout the interval ab, I f(x) dx has the sign of (6 — a)/(X), J a and if there is in ab a finite interval cd in which f(x) is not zero, rd then I f(x) dx is not zero. J c In particular, if the function is continuous in a whole interval ab, and the integral between every two values of x in the interval is zero, the function must be zero everywhere within the interval. If therefore two continuous functions give in every interval ab the same value of the integral, they must be equal everywhere in the interval. Suppose that the continuous function f(x) has in the interval ab a greatest value M and a least value m, the integral will have a value lying between M(b — a) and m(b — a) and we may write, J a where M > A >m. Since f(x) is continuous, it will take the value A for at least one value f of x between a and b, so that we may write (3) f/(*)e^ =/(£)(& -a), «<£<&. J a The above formula may be generalized. We have always (4) I (*/(*)*» S (*(x) are finite and inte- grable functions, and \f(x) \ lies always between M and m , (5) f J a If in the interval (x) always has the same sign, since M—f(x) and f(x) —m are positive, I* (M-fyj)(x)dx and [ (/(a?) - m) <£ (a?) das, J a J a or Jf f % (a?) ^ - f /(*) <£ (a?) cfo J a J a rb rb and I /(a?) (f> (x)dx — m I (f> (x) dx, J a J a rb have the same sign, and therefore f(x) (x) dx lies between J a rb rb rb Ml (j>(x)dx and ml (x)dx so that I f (x) (%) dx is equal to J a J a J a rb I (j> (x) dx multiplied by a factor A lying between M and m. J a If /"(a?) is continuous, there is some point f in ab for which /(f ) = A, and accordingly (6) f /(*) * (a?) ^ =/(£) f V («) (to, a < f < b. J a J a This important theorem is known as du Bois-Reymond's theorem of the mean. 24. Indefinite Integrals. Let f(x) be integrable between a and 6. The integral Cx I f(x) dx •' a is zero for x = a, and for every value of x between a and b it has a definite value. It is therefore a function of its upper limit x. Let us denote it by F(x). If x + h be another value of x in ab, rx+h rx rx+Ji F(x + h)= /(«)*-//(•)*•+;/ /(*)A>m. 42 DEFINITE INTEGRALS. [INT. III. Hence F (x) is a continuous function of x. If a is any number fx r x between a and 6, f(x)dx differs from f(x)dx only by a con- Ja Ja stant, namely the value of the integral I f(x) dx = C. The J a function F (x) + C is called the indefinite integral of /(#). Suppose that h approaches zero either positively or negatively, and let f(x) either be continuous at x, or have an ordinary discon- tinuity, i.e., by making a finite jump. Then for any positive number e however small we can find a number h^ of the same sign as h, such that for every x in the interval x, x + h^ (at most excepting x), the value of f(x) for any point differs by less than e from f(x + 0) or f(x - 0), according as h is positive or negative. Therefore the value /(£) in the expression differs from f(x ± 0) by less than e and we have lim J><« + *>-*<'>-/(. + 0) H^W)-^) rx That is, the integral I f(x) dx — F(x) is not only a finite and J a continuous function of x in the interval ab, but it has at all points where f(x) is continuous a finite and determined derivative f(x) and where f(x) has an ordinary discontinuity, though not having a determined derivative, F (x) has one on the right and left respectively equal to f(x + 0) and f(x — 0). If however f(x) has a discontinuity of the second kind, at x, the value of h as h decreases does not approach a limit and F (x) has no deriva- tive at x. The principle here proved enables us to calculate the definite integral whenever we can find a function F(x) whose derivative is /(#), for then i. The definition usually given of the definite integral, as deduced from the indefinite integral by the above formula, is unsatisfactory, 24, 25] DEFINITE INTEGRALS. 43 the true nature of the definite integral being that of the limit of a sum. 25. Infinite Integrand or Limit. The definition of the definite integral presupposed that the integrand was finite in the field of integration ab. If there should exist points in the region ab at which f(x) became infinite, the integral would in general have no meaning. In case however there is a single point c for which f(x) becomes infinite, if h^ and h2 are positive numbers rc-hi rb however small, the integrals I f(x)dx, /(#) das have a de- Ja J c+h2 finite meaning. If now as h^ and h2 approach zero independently of each other the sum £-&i cb f(x)dx+ f(x)dx, J C+hz approaches a definite finite limit, the value of that limit is what is meant by the definite integral, f /<*)*•»• J a For example, let then for x = c, f(x) becomes infinite. f6 das fc-fe> dx f6 das 7 -- \fc = lim 7 - 0- + ^lm / - \* Ja (X- C)" 7tj=0 J a . (X-c)k h2=Q J c+n, (* - C>* r (- h^-* -(a- c)*-k + (b- c)*-k - (h^~k l-k There is a limit as Aj and h2 approach zero only if 1 — k > 0. rx In like manner if the integral I f(x) dx approaches a finite J a limit when the limit x increases indefinitely, then this value defines the meaning of the definite integral ["/(*) d* J a Let, as before, r ^ Ja^ das C)k 44 DEFINITE INTEGRALS. [iNT. III. As x increases indefinitely, this approaches a finite limit only if k > 1, when dx (a — c)l~k a (x-C)k k-l 26. Differentiation of a Definite Integral. Suppose that the integrand is a function of a parameter u as well as of x. Then in the case of a function of x that is capable of representation by a curve, if we change the parameter u we change the curve, and if f(x, u) is a continuous function of u, to an infinitesimal change in u corresponds an infinitesimal change in the curve. The area rb represented by the definite integral I f(x, u) dx changes by the J a area of the narrow strip added to or included between the two curves, and we may find the ratio of this change to the given change in u. We thus get a geometrical notion of the meaning of the derivative of the integral with respect to u. Now by the definition of the derivative rb rb . f(x, u + h) dx - f(x, u) dx Cu [ /•/ v 7 -i' J a -J a •j- f(x, u) dx = lim - — r dujaj^ ' h=o h h=Q It now becomes a question whether we may change the order of taking the limits involved in the integration and in making h approach zero. If f(x, u) is a continuous function of x and u we may do this*, and since lim /(«,«+ A) -/(«,«> = a/ (*. u) h=o h du we have d f» i, fbdf(x,u) , •j- f(x, u)du= ^-V— - dx. du]aj ja du We have already considered the definite integral as a function of its upper limit, and have found, § 24, » d * So Kronecker, Theorie der einfachen und der vielfachen Integrate, p. 26, (the word gleichmassig being superfluous, vid. Harkness and Morley, Theory of Functions, § 64). For a more careful statement, see Tannery, Theorie des Fonctlons d'une Variable, § 166. 25 — 27] DEFINITE INTEGRALS. 45 In like manner ~ f dwj< If now u, v, w, are all functions of a variable £, we have for the derivative of the definite integral according to t, dF d f« ., dFdu dF dv dF dw c?w f ^ 27. Double and Multiple Integrals. Suppose we consider a continuous function of two variables, x varying from a to 6, and y varying from g to h. We may represent f(x, y) geometrically as the third coordinate z of a surface, erected perpendicular to the plane of xy. If now we subdivide the interval ab by points a y) dx dy = lim lim 2 2 (xs - x^} (yr - yr-i)fs, r> g n=oo w=oo5=l r=l We shall find by reasoning similar to that used in § 22 that the condition for the existence of a limit is that the sum s=n r=m 2 2 (sc8-x8,l)(y8-ys.l)D8r) s=l r=l rb r I J a J 46 DEFINITE INTEGRALS. [INT. III. where Dsr is the oscillation in the interval #s_1} xs, yr-\, yr, approaches the limit zero. In forming the double sum we may proceed with the summa- tion according to x first, in which case n rb (yr - yr-i) Km 2 (x§ - xs-^f(%s, r)r) = (yr - y^l f(x, rjr) dx, «=oo 1 Ja and the double limit is rh / rb \ ( f(a>,y)dx\dy. J g \J a / Or we may sum first with respect to y, in which case ra f rh (xs - a?M) lim S (yr - yr-i)/(&, Vr) = (#, - a?*-i) ?Jl=00 1 J gf and the double limit is /„*(/*/<«.?)%)<**• But we have always w I'n 2 (yr - 2/r-i) ^ (a?, - ^ 1 U n fm = 2 (X8 - Xs^) \ S (yr-2/ however small #s — ^-x and yr — yr-\> Accordingly, rh f rb 1 rb f rh ) rb rh j I J /(#> y) ^4 ^=Ja | J /fa ^ ^} ^= Ja J (In writing a double or multiple integral we shall write the integral signs with their limits in the same order as the differ- entials.) We might now deduce theorems for the double integral similar to those that we have already deduced for the single integral. In particular, the independence of the limit on the mode of sub- division, and the theorem of the mean may be demonstrated, and the extension of the definition made when the integrand or the limits become infinite. The definition of an integral may be extended to triple and multiple integrals in an obvious manner. 28. General Definition of Definite Integral. We have in the preceding definition of a double integral assumed that the limits of integration with respect to x and y were independent. 27, 28] DEFINITE INTEGRALS. 47 If instead of a rectangle in the XF-plane we should take any closed curve, given by an equation (x, y) = 0, we could in like manner divide its area into rectangles, erecting perpendiculars in each, and define the definite integral as the limit of a similar sum for the new field of integration. More generally, let f(M) be a function of a point M, moving either in a plane or in space. If we divide any area S in the plane or any volume r in space up into a number of parts, take the value of/(.M)at any point within each of those parts, multiply each value by the area or volume of the part in which it is taken, and add together for all the parts into which the area or volume is divided, the limit approached by this sum as the number of parts increases without limit in such a way that each dimension of every part approaches zero, if such a limit exists, is called the definite integral of f(M) through the region in question. We may write the integrals fjf(M)dS or jfjf(M)dr, respectively. In each case, the field of integration must be ex- pressly specified. It may be easily shown that this definition is equivalent to the preceding. A particular mode of subdivision is by drawing level lines or surfaces for two or three orthogonal coordinates qlt q2, q3. We have then, (§ 20), 7 « _ rfgi dq3 , dq,. dq2 dq3 U>O — — — - O-T = - ; - - ; - . Suppose that in two different sets of coordinates q^qz, and plypa,pa with parameters, h1}hz,h3) and when taken through any equal finite portions of a volume r. Then when we consider the meaning of the definite integral and its independence of the manner of subdivision, we see as in (§ 23) that the above integrals, being respectively equal to JJJ hlhzh3f(ql) q2> qs) dr and Jj J g^gz $ (pl} p2, p3) dr, 48 DEFINITE INTEGRALS. [INT. III. can be equal only if the point functions hlh.2hsf(q1) q2) q3) = g^g9 $ (p^ p2, ps\ everywhere in the volume r for the same point M, denoted by #2> #3 CT P\> P-2, PS in the respective coordinates. 29. Calculus of Variations. We shall frequently in what follows have to make use of the calculus of variations, which, since we shall use it always in connection with definite integrals, is introduced here. In the differential calculus, we have to consider questions of maxima and minima of functions. A function of one variable has a maximum or minimum value at a certain value of the variable if the change in the function is of the same sign for any change in the variable, provided the latter change is small enough. Since if f(x) is continuous at xt If h is small enough, the expression on the right will have the sign of the first term, which will change sign with h. Accordingly the condition for a maximum or minimum is /'<*) = o. Suppose that we change the form of the function — such a change may be made to take place gradually. For instance suppose we have a curve given in any way, e.g. a! = Fl(t)t y = Fa(t), z = F3(t), where the F's are any uniform and continuous functions of an independent variable t. If we change the form of the F's we shall change the curve — suppose we change to x = G1(t), y=G,(t), z=Gt(t). To every value of t corresponds one point on each curve, con- sequently to each point on one curve corresponds a definite point on the other. Such a change from one curve to the other is called a transformation of the curve. The change may be made gradually, e.g. / IUN7- 28, 29] DEFINITE INTEGRALS. 49 For every value of e we shall have a particular curve — for e=0 we shall have the original curve, for e = 1 the final curve, and for intervening values of e other curves. A small change in e will cause a small change in the curve, and if e is infinitesimal we shall call the transformation an infinitesimal transformation. The changes in the values of x, y, z, or of any functions thereof, for an infinitesimal change e, are called the variations of the functions, and are denoted by the sign 8. Suppose we denote dx d*x dkx _ _ _ PTif dt' dV ' dtk by the letters x'y x", «<*>, and by <£ any function 0 (t, x, y, z, x', y', z', ...... #<*>, y®, z^, ...... «<»>, y™, *<»>), and consider the change in made by an infinitesimal transfor- mation, where we replace x, y, z by where ^, 77, f are arbitrary continuous functions of t. „» dx ,. , , dx d% ,d^x, dkx Then Tt or x is replaced by ^ 4- ej and - by + e i.e., by a?*>+€f<*>. Hence <^> becomes (t, x + cf, y + 697, * + ef, ^' + ef, / + «/, which developed by Taylor's theorem for any number of variables, gives on collecting terms in equal powers of e e2 where W. E. 50 DEFINITE INTEGRALS. [INT. III. The terms e^, e22, €*$& are called the first, second, kih varia- tions of (j> and denoted by If for we put successively x, y, z, x', y'} z ...... we get Sy*> = ^*» =0. We thus see that the variations of x, y, z are infinitesimal arbitrary functions of t, the independent variable, and from the last equation that is, the operations of differentiation and variation are com- mutative, for the variables x, y, z. It is evident that <£& is the &th derivative of <£ with respect to e for the value e = 0. Since we may always change the order of differentiation, it is evident that the commutative property holds for any function. Let us now find the variation of the integral «,y,*,«',y,/, ...... )dt. Changing as to x + 8%, y to y + §y, x' to x' + Sat', etc., 7 + S/ + iS2/+ ...... = f(<£ + S + JS24>+ ...... )dt, Jto and the variations are to that is, the operations of variation and integration are commu- tative. (The limits have been supposed given, that is unvaried). These two principles of commutativity of 8 with d and I form the basis of the subject. 30. Line and Surface Integrals. If we consider any curve in space joining a point A to a point B, and if on the curve between A and B we place n — l points, ply p2 ..... .pn-i, whose 29, 30] DEFINITE INTEGRALS. 51 ^-coordinates are #13 #2 ^«-i> multiply the length of each chord PS-I PS, v - ^ by the value of the point function f(p) at some point TTS in the arc between ps-i)ps, and take the sum for all the arcs into which the curve has been subdivided, then if this sum approaches a finite limit as the number of subdivisions increases indefinitely, this limit is called the line-integral of the point-function f(p) along the curve AB, and is denoted by /(p)«fe-lii A n=oo i If y(jp) = 1., the integral represents the length of the curve AB 'B ds=sAB. If in forming the line-integral we had multiplied the values of /(TTS) by the ^-projection of the chord, instead of by the chord itself, we should have arrived at the integral already defined, (p) dx = lim S/(TTS) (xs — ^_a), n= T \ IT- dx Tr dy „ dz R cos (R, ds) = X -Y- + Y-2- + Z -y . ds ds ds The line integral of this resolved component dz -r + -f- + A may be written (B -D /D j \ j fs f-v-dx ^rdy ,7dz Rcos(R, ds).ds=\ (X-r + Y-f- + ZJ- } A i A \ ds ds ds /; with the understanding of the previous section. The functions X, F, Z, being given for every point x, y, z, the integral / will in general depend on the form of the curve AB. If we make an infinitesimal transformation of the curve, the 54 DEFINITE INTEGRALS. [INT. 111. integral will change, and we shall now seek an expression for the variation. We have ds ds dX ~ dX ~ dX ~ XT <> NOW & dx dy dz ^dx d (Sx) and & -y = -A— . ds ds We may perform upon the term A an integration by parts IB where XSx j signifies that from the value of the function X&K at the point B we subtract its value at A. Now dX dXdx dXdy dXdz i ^ _i ds dx ds dy ds dz ds ' Performing similar operations on the other terms we have *T fr* Lro i. 7* \ IB - ft^X , .dX^ .dX«\dx o 1 = (A ox + i oy + Zbz) . ds \dx dy ' dz J ds dx dX dy dX dz - _ fd x y z ___ f\fV» I _ _ T I _ _ L j -, _ _ V dx ds dy ds dz ds fdYdx dYdy dTdz\ — °y\*" j" + ^~ :j+^ — r ) 9 \dx ds dy ds dz dsJ (dZ_ clx 3Z dy dZ dz\] , *(fads~+tyds + dz ds)\ Now if in the variation the ends of the curve A and B are fixed, &», &/, Sz vanish for A and B, and the integrated part 31] DEFINITE INTEGRALS. 55 Y$y + Zbz I vanishes. Collecting those terms under the sign of integration that do not cancel, we have, Now the determinant Bydz — Szdy is the area of a parallelogram in the F^-plane the projections of whose sides on the Y- and ^-axes are dy, dz, §y> Bz. That is, if we consider the infinitesimal parallelogram whose vertices are the points s, s + ds and their transformed positions, the above determinant is the area of its pro- jection on the F^-plane. If the area of the parallelogram is dS and n is the direction of its normal, we as in § 30 Sydz — &zdy= dS cos (nx), Sz dx — Sxdz = dS cos (ny), Sxdy —§ydx = dS cos (nz\ and dY\ w cos (nx} dZ -- cos +(E-!?)««<«n« which is in the form of a surface integral over the strip of in- finitesimal width. If we again make an infinitesimal transformation, and so continue until the path has swept out any finite portion of a FIG. 11. 56 DEFINITE INTEGRALS. [iNT. III. surface S, and sum all the variations of /, we get for the final result that the difference in / for the two extreme paths 1 and 2 is the surface integral fiX dZ\ ft 7 dX\ :1 -: + a -- 5-) cos (ny) + {-= -- 5- 006(910) Y dS \dz dxj \da) dyj '] taken over the portion of the surface bounded by the paths 1 and 2 from A to B. Now — II may be considered the integral from B to A along the path 1, so that /2 — /i is the integral around the closed path which forms the contour of the portion of surface 8. We accordingly get the following, known as STOKES'S THEOREM*. The line integral, around any closed contour, of the tangential component of a vector R, whose com- ponents are X, Y, Z, is equal to the surface integral over any portion of surface bounded by the contour, of the normal com- ponent of a vector &>, whose components f, 77, f are related to X, Y, Z by the relations 8^_8F fmty a* ' dx dz ==_ 3# dy ' The normal must be drawn toward that side of the surface that shall make the rotation of a right-handed screw advancing along the normal agree with the direction of traversing the closed contour of integration. (ltd* = (Xdoi + Ydy + Zdz = jjco cos (am) dS = 1 1 (| cos (nx) 4- 97 cos (ny) + f cos (nz)) dS. The vector o> related to the vector point-function R by the differ- ential equations above is called the rotation, spin (Clifford), or curl (Maxwell and Heaviside) of R. Such vectors are of frequent * The proof here given is from the author's notes on the lectures of Professor von Helmholtz. A similar treatment is given by Picard, Traite d1 Analyse, Tom. i, p. 73. 31, 32] DEFINITE INTEGRALS. 57 occurrence in mathematical physics. The curl may be derived from the primitive vector by the application of Hamilton's vector differential operator V (§ 16) to a vector point-function R, dy /ax dY dz- dY dZ\ ./dZ j j _i_ ^ j dy dzj \]cjy = dz d_z to dz dx dz -o- dy curl R = WE = i So that the vector part resulting from the application of the operation V to a vector point-function gives its curl. The scalar part \ dx dy dz has an important interpretation to be given shortly. [The significance of the geometrical term curl can be seen from the physical example in which the y vector R represents the velocity of a point instantaneously occupying the position x, y, z in a rigid body turn- ing about the ^/-axis with an angular velocity o>. Then the vector R = wp is perpendicular to the radius p and o its components are FIG. 12. X = R cos (Eos) = - R sin (px) = -R- = - — = xco, P Y = R cos (Ry) — R cos (px) = where &> is constant, and dY dX_ dx dy ~ So that the ^-component of the curl of the linear velocity is twice the angular velocity about the ^T-axis.] 32. Lamellar Vectors. In finding the variation of the integral / in the previous section, since the variations &e, Sy, 8z are perfectly arbitrary functions of s, if the integral is to be inde- 58 DEFINITE INTEGRALS. [INT. III. pendent of the path, SI must vanish, which can only happen for all possible choices of SOD, By, Sz if _=_=_ = 'by dz dz dx ~ dx dy ~ that is, if the curl of R vanishes everywhere. In case this con- dition is satisfied, / depends only on the positions of the limiting points A and #, and not on the path of integration. If A is given, / is a point-function of its upper limit B, let us say c/>. If B is displaced a distance s in a given direction to B', the change in the function <£ is fa -B=(B (Xdx + Ydy + Zdz), J B and the limit of the ratio of the change to the distance lim*g-fr = j* Z!jg+ F^+ z £ s=0 s ds as as as is the derivative of $ in the direction s. If we take s successively in the directions of the axes of co- ordinates, <^_ Y ty-V fy-7 - -^A- j t-\ "~~ -*• * >"\ ^9 as dy dz that is, R is the vector differential parameter of the scalar function . Accordingly the three equations of condition equivalent to curl R = 0 are simply the conditions that X, Y, Z may be repre- sented as the derivatives of a point-function. In this case the expression is called a perfect differential. From the definition of the parameter of a scalar point-function, we see that the magnitude of the parameter is inversely propor- tional to the normal distance between two infinitely near level surfaces of the function. Such a pair of surfaces will be called a thin level sheet or lamina. For this reason a vector point-function that may be represented everywhere in a certain region as the 32, 33] DEFINITE INTEGRALS. 59 FIG. 13. vector parameter of a scalar point-function will be called a laminar, or lamellar vector (Maxwell). The scalar function c/> (or its negative) will sometimes be termed the potential of the vector R. 33. Connectivity of Space. Green's Theorem. We have supposed in § 31 that it was possible to change the path 1 from A to B into the path 2 by continuous de- formation, without passing out of the space considered. A portion of space in which any path between two points may be thus changed into any other between the same two points is said to be singly-connected. For instance, in the case of a two-dimensional space, any area bounded by a single closed contour will have this property. If, however, we consider an area bounded externally by a closed contour 0, and internally by one or more closed contours 7, Fig. 13, such as the surface of a lake containing islands, it will be possible to go from any point A to any other point B by two routes which cannot be continuously changed into each other without passing out of the space considered, that is traversing the shaded part. The space in Fig. 13 between the contour C and the island 7 is said to be doubly-connected. We may make it singly-connected by drawing a barrier connecting the island with the contour (7, represented by the dotted line. If no path is allowed which crosses the barrier the space is singly-connected. A three-dimensional space bounded externally by a single closed surface is not made doubly-connected by containing an inner closed boundary. For instance, the space lying between two concentric spheres allows all paths between two given points to be deformed into each other, avoiding the inner sphere. But the space bounded by an endless tubular surface, Fig. 14, is doubly-connected, because we may go from A to B in either direction of the tube, and the two paths cannot be deformed into each other. We may make the space singly-connected by the insertion of FIG. 14. 60 DEFINITE INTEGRALS. [INT. III. a barrier in the shape of a diaphragm, closing the tube so that one of the paths is inadmissible. The connectivity of a portion of space is denned as one more than the least number of barriers or diaphragms necessary to make it singly-connected. Thus the space in a closed vase with three hollow handles, Fig. 15, is quad- ruply-connected. We shall always suppose the spaces with which we deal in this book to be singly -connected, or to be made so by the insertion of dia- phragms, unless the contrary is expressly stated. Suppose that W is a point-function which, together with its derivative in any direction, is uniform and continuous in a certain portion of space r bounded by a closed surface S. Then its deri- vative dW/dx is finite in the whole region, and if we multiply it by the element of volume dr and integrate throughout the volume T, the integral is finite, being less than the maximum value attained by dW/dx in the space T multiplied by the volume r. We have at once FIG. 15. If, keeping y and z constant, we perform the integration with respect to x, the volume is divided into elementary prisms whose Y FIG. 16. sides are parallel to the .XT-axis, and whose bases are rectangles with sides dy, dz. 33] DEFINITE INTEGRALS. 61 The portion of the integral due to one such prism is dydz^dx. Now the integral is to be taken between the values of x where the edge of the elementary prism cuts into the surface S and where it cuts out from the surface. If it cuts in more than once, it will, since the surface is closed, cut out the same number of times. Let the values of x, at the successive points of cutting, be 1 ) 2 then \^-dx= TF2- W.+ W.-W,... + Wzn- J ox Wk being the value of W for #&, and -=— dxdydz ox -InW — W-4-W — W 4-W -W 1 tlii fly [_ "r 2 — ™ i TP S^2 ... dS2n denote the areas of the elements of the surface S cut out by the prism in question at a?l5 %2, ... xm — these all have the same projection on the F^-plane, namely dydz. Now if all these elements are considered positive, and if n be the normal always drawn inward from the surface 8 toward the space T, at each point of cutting into the surface S, n makes an acute angle with the positive direction of the axis of X, and the projection of dS is but where the edge cuts out n makes an obtuse angle, with negative cosine, and therefore dydz= — d8 cos (nx). We may accordingly write dydzWl = Wl cos^i^dSj^, — dydzW?. = W2 cos(n2x)dS2, - dydzWm = Wm cos 62 DEFINITE INTEGRALS. [iNT. III. and in integrating with respect to y and z we cover the whole of the projection of the surface 8 on the F-£f-plane. On the other hand we cover the whole of the surface S, so that the volume integral is transformed into a surface integral, taken all over the surface S. In like manner we may transform the two similar integrals cos Applying this lemma to the function where both Ut V and their derivatives in any direction are uniform -and continuous point-functions in the space r, we have Similarly for FF= U dy and for W=U~- Adding these three, and performing the differentiations, rrrr /92F 92F 82F\ dUdV difdV d_UdV~] , JJJ L \W*W* W) + fafo ty ty fa ^\ = - II U f •=— cos (na&) + ^- cos (ny) + y- cos («f)J dS, 33] DEFINITE INTEGRALS. or, transposing, and denoting the symmetrical integral by J, mtv ,dUW-\, " ~" I / l a" cos ^nx^ + 7T cos (ny) + £T~ cos ^"^ I This result is known as GREEN'S THEOREM*. By the definition of differentiation in any direction the paren- thesis in the surface integral on the right is if Pv is the parameter of V. Since the integral on the left is symmetrical in U and F, we may interchange them on the right, so that Writing this equal to the former value, and transposing, we obtain which will be referred to as Green's theorem in its second form. It will be noticed that the integrand on the left in the first form is the geometric product of the parameters of the functions £7 and F, We shall, unless the contrary is stated, always mean by n the internal normal to a closed surface, but if necessary shall dis- tinguish the normals drawn internally and externally as ^ and ne. If we do not care to distinguish the inside from the outside we shall denote the normals toward opposite sides by n^ and n2. * An Essay on the Application of Mathematical Analysis to the theories of Electricity and Magnetism. Nottingham, 1828. Geo. Green, Reprint of papers, p. 25. 64 DEFINITE INTEGRALS. [INT. III. 34. Second Differential Parameter. If for the function U we take a constant, say 1, S=f =f =°> $& and we have simply The function which, following the usage of the majority of writers, we shall denote by AF, was termed by Lame* the second differential para- meter of V. As it is a scalar quantity it will be sufficiently distinguished from the first parameter if we call it the scalar parameter. We have accordingly the theorem giving the relation between the two : — The volume integral of the scalar differential parameter of a uniform continuous point-function throughout any volume is equal to the surface integral of the vector parameter resolved along the outward normal to the surface $ bounding the volume. We may obtain a geometrical notion of the significance of AF in a number of ways. Applying the above theorem to the volume enclosed by a small sphere of radius R, we have, since n is in the direction of the radius, but drawn inwards, 8F .. FS-FO -3- = lim -£= — !, v'R' jR=0 -^ where F0 is the value of F at the centre of the sphere, Vs on the surface. Now remembering the significance of a definite integral as a mean, we have lim -^ {Mean of F on surface — F at center} x Area of surface R=o-tt = / 1 1 A Vdr = (Mean of A F in sphere) x Volume of sphere. * G. Lam6. Legons sur les Coordonnees cwvilignes et lews diverses Applica- tions. Paris, 1859, p. 6. 34] DEFINITE INTEGRALS. 65 Now since the volume of a sphere is J the product of the surface by the radius, we have, on making R approach zero, A T7. T . (Mean V on surface — F at center} A F= 3 Lim l— ->- . .8=0 -ft The negative scalar parameter — A F was accordingly called by Maxwell the concentration of F, being proportional to the excess of the value of F at any point over the mean of the surrounding values. It is evident from this interpretation of AF that if the concentration of a function vanishes throughout a certain region, then about any point in the region the values at neighbouring points are partly greater and partly less than at the point itself, so that the function cannot have at any point in the region either a maximum or minimum with respect to surrounding points. A function that in a certain region is uniform, continuous, and has no concentration is said to be harmonic in that region. The study of such functions constitutes one of the most important parts, not only of the theory of functions, but also of mathematical physics. By means of the same theorem we may obtain another repre- sentation of A F. Let us apply the theorem to the space included between two small concentric spheres of radii RL and R2 = jRx + h. Then at the outer sphere IF and the surface integral being taken over the surface of both spheres, with the normal pointing in each case into the space between them, As we make h approach zero, the first term of the second integral destroys the first, and [fdVja T. rr 92F7 ,0 - ^-dS = Lim ^— hdS, JJ dn fc=0 }JM> 9r2 so that fff&VdT = Lim ([ ~£kd& JJJ h=v HE, dr* Now hdS is the element of volume dr, so that AF may be 92F defined as the mean value of the second derivative -«--- for all W. E. 5 66 DEFINITE INTEGRALS. [INT. III. directions as we leave the point. This interpretation is due to Boussinesq*. We may derive the parameter AF by applying Hamilton's operator V twice to V, das J dy dzj \ dx J dy dz 35. Divergence. Solenoidal Vectors. If the components of the vector parameter are Pcos(P*) = Z = ^, we have AF= — — — dx dy dz ' and the above theorem becomes - JJP cos (Pn) dS = -l l(X cos (nx) + Fcos (ny) + Z cos (nz)) dS -///§ 'dX . d_Y d& dy dz. If P is everywhere directed outward from the surface S, the integral is positive, and fdx dY dz\ _ mean •= — \- -5 — h •«- > 0. \ox oy oz/ Accordingly r— +•«— + «— is called the divergence of the vec- 7 dx dy dz tor point-function whose components are X, Y, Z, and will be denoted by div. R. Comparing with § 31 we find that the divergence of a vector is minus the scalar part of the V of the vector, div. R = - SVE. The theorem as just given may be stated as follows, and will be referred to as the DIVERGENCE THEOREM : The mean value of the normal component of any vector point-function outward from * Boussinesq, Application des Potentiels a Vetude de Vequilibre et du mouve- ment des solides elastiques, p. 45. 34, 35] DEFINITE INTEGRALS. 67 any closed surface S within which the function is uniform and continuous, multiplied by the area of the surface, is equal to the mean value of the divergence of the vector in the space within S multiplied by its volume. The theorem was proved for a vector which was the parameter of a scalar point-function V, but it is evident that it may be proved directly by partial integration whether this is the case or not. Let us consider the geometrical nature of a vector point- function R whose divergence vanishes in a certain region. In the neighbourhood of any point, the vector will at some points be directed toward the point and at others away. We may then draw curves of such a nature that at every point of any curve the tangent is in the direction of the vector point-function R at that point. Such curves will be called lines of the vector function. Suppose that such lines be drawn through all points of a closed curve, they will generate a tubular surface, which will be called a tube of the vector function. Let* us now construct any two surfaces S± and $2 cutting across the vector tube and apply the divergence theorem to the portion of space inclosed by the tube and the two surfaces or caps $j and $2. Since at every point on the surface of the tube, R is tangent to the tube, the normal component vanishes. The only parts contributing anything to the surface integral are accordingly the caps, and since the divergence everywhere vanishes in r, we have Rcos(Rn1)dSl+jl R cos (Rn,) dS2 = 0. If we draw the normal to S2 in the other direction, so that as we move the cap along the tube the direction of the normal is continuous, the above formula becomes Rcos(Rnl)dS1-jj or the surface integral of the normal component of R over any cap cutting the same vector tube is constant. Such a vector will be termed solenoidal, or tubular, and the o -y f\v ^\y condition -= — h ~ — H ^-=0 will be termed the solenoidal condition ox dy dz (Maxwell). We may abbreviate it, div. R = 0. If a vector point- 5—2 68 DEFINITE INTEGRALS. [INT. III. function R is lamellar as well as solenoidal, the scalar function V of which it is the vector parameter is harmonic, for dx dy A solenoidal vector may be represented by its tubes, its direction being given by the tangent to an infinitesimal tube, and its magnitude being inversely proportional to its cross-section. As an example of a solenoidal vector we may take the velocity of particles of a moving fluid. If the velocity is R, with components X, Y, Z, the amount of liquid flowing through an element of surface dS in unit time is that contained in a prism of slant height R, and base dS, whose volume is R cos (Rn) dS. The total flux, or quantity flowing in unit time through a surface S, is the surface integral I JB cos (Bn) dS = ff(X cos (nx) + T cos (ny) + Z cos (m)) dS. Such a surface integral may accordingly be called the flux of the vector R through 8. A tube of vector R is a tube such that no fluid flows across its sides, such as a material tube through which liquid flows, and the divergence theorem shows that as much liquid flows in through one cross-section as out through another, if the solenoidal condition holds. If the liquid is incompressible, this must of course be true. As a second example of solenoidal vectors we have any vector which is the curl of another vector, for d_ (d_z _ « identically. The equation 32 32 32 is called Laplace's equation, and the operator A = ^- • + ^-, • + ^-. dx? dy2 dzz Laplace's operator. _a_ ,az _dz\ + d_ rar _ az|= Q dy\dz dx] dz\dx dy } 35, 36] DEFINITE INTEGRALS. 36. Representation of Solenoidal Vector. Multiplier. We have obtained in § 32 a means of representing a lamellar vector- function by means of the level surfaces of its ' potential function. By means of Jacobi's multiplier we may find a some- what similar representation for a solenoidal vector. If we suppose the curves drawn whose tangent at every point has the direction of the vector function R whose components are X, Y, Z, since the direction cosines of the tangent are dx dy dz ~ds' ds' ds' the curve is defined by the differential equations (I) dx:dy:dz = X:Y:Z. The integrals of these equations will each contain an arbitrary constant. Let us suppose that an integral is of the form X (x, y, z) = const. Then we must have ax 7 ax 7 ax 7 -. ^- dx + ~- dy + — dz = 0, dx dy ' dz and since dx, dy, dz are proportional to X, Y, Z, (2) ~ dx - -- \- dy — = U. dz This partial differential equation may serve as a definition of an integral of the system of differential equations (i). Geometrically it shows that the vector R is perpendicular to the normal to the surface \ = const., that is, is tangent to the surface. If //, = const, is a second integral, then (3) o ox ^— dy TT- dz and since R is tangent to a surface of each family X = const., //, = const., the lines of the vector R are the intersections of the surfaces X with the surfaces p. From (2) and (3), linear equations in X, Y, Z, we may determine their ratios. We obtain (4) X:Y:Z= ~ ax ax ax ax ax ax dy' dz " dz ' dx da' dy dfJb d/jb d/ju d/jL dp dp dy' ^z ~dz' dx dx' dy 70 DEFINITE INTEGRALS. [INT. III. If M be a factor to be determined, we may put (5) MX = A, MY=B, MZ=C, where A, B, C are the above determinants. But the determinants A, B, C, if differentiated by x, y, z, respectively and added, are found to satisfy identically the solenoidal condition (6) :, ¥ + *i-+f^ dx dy dz so that we have the equation for M, d(MY) (7) dx dy Consequently for any continuous vector function R it is possible to find a scalar multiplier M that shall make the vector whose components are MX, MY, MZ, solenoidal. If the vector R is itself solenoidal, the equation for M is satisfied by any constant, say 1, so that in this case we have y _ d\ d/JL d\ dfi ~~ dy dz dz dy ' ^=8\a/^_8\a/A dz dx dx dz ' „ _ 9X 3//, d\ d/j, dx dy dy dx ' But if PX, PM denote the vector parameters of the functions X, IJL we see by the definition of the vector product, R If we consider two infinitely near surfaces of the first family for which X has the values X and X + d\ respectively, the normal distance between which is dn^, we have by §§ 16 and 20 Considering two infinitely near surfaces of the other family and fj, -t- dp, we have in like manner for their normal distance , d/j, 36, 37] DEFINITE INTEGRALS. 71 The area of a right section of the four-sided tube thus formed (Fig. 18) is / . sin (nK n^ sin (Px PM) ' A / ds and multiplying this by the value offl, /+rfx A RdS = Px PM dn^ drip — d\ dp, FIG. 18. which is constant for the whole tube. Consequently we obtain a new proof of the fundamental property of a solenoidal vector, for any tube may be divided up into infinitesimal tubes defined by surfaces of the two families. 37. Principle of the Last Multiplier. If we have two functions M and N, each of which is a multiplier for the equations (i), they must each satisfy the partial differential equation (7) so that dy dz (dx dy dz Multiplying the second of these by M, the first by N, and subtracting, dz dz and dividing by M 2, ++= ox oy dz That is, the quotient of the two multipliers is an integral of the differential equations (i). This result is of particular importance when we have found one integral X = const, and any multiplier, for we may then find a last multiplier, which shall give us at once the remaining integral. By means of the integral equation X (x, y, z) — const, let us, by solving for one of the variables, say z, express z as a function of x, y, X, z = z(x, y, X). 72 DEFINITE INTEGRALS. [INT. III. If fj, — const, is a new integral, let us by introducing the value of z just found, express JJL in terms of x, y, X, fi = fi(a}tyt X). We shall distinguish the partial derivatives of //, thus expressed from its partial derivatives when expressed in #, y, z, by brackets, so that we have ty = [VI |~ VI ?± dp = ra/ti rev] ax a^ = r^i ax dx ~ [_a^J [_axJ ^ ' 92/ ~ L?2/ J L^J ty* Tz~ [axj a^ * Accordingly we obtain for the values of A, B, C _ ~te\?y\' a^L^J' Now /x being expressed in terms of x, y, X, we have and since X = const, is an integral, d\ = 0, But from the values of -4 and B £ ax5 dy ax' a^ , Bdx — Ady so that cut = - ... y . CA, a? But since ^1 = J/Z, B = MY, this becomes c?/i, = r- ( FcZa; — JT a* Accordingly although the expression is not a perfect differential, the factor M ax dz 37, 38] DEFINITE INTEGRALS. 73 f a function /<&, and ( Ydx — Xdy)— const. makes it the differential of a function /<&, and «M I is a second integral of the equations (i). X, Y and =-- must of course be expressed in terms of x, y, \. Consequently if we have the system of differential equations dx : dy : dz = X : Y : Z, and we have found one integral X = const, together with a mul- tiplier satisfying the partial differential equation \J ox dy oz then the expression M 8X dz is an integrating factor, or last multiplier* for the equation When X, Y, Z satisfy the solenoidal condition, the last multi- plier is J_ 8X* dz This result will be used in § 103. 38. Variation of a Multiple Integral. In illustrating the method of the Calculus of Variations we have found the varia- tion of a single integral, and in the example taken the functions varied were the coordinates x, y, z, of points of a curve, the variable of integration being t. We may in a similar manner vary a surface or volume integral, by causing the functions entering into the integrand to change their forms by an infinitesimal trans- formation, while the variables of integration are unchanged. For instance let Vdxdydz * Jacobi, Vorlesungen tiler Dynamik, p. 78. 74 DEFINITE INTEGRALS. [INT. III. be a volume integral, we may define its variation by the equation (V + SV)da;dydz, where 8V is any arbitrary function of x, y, z multiplied by an in- finitesimal constant e. We may also vary an integral in another manner. Suppose we consider the volume in question to be occu- pied by material substance, and that to each material point belongs a value of the function V. Now let every material point be displaced in any manner by an infinitesimal amount defined by the projections Bx, By, Bz. The material point which arrives at x, y, z brings with it a different value of V, and the value of the integral through the same portion of space, since the latter is filled with different material points, is different. It is to be noticed that this is the exact converse of the process exemplified in §§ 29, 31 for there the functions X, F, Z were associated with fixed points in space, while the integral was over a field which was varied, whereas here the function V goes with the varied point, while the field of integration is fixed. As an example, let us consider the integral m = 1 1 1 pdxdydz representing the mass of a body r whose density at any point is p, the density being defined as the limit of the ratio of the mass of a portion of the body to its volume, both being decreased in- definitely. Let us consider the mass in an infinitesimal rect- angular parallelepiped, whose sides are dx, dy, dz, and whose mass is dm — pdxdydz. When all points are displaced by the amounts Bx} By, Bz, particles in the face normal to the Jf-axis and nearest the origin move to the right a distance Bx, and the volume of new matter that enters the parallelepiped through that face is dydzBx, whose mass is pdydzBx, p and Bx having the values belonging to the face in question. At the opposite parallel face, farthest from the origin, pBx has the value and the amount of matter that moves out of the parallelepiped to the right is j j ( * d(p&r) , ) ay dz •< p ox -I — ^ ax j- . 38, 39] DEFINITE INTEGRALS. 75 The total gain through these two sides is, accordingly, the difference -- -^ — - dxdydz. Similarly through the sides normal to the F-axis the gain is and through the sides normal to the 9(p&) , -- ^ — - ax ay dz. oz The total increase of the mass in the parallelepiped is therefore ,, 8dm = - — — f + --^ + -* a dy dz } and this being taken for an element of our integral, the total increase of mass, or variation of the integral, is f/Y (d(p8x) d(pfy) d(pSz)] , cm = - \ -^ — '- + \ v; + ;r '\ dxdydz. JJJr( fa dy dz j We may obtain this result in a more rigid manner by the use of Green's Theorem. Through each element of surface dS of the boundary of the space in question there moves inwards an in- finitesimal prism of matter whose volume is dS [§x cos (nx) + &y cos (ny) + Sz cos (nz)}. The mass of this is {pSx cos (nx) + p§y cos (ny) + pSz cos (nz)} dS, so that the total gain of mass in the space r is 8m = 1 1 {p §x cos (nx) + p Sy cos (ny) + pSz cos (nz)} dS. But by Green's Theorem this is equal to This result will be of frequent use. 39. Reciprocal Distance. Gauss's Theorem. Consider the scalar point-function, V = - , where r is the distance from a 76 DEFINITE INTEGRALS. [INT. III. fixed point or pole 0. Then the level surfaces are spheres, and the parameter is and since hr = l, R = —} drawn toward 0. (§ 16.) Consider the surface integral of the normal component of R directed into the volume bounded by a closed surface S not containing 0, or as we shall call it, the flux of R into S, ( i ) (JR cos (Rn) dS = - ffi cos (rn) dS. FIG. 19. The latter geometrical integral was reduced by Gauss. If to each point in the boundary of an element dS we draw a radius and thus get an infinitesimal cone with vertex 0, and call the part of the surface of a sphere of radius r cut by this cone d%, dZ is the pro- jection of dS on the sphere, and as the normal to the sphere is in the direction of r, we have d% = ± dS cos (rn), the upper sign, for r cutting in, the lower for r cutting out. If now we draw about 0 a sphere of radius 1, whose area is 4-Tr, and call the portion of its area cut by the above-mentioned cone dco, we have from the similarity of the right sections of the cone The ratio da) is called the solid angle subtended by the infinitesimal cone. 39] DEFINITE INTEGKALS. 77 Accordingly dSc< r* d8cos(rn)_ d% _ (2) ^ - ± ^ ~ ± and Now for every element da), where r cuts into S, there is another equal one, — day, where r cuts out, and the two annul each other. Hence for 0 outside S, (4) If on the contrary, 0 lies inside S, the integral is to be taken over the whole of the unit sphere with the same sign, and consequently gives the area 4?r. Hence for 0 within 8, (5) These two results are known as Gauss's theorem, and the integral (3) will be called Gauss's integral*. These results could have been obtained as direct results of the divergence theorem. For the tubes of the vector function R are cones with vertex 0. If 0 is outside 8t R is continuous in every point within 8, and since the area of any two spheres cut out by a cone are proportional to the squares of the radii of the spheres, we have the normal flux of '-* equal for all spherical caps. Consequently R is solenoidal, and the flux through any closed surface is zero. If 0 is within S, R is solenoidal in the space between S and any sphere with center 0 lying entirely within 8, and the flux through S is the same as the flux through the sphere, which is evidently — 4-Tr. The fact that R is solenoidal and V harmonic may be directly shown by differentiation. If the coordinates of 0 are a, 6, c, (6) r2 = (x - a)2 + (y - 6)2 + (z - c)2, * Gauss, Theoria Attractionis Corporum Sphaeroidicorum Ellipticorum homo- geneorum Methodo nova tractata. Werke, Bd. v., p. 9. 78 DEFINITE INTEGRALS. [INT. III. /~\ dr _ x — a dr _y — b dr z— c dx r ' dy r ' dz r — f-\- ----- x~a • dx \rj r2 ox r3 - _ _ g^l^-J- ~3+ ^4 3^- ^ a'2 3 (y - 6)3 - r2 a2 / 1\ _ 3 (z - c)2 - r2 > 2 a2 / 1\ _ ' dz*(r)~ r5 ' dz*r r5 82 0 KS a>2£ > + "ap"+ 1^2" 3 {(as - cbf + (y- &)2 + (z- c)2} - 3r2 r5 and — is harmonic, except where r = 0. r CHAPTER IV. FUNCTIONS OF A COMPLEX VARIABLE. 40. Multiplication of Complex Numbers. We have seen in (5) how the two-dimensional complex number a + ib may be represented in the plane by Argand's diagram. From the definition of addition of complex numbers it follows that two complex numbers are added by the parallelo- gram construction, that is the sum of the two complex numbers p = ax + H>i and q = a2 + ib2 is represented by the FlG- 20- diagonal of the parallelogram constructed on lines whose lengths are equal to the moduli of p and q, and which make angles with the X-axis equal to the arguments of p and q. Hence \ P±<1\ ^ \ P \ + \q\ • If we introduce the polar coordinates -i& we have a = r cos (/>, b = r sin 0, p = a + ib = r (cos 0 + i sin 6). 80 FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. Now since 2T 4! 6! it follows that cos cf) + i sin <£ = e**, jp » | J? | d**. It is easy to show that the modulus of the product of two complex numbers is equal to the product of their moduli, and that the argument is equal to the sum of their arguments. For if p = &! + ibi = TI (cos ! + i sin fa) = r^, q = a2 + ^2 = ?*2 (cos <£2 4- * sin <£2) = r*^'*3, then £>£ = r^e* (*l+*a) = r^ [cos (0! + <£2) + i sin (^ + <£2)]. In like manner for the quotient, substituting the words quotient for product, and difference for sum. A complex number vanishes only when its modulus vanishes, and is considered infinite when its modulus is infinite, whatever its argument. 41. Function of Complex Variable. A function of the complex variable z = x + iy, if given as an analytic expression containing z, will be a certain function of the two real variables x and y and will contain a real part, which we shall denote by u (x, y), and an imaginary part, which we shall denote by iv (x, y). Hence the study of functions of a complex variable may be made to depend on the study of functions of two real variables. Let The representation of variable and function by means of abscissa and ordinate of a curve is not here applicable, for both variable and function have two degrees of freedom. The function may be otherwise represented by means of another plane in which we mark off lengths u and v as the rectangular coordinates of another point representing w on another Argand's diagram. To every point x, y in the first plane will then correspond a point u, v in the second plane. As the point x, y moves, so will the point u, v. As the point x, y representing the variable z, describes any curve, u, v, 40 — 42] FUNCTIONS OF A COMPLEX VARIABLE. 81 representing w —f(z) describes another curve, if f(z) is continuous, otherwise the point u, v may jump from one point to another. The definition of continuity is that two points on the function curve may be made to approach each other as nearly as we please by taking the corresponding points on the curve of the variable sufficiently near. Or, a function is continuous in a region of the ,3-plane continuing ZQ if to every real positive quantity e as small as we please, we can find a corresponding quantity 8 such that I/W-/WI <* if | *-*!<«• In considering the representation by means of curves, it is of importance to inquire whether, if the curve of z starting from an arbitrary point z0, returns to it after describing a closed curve, the curve representing w =f(z) also returns to its point of departure. If this is the case, the function f(z) within the region in which this property holds, is said to be uniform, or single-valued, for to every value of z corresponds one value of w. 42. Derivative. Analytic Function. Let us examine the relation between an infinitesimal change in z and the corre- sponding change in f(z). The change dz = dx -t- idy has the modulus | dz \ = */dx* -f dy2, and the argument w = tan"1 — . The change dw = du + idv has the modulus | dw \ = Vdu2 -f dv2 and the argument 6 = tan"1 -j- . Also , du j du j du = dxdx + ^jdy' -, dv 1 dv 7 dv = 5- dx + — dy, dx dy ' 7 7 . 7 du -, da 7 . (dv 7 dv dw = du + idv = —-dx + ^- dy + i-<~-dx + ^-c ox dy ' (ox oy The ratio du du j . (dv j dv 7 7 .7 TT~ dx -f -^r- dy -f- 1 •< ^- dx 4- — dy , . dw _ du + idv _ dx dy * (dx dy ' I dz i =0 dz dx + idy ~ dx -f idy du .dv_ du ~ + l *Tx W. E. 82 FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. is in general dependent on -— , that is on the direction in which we leave the point z. The value of the derivative will not then be determined for the point z irrespective of the direction of leaving it unless the numerator is a multiple of the denominator and the expression containing -^ divides out. In order that this may be true we must have fdu .dv\ /du .dv\ . «- -M 3- 1 : 1" •= ( 57 -|- i «- 1 : tj \dx dxJ \dy dyj . . . /du .dv\ du .dv that is Mo~ + *o~==3~~ + io~' \dx dxJ dy dy Putting real and imaginary parts on both sides equal, du _ dv dv _ du dx dy' dx~ dy' IS t/ dw du .dv dv . du and -j- = r- + * 5- = 5 l 6~ > dz ox ox oy oy i dw 2 /du\2 /du\2 fdv\* fdv \2 In this case the function w has a definite derivative, and it is only when the functions u and v satisfy these conditions that u+iv is said to be an analytic function of z. This is Riemann's definition of a function of a complex variable*. (Cauchy says monogenic instead of analytic.) The real functions u and v are said to be conjugate functions of the real variables x, y. It is obvious that if w is given as an analytic expression involving z, w —f(z\ then w always satisfies this condition. For dw_ df(z)dz_j,,^ dw_dj\2)dz_ d^~~d7~dx~^ (Z)' d~ dz d-V Accordingly .dw dx du dx du ^x + _dv .dv\_ dxj dv dx dw dy = du dy' du dy + .dv ldy' * Biemann, Mathematische Werke, p. 5. 42, 43] FUNCTIONS OF A COMPLEX VARIABLE. 83 43. Orthogonal Coordinates. Conformal Representation. We may apply the considerations of §§ 15 and 20 to the case of orthogonal coordinates in a plane. If a set of point-functions are independent of one rectangular coordinate, the geometry of all planes perpendicular to the axis of that coordinate is the same, and we have the uniplanar, or two-dimensional case involving only two variables which we will take as x, y. If we take u and v as any two point-functions, whose parameters are hu, hv " V "* 1 2T ) + VST 1 > \dxj \oy/ \dxj \dyj their level lines u = constant and v = constant may be taken for coordinate lines. Their normals have the direction cosines 1 du 1 du l Sv and the condition that u and v shall form an orthogonal system is du dv du dv _ - dx fix dy dy The lengths of infinitesimal arcs of curves, forming the sides of a rectangle whose opposite vertices have coordinates u, v, u + du, v + dv. are as in § 20 du dv hu' hv} and the length of the diagonal ds, or element of length of a curve whose ends have the above coordinates, is given by du2 dv2 ds = n + O ' AM2 hf If now we take for curvilinear coordinates in the #, y plane two functions u and v such that u + iv is an analytic function of as + iy, in virtue of the equations (A) of § 42 we have du dv du dv _ - dx dx dy dy and u and v form an orthogonal system. Now in any orthogonal system if we construct a set of level curves for equal small incre- ments of u and v, they will divide the plane up^into small (( UNIVE! FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. curvilinear rectangles the ratios of whose sides at any point are given by the ratio of the parameters hu and hv. But from the equations (A), we have /du\* , fdu\* idv\* fdv hu 2 = ~- + U- = U- + 1 5~ so that in this case the plane is divided into small squares. Let us now construct in the second plane, in which u and t; are FIG. 21. rectangular coordinates, the curves corresponding to u = constant and v = constant. These are of course straight lines dividing their plane into small squares. Moreover the length of any arc da of a curve in their plane, is given by da* = du2 4- dv\ But in virtue of the above relations, this gives h = dw is accordingly the ratio of magnification at the point in question, and varies for different points of the plane. Let us now construct, (Fig. 21,) at a point in the x, y plane an infinitesimal triangle made by the intersection of any three curves, and let the lengths of its sides be dsl} ds2, ds3. Construct the corresponding curves in the u, v plane, intersecting to form an infinitesimal triangle with sides dal} d = I vdx + udy, ^jr = I udx — vdy, give two new point-functions , ty which in virtue of the equations d(f> 9"^ d4> 9^ dx dy ' dy dx ' are conjugate to each other, and give a new analytic function of 2, (f> + ity,or ^ + i -H y2 - Cjx = 0, #2 + ?/2 + give two sets of circles, the first all tangent to the F-axis at the origin, the second all tangent to the X-axis, Fig. 23. The power zn = (x + iy)n = rn {cos n + i sin ??$), gives the two functions u — rn cog n( v and a sum of any number of such terms each multiplied by a con- stant 2rw { An cos n of the last section. This is on the supposition that the functions u, v are uniform and continuous in the whole region considered. If this is the case the function w — u + iv is called holomorphic. If w becomes discontinuous in the region considered it ceases to be true that the integral is the same over two paths AB between which lies a point of discontinuity of the function w. 45] FUNCTIONS OF A COMPLEX VARIABLE. 89 For example the function w = - is discontinuous at the point z = 0. Accordingly the integral I — around a closed contour con- taining the origin within' it is not zero, for it may be taken as the difference between the integrals between two points AB on the contour along two paths between which lies the point of dis- continuity of the function w. The integral around any closed contour embracing the origin is however the same as around a circle of radius R with center at the origin, for between the two curves there is no point of discontinuity of the function. Now since z — x + iy = re**, if r is constant = R, dz = i^ and the integral from z = dz which taken around the circle is 2-7T*. rz fa The integral I - is denned as the logarithm of z, and it pos- sesses the property that as z describes any closed path enclosing the origin, the function instead of returning to its original value increases by a constant 2?ri. The function is then not uniform, but has at any point an unlimited number of values, depending upon the path by which we arrive at the point. These values all differ by integral multiples of the constant 2?™. We see that this accords with the ordinary definition of the logarithm, log z — log (x + iy) = log (re*(*+anir)) = log r + i + 2rwn', for if we increase the argument of a complex number z by any multiple of 2?r, the number is unchanged. A point such that a function f(z) assumes a new value when the variable traverses a closed circuit about the point is called a critical, or branch point In this case the conformal representation given by the function f(z) is multiple in character, for in the [/"F-plane we are to take a point for each of the values of the function f(z). Each of these repre- sentative points gives a conformal representation of the whole of the JTF-plane on a part of the £/"F-plane. 90 FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. For instance, in the case of the logarithm log z = log r + i + Zmri, u = log r, v = + 2mr, as z takes all possible values in the JT F-plane, u = log r varies from — oo to + oo but v varies only in limits differing by 27r, so that the whole X F-plane is entirely represented on a strip of the UV- plane infinite in one direction but of the finite width 2?r in the other. This strip is repeated an infinite number of times each giving the same conformal representation of the whole X F-plane. For instance the radii (j> = const, and the circles r = const, cutting them orthogonally in the X F-plane correspond to the lines v = const, u = const, in the CTF-plane. FIG. 24. Corresponding regions of the figures are similarly shaded. PART I. THEORY OF NEWTONIAN FORCES. CHAPTER I. PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 46. Matter and Energy. Dynamics. Physics is the science of Matter and of Energy. Its laws are found to be invariable and capable of exact statement, that is of presentation in the language of Mathematics. The application of mathematical analysis to the treatment of physical phenomena, enabling us to deduce general laws from the results of experiment, and to infer the consequences of general laws, forms the subject of Mathe- matical or Theoretical Physics. Matter has the essential property of occupying space. It has in addition universally only the property of Inertia, to be defined below. In order to define Energy, we must consider the motion of matter in space. That portion of mathematical physics which treats of the motion of matter is called Mechanics, or Dynamics. It is the object of physicists to reduce the explanation of all physical phenomena to descriptions of motion of matter, and accordingly the study of the principles of Dynamics is indis- pensable to the study of any branch of theoretical physics. Before considering the nature of electrical and magnetic phenomena we shall therefore devote a few chapters to Dynamics. 47. Scalar and Vector Quantities. Physical quantities are of two kinds. Quantities whose complete specification involves 92 THEORY OF NEWTONIAN FORCES. [FT. I. CH. I. no idea of direction are called scalar quantities, for they may be conceived as arranged on a scale according to their magnitude. Such are time, temperature, size, density. Quantities whose specification involves the idea of direction as well as of magnitude are called vector quantities. They may be represented by geometrical directed lines, and all that has been said of vector quantities and their addition, etc. applies to them. 48. Degrees of Freedom. A set of magnitudes or para- meters which completely specify a quantity are called its co- ordinates. The number of coordinates required is called the number of degrees of freedom of the quantity. For instance, a point in a plane may be defined by two rectangular, or two polar coordinates, and has two degrees of freedom. We may also say that there is a double infinity or oo 2 of points in a plane. A point in space requires three coordinates of any sort, and has three degrees of freedom. Every independent relation that the coordi- nates of a quantity are made to satisfy diminishes the number of its degrees of freedom by one. For instance, a relation between the rectangular coordinates of a point restricts it to lie on a certain surface, — it then has two degrees of freedom instead of three, and requires but two coordinates to specify it. For example, a point satisfies the condition #2 + y2- + z* = a2. It lies on the sphere of radius a, and may be fully specified by giving its lati- tude and longitude. For the coordinates of a vector R we may take its projections on the three coordinate axes. If we choose its length, or modulus, and its three direction cosines, a = cos (Rx), /3 = cos (Ry), 7 = cos (Rz), one of the four coordinates R, a, & 7 is redundant, for the latter three satisfy the identical relation This furnishes us an example of the general case where we give n coordinates of a quantity satisfying k independent identical relations, or equations of condition. The quantity then has only n — k degrees of freedom, and we may find n — k independent coordinates which completely specify it. 47 — 49] PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 93 49. Velocities. If a point change its position in space, its motion may be described by giving the values of its coordinates for every instant of time, by means of equations such as The functions / must be continuous, since the point cannot jump from one position to another. We may describe the motion otherwise by giving two equations F1 (x, y, z) = 0, F2 (x, y, z) — 0, which denote the curve of intersection of two surfaces along which the point moves. This curve is called the path of the point. We must further give the distance s measured along the curve, which the point has traversed, counting from a fixed point on the curve. We must know s at all times t, which is expressed by giving s as a continuous function of t, s =

ds dt ds dt dz ds dz dz V=Vry— V— - = —..— ——- ds dt ds dt 94 THEORY OF NEWTONIAN FORCES. [PT. I. CH. I. and vz = vJ* + vJ + vJ2 •• We might have defined the vector velocity as the resultant of the three vectors _ _dx - _dy _ _dz dt ' dt ' dt 50. Accelerations. If the velocity of a point is variable with the time we define the acceleration of the point as the limit of the ratio of the increment of velocity Ay to the increment of time A£, as both approach zero. We may consider either the numerical change iV dv d2s or the geometrical change. If we draw a vector AB to represent the velocity at the time t and the vector AC to represent the velocity at the time t + A£, and draw the arc of a circle BD, DC will represent the numerical change of velocity, Av, not considering its direction, while BG represents its geometrical, or vector change, A#, for AS + BG = AC, FIG. 25. Accordingly Lim — = Lim EG is the vector acceleration a. Since the projections of the geometrical difference of two vectors are the differences of the projections, the components of a in any direction will be proportional to the changes of the corre- sponding components of the velocities, that is dvx d2x _ dvy ~ dvz dt 49 — 51] PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 95 The vector acceleration, a, may be defined as the resultant of the components ax, ay, az, and accordingly its modulus is This is not in general equal to -y- which is the acceleration of ctt the scalar velocity. The direction of a is given by its direction cosines a a a 51. Physical Axioms. The results of universal experience with regard to motion are summed up by Newton in his three Laws of Motion or Axioms of Physics. An axiom is defined by Thomson and Tait* as a proposition, the truth of which must be admitted as soon as the terms in which it is expressed are clearly under- stood. These physical axioms rest, not on intuitive perception, but on convictions drawn from observation and experiment. LEX I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum suum mutare. Every body persists in its state of rest or of uniform motion in a straight line, except in so far as it may be compelled by force to change that state. The property of persistence thus defined is called Inertia. This gives a criterion for finding whether a force is acting on a body or not, or in other words a negative definition of force. Force is acting on a body when its motion is not uniform. By uniform we mean such motion that the vector velocity is constant. If the body be a material point, that is a body so small that the distances apart of its different parts may be neglected, the motion is uniform if xx dx _ dy _ dz _ ~dt~Cl> ~dt~^ di = C3) that is ^ = ^-^£-0 dt* dt*~dt*~ * Thomson and Tait, Natural Philosophy, § 243. 96 THEORY OF NEWTONIAN FORCES. [PT. I. CH. I. Accordingly we see that the force and acceleration vanish together. Integrating the equations (i), x — di y — c?2 z — c?3 GI C2 C3 the path is a straight line, and since it is traversed with constant velocity. We may on the other hand interpret the statement as giving us a means of measuring time. Intervals of time are proportional to the corresponding distances traversed by a point not acted on by forces. The second law gives the measure of a force. LEX II. Mutationem motus proportionalem esse m motrid impressae, et fieri secundum lineam rectam qua vis ilia im- primitur. Change of motion is proportional to force applied, and takes place in the direction of the straight line in which the force acts. By change of motion is meant acceleration. If we have to do with different bodies, however, the factor of proportionality will be different for each. LEX III. Actioni contrariam semper et aequalem esse reac- tionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. To every action there is always an equal and contrary reaction : or, the mutual actions of any two bodies are always equal and oppositely directed. If we have an action between two bodies 1 and 2, if the forces were proportional only to the accelerations, we should have ggi _ _ cgg_a dfyi _ _ dfyz d2^ _ cfrga W~ ~W ~di?~ ~W dtz~ dt*' This is not the case, but we must introduce factors of proportionality, so that 51, 52] PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 97 The factors m^ ra2, are called the masses of the bodies 1 and 2. This gives us a means of comparing masses. If we make two bodies act upon each other in any manner, their masses are inversely proportional to the accelerations they have at the same instant. The vector whose components are is called the impressed force acting on the mass ra. If the quantities X, Y, Z are given functions, the above are called the differential equations of motion of the material point m. 52. Units. The specification of any quantity, scalar or vector, involves two factors, first a numerical quantity (integer, fraction or irrational) or numeric, and secondly a concrete quantity in terms of which all quantities of that kind are numerically expressed, called a unit. The simplest unit is that of the geometrical quantity; length. We shall adopt as the unit of length the centi- meter, defined as the one-hundredth part of the distance at tem- perature zero degrees Centigrade, and pressure 760 millimeters of mercury, between two parallel lines engraved on a certain bar of platinum-iridium alloy, deposited in a vault in the laboratory of the " Comitd International des Poids et Mesures" at Sevres, near Paris. This bar is known as the "Metre Prototype" and serves as the basis of length measurements for the civilized world (except the British Empire and Russia*). It was proposed by Maxwell to use a natural unit of length, namely the length of a wave of light corresponding to some well- defined line in the spectrum of some element, at a definite tem- perature and pressure, as it is extremely probable that such a wave-length is extremely constant. Measurements were carried out at Sevres by Michelson, with this end in view, which established the ratio between the above meter and the wave-length in air of a red cadmium ray as l,553,164f. * The United States yard is defined as 3600/3937 metres. t Michelson, Journal de Physique, Jan. 1894. W. E. 7 98 THEORY OF NEWTONIAN FORCES. [PT. 1. CH. I. The unit of mass will be assumed to be the gram, defined as the one-thousandth part of a piece of platinum-iridium, deposited at the place above mentioned and known as the "Kilogramme Prototype" As the unit of time we shall take the mean solar second, obtained from astronomical observations on the rotation of the earth. The unit of time cannot be preserved and compared as in the case of the units of length and mass, but is fortunately preserved for us by nature, in the nearly constant rotation of the earth. As the earth is gradually rotating more slowly, however, this unit is not absolutely constant, and it has been proposed to take for the unit of time the period of vibration of a molecule of the substance giving off light of the standard wave-length. To obtain such a unit would involve a measurement of the velocity of light, which cannot at present be made with sufficient accuracy to warrant the change. 53. Derived Units. Dimensions. It can be shown that the measurements of all physical quantities with which we are acquainted may be made in terms of three independent units. These are known as fundamental units, and are most conveniently taken as those of length, mass, and time. Other units, which depend on these, are known as derived units. If the same quantity is expressed in terms of two different units of the same kind, the numerics are inversely proportional to the size of the units. Thus six feet is otherwise expressed as two yards, the numerics 6 and 2 being in the ratio 3, that of a yard to a foot. If we change the magnitude of one of the fundamental units in any ratio r, the numeric of a quantity expressed in derived units will vary pro- portionately to a certain power of r, r~n ; the derived unit is then said to be of dimensions* n in the fundamental unit in question. For instance, if we change the fundamental unit of length from the foot to the yard, r = 3, an area of 27 sq. ft. becomes expressed as 3 sq. yds., the numeric has changed in the ratio 3 : 27 = 1 : 32= r~2, and the unit of area is of dimensions 2 in the unit of length. We may express this by writing [Area] = [Z2]. * The idea of dimensions of units originated with Fourier: vid. Thforie analytique de la Chaleur, Section ix. 52, 53] PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 99 The derived unit increases in the same ratio that the numeric of the quantity decreases. In our system the unit of area is the square centimeter, written 1 cm2. In like manner the unit of volume is of the dimensions [Z3] and the unit is 1cm3. The dimensions of velocity are ™ , or as we write for convenience, velocity = length /time. Two quantities of different sorts do not have a ratio in the ordinary arithmetical sense, but such equations as the above are of great use in physics, and give rise to an extended meaning of the terms ratio and product. The above equation is to be interpreted as follows. If any velocity be specified in terms of units of length and time the numerical factor is greater in proportion directly as the unit of length is smaller, and as the unit of time is greater. For instance we may write the equation expressing the fact that a velocity of 30 feet per second is the same as a velocity of 10 yards per second or 1800 feet per minute 30^=10^ = 1800-^. sec. sec. mm. We may operate on such equations precisely as if the units were ordinary arithmetical quantities, for the ratio of two quantities of the same kind is always a number. For instance 30 _ yd. sec. 10 ~ ft. sec. ' vd sec The ratio ~-^ is the number 3, while - — = 1. Also ft. sec. . 10 ft. sec. Such an expression as - - is read feet per second. The unit of velocity is one centimeter-per-second, cm. - = cm. sec"1. sec. Since acceleration is defined as a ratio of increment of velocity to increment of time, we have [Acceleration! - [Velocityl - tLen£th] - f ^1 - [Time] " [Time2] " \_T\\ ' 7—2 100 THEORY OF NEWTONIAN FORCES. [PT. 1. CH. 1. or the numeric of a certain acceleration varies inversely as the magnitude of the unit of length, and directly as the square of the unit of time. For instance, an acceleration in which a velocity of 10 feet per second is gained in 2 seconds is equal to one in which a velocity of 9000 feet per minute is gained in a minute, 10 ft. =ioft. 9000 ft. (2 sec.)2 4 sec.2 min.2 ' The unit of acceleration is one centimeter-per-second per second. Since force = mass x acceleration, rForcel - [Mass]. [Length] _ [(Time)*] - The unit of force is one gram-centimeter-per-second-per-second. It is called a dyne. All physical equations must be homogeneous in the various units, that is, the dimensions of every term must be the same. This gives us a valuable check on the correctness of our equa- tions. 54. Absolute Systems. The above system of units, which has for its fundamental units the centimeter, gram, and second, is called the C. G. S. system, and was recommended by a committee of the British Association for the Advancement of Science in 1861. It is sometimes incorrectly spoken of as the absolute system of units. An absolute system is any system, irrespective of the magnitudes of the units, by which physical quantities can be specified in terms of the least number of fundamental units, which shall be independent of time or place, and reproducible by copying from standards. A system based on the foot, pound, and minute is just as much an absolute system as the c.G.s. system. The idea of an absolute system is due to Gauss *. The ordinary method of measuring force, used by non-scientific persons and (or including) engineers, does not belong to the abso- lute system of measurements. The unit of force is taken as the weight of, or downward force exerted by the earth upon, the, mass of a standard piece of metal, such as the standard pound or kilogram. To measure the force in absolute units, we must know * Gauss. Intensitas vis magneticae terrestris ad mensuram absolutam revocata* Gottingen, 1832. Ges. Werke, v. p. 80. 53, 54] PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 101 what acceleration the earth's pull would cause this mass to receive, if allowed to fall. Experiment shows that in a given locality on the earth's surface all bodies fall in vacuo with the same acceleration. The value of this acceleration is denoted by g, and its value at the sea-level in latitude 45° is OTY1 #=980-606 '-. sec.2 Accordingly the force exerted by the earth on a mass of m grams is mg dynes, or the weight of a kilogram in latitude 45° = 980,606 dynes. Now the value of the acceleration g is not constant, but varies as we go from place to place on the earth's surface, ascend moun- tains or descend into mines. Accordingly, the weight of a kilogram is not an invariable, or absolute standard of force. At the center of the earth, a kilogram would weigh nothing. Its mass is, how- ever, invariable. The value of g at points on the earth in lati- tude X and h centimeters above the sea-level, is given by the formula, originally given by Clairaut*, g = 980-6056 - 2*508 cos 2X - '000003/L For further information with regard to units, the reader may consult Everett's Units and Physical Constants. * Everett, Units and Physical Constants, Chap. in. CHAPTER II. WORK AND ENERGY. 55. Work. If a point be displaced in a straight line, under the action of a force which is constant in magnitude and direction, the product of the length of the displacement and the resolved part of the force in the direction of the displacement, that is, the geometrical product of the force and the displacement (§ 7), is called the work done by the force in producing the displacement. If the components of the force F are X, F, Z, and those of the displace- ment s are sx, syt sz, the work W is (i) W = sFcos (Fs) = Fs = Xsx + Ysy + Zsz. Since work is defined as force x distance, we have for its dimensions, The C.G.S. unit of work is the work done when a force of one dyne produces a displacement of one centimeter in its own direction. This unit is called the erg = gm . cm2 . sec~2. If the displacement be not in a straight line, and the force be not constant, the work done in an infinitesimal displacement ds is (2) ds ds ds and the work done in a displacement along any path AB is the line integral f*\ (3) 55, 56] WORK AND ENERGY. 103 The components of the force are supposed to be given as functions of s and the derivatives -y- , -~ , 7— are known as func- ds as as tions of s from the equations of the path. Understanding this, we may write (4) WAB = [ Xdx + Ydy + Zdz. J A 56. Virtual Work. Suppose that we have a system of n material points. If they are entirely free to move, they require 3n coordinates for their specification. They may be mechanically constrained, however, in such a manner that there must be certain relations satisfied by their coordinates. Let these equations of condition or constraint be <£i Oi, 2/1, *i, #2, 2/2, *2, #n, yn, -O = o, 02 (#1, 2/i> ^' ^2, 2/2.- *2, #n> 2/n, *n) = 0, (5) &Ol, 2/1, 2l, #2, 2/2, *2, #n> 2/«, %) = 0. Such constraints may be imposed by causing the particles to lie on certain surfaces. For instance, if two particles 1 and 2 are connected by a rigid rod of length I, either particle must move on a sphere of radius I of which the other is the center, and we have the equation of condition *0 = Zr + bz, ....... ) = 0, and if be a continuous function, developing by Taylor's Theorem, and accordingly, taking account only of the terms of the first order in the small quantities &z?r> 8yr, &zr, and using equations (6), we have If a number of particles are displaced, we must take the sum of expressions like the above for all the particles, or as the conditions which must be satisfied by all the displacements 8xrt Syr, Szr. There must be one such equation for each function <£. Such displacements, which are purely arbitrary, except that they satisfy the equations of condition, are called virtual, being possible, as opposed to the displacements that actually take place in a motion of the system. The Principle of Virtual Work is an analytical statement of the conditions for equilibrium of a system. A system is in equilibrium when the forces acting on its various particles, together with the constraints, balance each other in such a way that there is no tendency toward motion of any part of the system. If the system consists of a single free point, in order for it to be in equilibrium, the resultant of all the forces applied to it, whose components are X, F, Z, must vanish, (9) X=Y = Z=0. If we multiply these equations respectively by the arbitrary small quantities Sx, By, Bz and add, we get (10) XSx+YSy + ZSz = 0, which expresses that the work done in an infinitesimal displace- ment of a point from its position of equilibrium vanishes. The 56] WORK AND ENERGY. 105 equation (10) is equivalent to the equation (9), for since the quantities Bx, by, Bz, are arbitrary, if X, Y, Z, are different from zero, we may take Bx, By, Bz respectively of the same sign as X, Y, Z, — each product will then be positive, and the sum will not vanish. If the sum is to vanish for all possible choices of Bx, By, Bz, X, Y, Z must vanish. If the particle is not free, but constrained to lie on a surface — 0, So;, by, Bz are not entirely arbitrary, but must satisfy (7) dx dy ' dz Let us multiply this by a quantity X and add it to (10), obtaining CD We may no longer conclude that the coefficients of Bx, By, Bz must vanish, for Bos, By, Bz are not arbitrary, being connected by the equation (7). Two of them are however arbitrary, say By and Bz, X has not yet been fixed — suppose it determined so that Then we have + X By + z + X Bz = 0, in which By and Bz are perfectly arbitrary, it therefore follows of necessity that the coefficients vanish. , dy By the introduction of the multiplier X we are accordingly enabled to draw the same conclusion as if Bx} By, Bz were arbitrary. Eliminating X from the above equations we get dy dz Now the direction cosines of the normal to the surface = 0 are proportional to -^ , ~, ^ , consequently, the components ox oy oz X, F, Z being proportional to these direction cosines, the resultant 106 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. is in the direction of the normal to the surface. But under these conditions the particle is in equilibrium. In like manner we may show that if the forces Xlt Ylt Zl} act upon the particle 1, X2) Y2, Z2) upon the particle 2, etc., the condition of equilibrium is ( 1 2) XM\ + Fifyx + Z^Z, + X£a± + K%2 + ZJlZt ...... + ZnSzn = 0, where the displacements satisfy (13) Multiplying the equations (13) respectively by \lt X2, . . . X^, and adding to (12) we have Of the 3n quantities 8xl} Szn, only 3^-A; are arbitrary, we may however determine the k multipliers X so that the coeffi- cients of the k other S's vanish, then the coefficients of the 3n — k arbitrary S's must vanish, so that we get the 3n equations 56, 57] WORK AND ENERGY. 107 (15) ...... . ozn ozn czn Eliminating from these the k quantities X, we have 3ft - A; equations expressing the conditions of equilibrium, being as many as the system has degrees of freedom. The equation (12), or as we may write it ( 1 6) 2 (XSx + YSy + ZSz) = 0, expresses the fact that the work in a virtual displacement vanishes, and is the condition for equilibrium. This is the Principle of Virtual Work. 57. D'Alembert's Principle. The equations of motion of a point are (§51) (17) ' «r mr or (18) Multiplying the equations (18) respectively by the arbitrary quantities 8#r, fyr> §ZT adding, and taking the sum for all values of the suffix r, 2) / ' ^ ™ ^r V \ 5- r 1 \mr ~T^r — ^ r) O®r ' 108 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. This equation may be called the fundamental equation of dynamics, and is the analytical statement of what is known as d'Alembert's Principle. Lagrange made it the basis of the entire subject of dynamics*. Interpreted by means of the principle of virtual work, equation (19) states: — If, the motion of a system of particles being given, we find the acceleration of every particle, and apply to each particle a force whose components are then the system of forces X', Y, Z', together with the impressed forces X, Y, Z, will form a system in equilibrium. The forces X', Y, Z' are called the forces of inertia, or the reversed effective forces. D'Alembert's principle is thus only another form of stating Newton's third law of motion. We have now a measure of the inertia of a body, namely the force of inertia above defined f. We may now define matter as whatever can exert forces of inertia. 58. Energy. Conservative Systems. If in the equation of d'Alembert's principle, (19), we put for &c, Sy, Sz the displacements which take place in the actual motion of the system in the time dt, s- • d&r 7 . cs dyr -, % dzr , r==~dt yr==~3t r==~dt ' we obtain / x v f (d2xr dxr d?yr dyr d*zr dzr\ (20) ±r jmr ^jrjf + -fa ~dt + W ~dt) x dxr Y dyr „ dzr[ , Q ~Xr~dt Yrdl Zrdt}d d*xr dxr d dxr Smce the sum of the first three terms is the derivative of the sum * Lagrange, Mecanique Analytique. (Euvres, t. 11, p. 267. f The inertia of a body is sometimes considered as the factor of the negative acceleration in the expression for the force of inertia, thus making inertia synonymous with mass. 57, 58] WORK AND ENERGY. 109 and the equation may be written Integrating with respect to t between the limits t0 and tl} T1V \(dxr\ (dyr\* , [dzr\Y\^ (22) ^.|^)+^j+^jj^ d(Cr,y fyr 7 r r r The square brackets with the affixes t0, ti denote that the value of the expression in brackets for t = tQ is to be subtracted from the value for t = ti. The integral on the right of (22), which may be written Xrdxr + Yrdyr + Zrdzr> \ denotes the work done by the forces of the system on the particle mr during the motion from t0 to tl} and the sum of such integrals denotes the total work done by the forces acting on the system during the motion. The expression /7'r \2 frlii \2 / r) y \1\ U + =M the half-sum of the products of the mass of each particle by the square of its velocity, is called the Kinetic Energy of the system. If we denote it by T, the equation (22) becomes (23) Ttl - Tto = 2rj\Xrdxr + Yrdyr -h Zrdzr). This is called the equation of energy, and states that the gain of kinetic energy is equal to the work done by the forces during the motion. The equation of energy assumes an important form in the particular case that the forces acting on the particles depend only on the positions of the particles, and that the components may be 110 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II represented by the partial derivatives of a single function of the coordinates 7 ?'~* In this case the expression 2r {Xrdxr + Yrdyr + Zrdzr] is the exact differential of the function U, and the integral [tlZ(Xrdxr + Yrdyr + Zrdzr) = Utl - Uto. J t0 The equation of energy then is The function U is called the force-function, and its negative W = — U is called the Potential Energy of the system. Inserting W in (25) we have (26) Ttl+Wtl=T,.+ Wta. The sum of the kinetic and potential energies of a system possessing a force-function is the same at all instants of time. This' is the principle of Conservation of Energy. Systems for which the conditions (24) are satisfied are accord- ingly called conservative systems. The potential energy, being defined by its derivatives, contains an arbitrary constant. Conservative systems possess the property, since W depends only on the coordinates, and T + W is constant, that T, the kinetic energy, depends only on the coordinates, or if in the course of the motion all the points of the system pass simultaneously through positions that they have before occupied, the kinetic energy will be the same as at the previous instant, irrespective of the directions in which the points may be moving. For instance, a particle thrown vertically upwards, or a pendulum swinging, have the same velocity when passing a given point whether rising or falling. The principle of virtual work, § 56, may evidently be expressed by saying that for equilibrium the potential energy of the system 58, 59] WORK AND ENERGY. Ill is a maximum or minimum, and a little consideration shows that for stable equilibrium it is a minimum. Examples of non-conservative systems are found whenever the forces depend upon the velocities as well as upon the coordinates ; for example, bodies moving through the air or other resisting medium or bodies whose motion is opposed by friction of any sort, form non-conservative systems. Even if the friction be constant in magnitude, its direction will depend on the direction of the velocities, being in such a direction as always to oppose the motion, and to diminish the total energy of the system. The dynamical theory of heat accounts for the energy that apparently disappears in non- conservative systems. Kinetic energy being defined as ^^mv2 is of the dimensions -„£- , the same as those of work. Potential energy is defined as work. The unit of energy is, therefore, the erg. 59. Particular case of Force-function. Newtonian Forces. In the particular case in which the only forces acting on the system are attractions or repulsions by the several particles directed along the lines joining them and depending only on their mutual distances, a force-function always exists. For let the force between two particles mr and ms at a distance apart rrs be It will be convenient to consider F positive if the force is a repulsion. Consider now the force Fs(r) acting on ms and acting in the direction FlG- 26> from mr to ms. Its direction cosines are those of the vector rrS) 112 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. Now since rf . = (ccs - #r)2 + ( ys - 2/r)2 + (z9 - zr)\ differentiating partially by or8, i-,K\ Srn_x,-xr drrs_ys-yr Srn _ze-zr fo»~ rn ' dy~s rn Szs~ rn and accordingly fa,' F.M dy,' F.v-di,,' If we put Urs such a function of rr^ that dU drr8 dUrsdr rsr8 dr drr8 Such forces are called Newtonian forces, the fra most familiar examples of which are the mutual attractions of the sun and the planets. Then , , mrms n mrms (30) (rn) = — j- , Un = - —— , (31) P.=- ^ ' IS '28 and the symmetrical function U will be u=-^\r^2 + ~ri3+., / x a! 32 t 3n ...... or more briefly r=l s=i Trs understanding that terms in which r = s are to be omitted. w. E. 8 114 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. The factor -J is introduced because in the above summation every term appears twice. But in V each pair of particles is to appear only once. If no constant be added to U as defined above, both it and the potential energy (33) F = i2A!^, ' will vanish when every rrs is infinite, that is when no two particles are within a finite distance of each other. This furnishes a con- venient zero configuration for the potential energy, and is the one generally adopted. We may accordingly define the potential energy of the system in any given configuration as the work that must be done against the mutual repulsions of the particles in order to bring them from a state of infinite dispersion to the given configuration. CHAPTER III. HAMILTON'S PRINCIPLE. GENERALIZED EQUATIONS OF MOTION. CYCLIC SYSTEMS. 60. Hamilton's Principle. If in d'Alembert's equation we consider &», Sy, §z variations consistent with the equations of condition, we have d?x -, _ d fdx £ \ dx d§x dt* ~dt(~didx~ _ d fdx ~\ dx §dx ~~di(dt )~di ~dt d (dx j, \ ., 1 /dso =Ba:- Treating each term in this manner, dy ~ dz s+ If there is a force function U we have hence the right-hand member of (1) is ST+SU. 8—2 116 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. The left-hand member being an exact derivative we may inte- grate with respect to t, Jt t0 If the positions are given for t0 and tlt that is if the variations So;, By, Bz vanish for t0 and ti, then the integrated parts vanish, and or (3) This is known as Hamilton s Principle *. It may be stated by saying that if the configuration of the system is given at two instants t0 and t1} then the value of the time-integral of T+ U is less (or greater) for the paths actually described in the natural motion than in any other infinitely near motion. Hamilton's principle is broader than the principle of energy, inasmuch as U may contain the time as well as the coordinates. It is true even for non-conservative systems (where a force- function U does not exist), if we write instead of 8 U 61. Lagrange's Generalized Equations. By means of Hamilton's Principle we may deduce the generalized equations of motion. Suppose that by means of the equations of condition, if there are any, we express all the coordinates as functions of m = 3n — k parameters qlt q.2, ... qm, which are known as the generalized coordinates of the system, asl = a;1(qlt 0a,... qm) Then TF, if the system is conservative, becomes a function of the parameters q. * Hamilton. On a General Method in Dynamics. Phil. Trans. 1834. 60, 61] HAMILTON'S PRINCIPLE. 117 Differentiating the above by t, daci _ ctai dqi 8#i d we get the same equations as before. If there is a force-function and _ ^ dw dqr' 61, 62] HAMILTON'S PKINCIPLE. 119 There are m of the equations (7), one for each coordinate q. These are Lagrange's equations of motion in generalized co- ordinates. 62. Proof independent of Hamilton's Principle. We will verify these equations by direct transformation of the equations in rectangular coordinates .1. 1 which are obtained from equation (15) of Chapter IT. by means of d'Alembert's principle. Multiplying these respectively by dxr dyr dzr dqs ' dqs ' dqs ' adding and summing for all values of r, the coefficient of Xx be- comes . r dqs dyr dqs £zr d If there are no relations between the q's, the expression <£i (?!>••• 2m) = ° is an identity, and all its partial derivatives -^ are equal to zero. vqs Accordingly the terms in Xl5 X2, ... disappear. We have then T r yr Z JLr= -- h I ro~" "r ^r oT~ I 9^8 dqs dq Now T= (12) 120 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. but by (i) *•-§••+!*•*•-+£•.•• '".. hence Differentiating #/ by qs (13) J^L — — ^L-q^. Inserting these values in ar 821 „ , , , + yr ,far\ ,d ftxr which, since is equal to V I V ^Xr V tyr 7 <^zr\ p *r \ r ^Qs r dq& r dqs) Hence we have proved by direct transformation the expression dt \dq8/ dqs to be equal to P8. 62, 63] HAMILTON'S PRINCIPLE. 121 7\T The derivative =—, , which i oqs the q"s, is generally denoted by The derivative =—. , which is a homogeneous linear function of oqs In the case of rectangular coordinates, the ^-component of the momentum of one of the particles. In general, pg may be called the generalized component of momentum, belonging to the coordinate qs and velocity qs'. The equations of motion may be written dp, 3T D dT = P or if, as we shall in future do, we denote by Ps simply that part of the impressed force which is not derived from the potential energy, under which are included all non- conservative forces, »> 63. Theorem on Reciprocal Functions. The ordinary notation for partial derivatives of functions of several variables sometimes gives rise to a certain confusion, from the lack of indication of what variables are to be considered as constant during the differentiation. For instance, suppose we have a function F of any number of variables, which for convenience we will divide into two classes, denoting them by the letters Suppose now we have n functions of these variables, given by the equations 122 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. Let us now consider the function 2...xn, zl9 zz ... zm) - By means of the equations (i) we may insert the values of the i/'s in terms of the a?'s and ^'s, so that G is explicitly given as a function of the variables x^...xn and z^...zm. On the other hand let us solve the equations ( I ) for the #'s obtaining #1 = & fyi> 2/2 ... 2/n, zlt z* ... sm) (2) ....................................... #n = 0n (2/1, 2/2 • • • 2/n, *i, ^2 ... *,,»), and by means of the latter let us insert in G the values of the a?'s in terms of the y's and z'a. Let the function in this form, that is, explicitly given as a function of y^...yn and z± . . . zm be de- noted by G. Then for all values of z*s and of #'s and ys com- patible with the equations (i) or (2), we have identically G(xl...scn, zlf..zm)= G(yl...yn) ^ ... zm). Differentiating both 6^ and G totally by varying all the variables that occur, we have n flf mdF n = 2 x- dxs + S ^- dz8 - 1 Oxs I VZs 1 1 but as these are identically equal, we get by transposing, In this equation there appear 2n + m differentials, only n + m of which are independent, in virtue of the equations ( i ), or their equivalents (2). The equation (3) assumes importance when we define the functions y in a particular way, namely as the partial derivatives of the original function F with regard to the variables x, dF 63] HAMILTON'S PRINCIPLE. 123 Then the coefficients of every dxs vanish, and since we may take the dy's and dz's arbitrarily, in order for the sum to vanish we must have for every dys and dzs, dG_ dF_dG ty.~ '*" dz.-**.' The function -G is called the reciprocal function to the function F with respect to the variables x^ ...tcn, fc>r we have the reciprocal relations or: — Two reciprocal functions have the property that the partial derivative of either with respect to any variable of reciprocation contained in it is equal to the corresponding variable replacing the original in the other function, whereas the partial derivative of one function with respect to any variable not of reciprocation is the negative of the derivative of the other function with respect to the same variable. In case the function F is homogeneous of degree /c in the variables of reciprocation *^i j *^2 > • • • ^n the theorem becomes more striking, for then, by Euler's theorem dF dF dF and the reciprocal function is simply a multiple of the original function. If the original function is of degree two, the reciprocal function is identically equal to the original function. We have thus a striking example of the remark made at the beginning of this section, for here the derivative of the function when expressed in one form by a variable z is exactly the negative of the derivative by the same variable of the function expressed in the other form. In this form the theorem will be frequently used hereafter. By means of it the equations of motion may be transformed from Lagrange's form to that given them by Hamilton. 124 THEORY OF NEWTO^TIAN FORCES. [PT. I. CH. III. 64. Hamilton's Transformation. We have seen, § 61 (3), that the kinetic energy is a homogeneous quadratic function of the variables q representing the velocities, the coefficients being functions of the coordinates q. If we call the reciprocal function with respect to the g"s, T, by the last section this is also the kinetic energy, expressed not in terms of the velocities, but of the momenta p. Any ps is a homogeneous linear function of the q"s, so that solving the equations dT ^ . ^ r\ i • • • + (Jmqn , (1) - 8— - 0 ' dqn for the q'*s, every q is a homogeneous linear function of the p's, and T is therefore a homogeneous quadratic function of the momenta p. By virtue of the two properties of the reciprocal function we have for every qs' (variable of reciprocation), and every qg (not of reciprocation), (2) 5~= — o~> dps dqs dqs so that Lagrange's equations, § 62 (17), are transformed to dps dT dW -n , dqg dT / ft\ _•* I . I TJ ry ' J.6 If we put H—T-\- W , this is the reciprocal function to the Lagrangian function and the equations take the nearly symmetrical form, dps = _dH p dqs = dH dt 'dqs dt dp8 ' These are Hamilton's equations of motion. From these equations we may immediately deduce the integral equation of energy. By cross-multiplication of the above equa- tions, after transposing and summing for all the coordinates, we get dqs 64, 65] HAMILTON'S PRINCIPLE. 125 But H is a function only of the ps and qs, so that the left-hand member is — j- ; and since H is equal to T + W it represents the (MI total energy. Also P8dqs is the work done by the external impressed force component Ps in the displacement dqS) so that the right- hand side is the time-rate at which the external forces do work on the system, or the activity of the external forces. The equation (6) Wl/ ]_ \AJ\J is accordingly sometimes called the Equation of Activity, while if there are no external forces, but only conservative ones, we have the equation of Conservation of Energy, 0, H=T+W= const. A case of frequent occurrence is that where there are non- conservative forces proportional to the first powers of the velocities q', so that any Ps=—tcsq8'. We may then form a function F which is also a homogeneous quadratic function of the velocities i dqs' ' and since in this case F represents one-half the time-rate of loss, or dissipation of energy. F is called the Dissipation Function. It was introduced by Lord Rayleigh *, and, like the other function used above, is of use in the theory of electric currents. 65. Transformation of Routh and Helmholtz. We •shall in general find Lagrange's form of the equations of motion more convenient than those of Hamilton. An intermediate form, introduced by Routh f, and afterwards by Helmholtz J, is of great importance. * Proceedings London Mathematical Society, June, 1873. t Bouth. Stability of a given State of Motion, p. 61. Rigid Dynamics, i. p. 318. £ Helmholtz. Ueber die physikalische Bedeutung des Princips der kleinsten Wirkung. Borchardt's Jour, fur Math. Bd. 100, 1886. Wissensch. Abh. in. p. 203. 126 THEORY OF NEWTONIAN FORCES. [PT I. CH. III. Suppose that instead of reciprocating with regard to all the velocities q as in Hamilton's transformation, we do so with regard to only a number r of them which we will choose so that they shall have the indices from 1 to r, while the q"s with indices r + 1, ... n, remain in the reciprocal function, and with all the coordinates q play the part of the variables z in § 63. Then calling the negative of the reciprocal function (1) T=T-i.qs'ps, 1 we have dT dT — = — , for s=l, 2, ... n, (2) fy* 9f« dT dT w=*Z'**8ssr+1""*' and dT , dT f -to *-&' -?.=^.fo"=i.2,...r. Replacing T in Lagrange's equations by T, we obtain d_(W\ ST_ dW '~- "* so that we may use for the suffixes corresponding to the un- eliminated velocities Lagrange's equations, using the function, 4> = T- W instead of the Lagrangian function L = T-W, and obtaining For the suffixes corresponding to the eliminated velocities we must use the Hamiltonian form of the equations (6) - dt s dq8 d(T-W) f --ff= v a -, for s=l, 2 ... r. dt dps If r = 7i, T becomes — T, and we have the complete Hamiltonian form, § 64 (4). 65, 66] HAMILTON'S PRINCIPLE. 127 The function is called by Routh the modified Lagrangian function, and on account of its importance has received from Helmholtz the special name of the Kinetic Potential, by which we shall designate it. (Helmholtz calls — the kinetic potential*.) It is to be noticed that the equations for the elimination of the velocities, the equations (2) of § 63 are now, instead of § 64 (i) Qliqi + Quqi .•• + Qirqr = Pi- Qir+iq'r+i ••• - Qmqn, Qnqi + Qrsq* ' • • + Qrrqr' = Pr - Qrr+i q'r+l • • • ~ Qm^n ', so that the q"s become linear functions of the right-hand sides of these equations and hence of Pi,p2, ... pr, q'r+i ••• qn'> thus T becomes a homogeneous quadratic function of P!...pr and q'r+i...qn', but is not homogeneous in either the p's or the V>'2 > • • • nJr by the determinant itself, are functions of the coordinates only, and since by hypothesis the function T did not contain the cyclic coordinates, the R's are functions of only the non-cyclic coordinates. The kinetic potential consequently is a function only of the non- cyclic coordinates and velocities, but on account of the presence of the constants cs, it is not a homogeneous function of the velocities, but contains a linear function of them, as was remarked in § 65. Cases in which the kinetic potential contains a linear function of the velocities may thus be considered as cases with concealed motions. A case of this nature will be found in considering the mutual actions of magnets and electric currents. Physically the difference between the two cases is that while if <& contains only terms of the second degree in the velocities, if every velocity is reversed the kinetic potential is unchanged, and hence the motion may be reversed without change of circumstances, but if on the other hand there are terms of the first degree in the velocities, the motion cannot be reversed unless the concealed motions are reversed as well. As an example we will take the case of a gyrostat hung in gimbals. Let the outer ring of the gimbals A, Fig. 27, be pivoted on a vertical axis, and let the angle made by the plane of the ring with a fixed vertical plane be i/r. Let the inner ring B be pivoted on a horizontal axis, and let its plane make an angle 6 with the plane of the outer ring. The gyrostat is pivoted on an axis at right angles with the last, and let a fixed radius of the gyrostat make an angle <£ with the plane of the inner ring. It is shown in the theory of the dynamics of a rigid body that the energy of a body revolving about an axis is one-half the product w. E. 9 130 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. of a constant called the moment of inertia of the body multiplied by the square of its angular velocity, and also that if we find the FIG. 27. angular velocities about three mutually perpendicular axes of symmetry the energy may be found by adding the three parts obtained for the energy of rotation about the three axes. We will resolve the motions of the gyrostat into three angular velo- cities, about the axis of the top, the axis of the inner ring, and an axis perpendicular to both. About the axis of the gyrostat the angular velocity is -^- = ', but there is also the angular cut velocity -J = -v/r' about the vertical axis, which has the com- Cut ponent ty' cos 6 about the axis of the gyrostat. The velocity 7/J about the second axis is -7- = 6', and about the third is the other dt component of the velocity about the vertical, ijr sin 0. If A is the moment of inertia of the gyrostat about its own axis, B that about either of the other two, we have for the kinetic energy T = J [A (

and ty are cyclic coordinates. For the components of the forces tending to increase T/T, 0, cf>, 67] CYCLIC SYSTEMS. 131 *- = \4 (*' + *' cos 0) cos + A (' + y cos 0) sin 0 - .Bip sin d cos 0, If there is no force tending to change the rotation of the gyrostat in its ring P*=0, 4(f + ^r'cos0) = c, and eliminating = T-c0' = -|^ + ^ £ (<9'2 + i|r' 2sin20) + c^' cos 0. the last term containing ^r' in the first power. Using this form of 4> to determine the forces, we obtain ' - ^'2 sin ^ cos 0 + of sin A The influence of the cyclic motion may be most simply shown if the vertical ring be held fixed. Then -fy = const., and i/r7 = 0, = — c sm V -r- , at Spinning the inner ring about the horizontal axis requires the same force whether the cyclic motion exists or not, whereas a force is developed tending to make the vertical ring revolve about its rlf) axis, which must be balanced by the force — c sin 6 -j- . This force at once shows that there is a concealed motion, even if the disposition of the concealed parts be unknown. This is exem- plified in the gyroscopic pendulum, which is simply a pendulum with two degrees of freedom, containing a gyrostat whose axis is 9—2 132 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. rigidly fixed in the axis of the pendulum. An ordinary pendulum set vibrating in a plane continues to vibrate in a plane, with a periodic reversal of its motion. The gyroscopic pendulum on the other hand describes a curious looped surface, never remaining in a plane nor returning on its course. This example is worked out in Thomson and Tait's Natural Philosophy, § 319, Example (D). 68. Cyclic Systems. A system in which the kinetic energy is represented with sufficient approximation by a homogeneous quadratic function of its cyclic velocities is called a Cyclic System. Of course the rigid expression of the kinetic energy contains the velocities of every coordinate of the system, cyclic or not, for no mass can be moved without adding a certain amount of kinetic energy. Still if certain of the coordinates change so slowly that their velocities may be neglected in comparison with the velocities of the cyclic coordinates, the approximate condition will be ful- filled. These coordinates define the position of the cyclic systems, and may be called the positional coordinates or parameters of the system. In the case of the gyrostat the two coordinates of the gimbal rings may be taken for the positional coordinates, while the cyclic coordinate determines the rotation of the gyrostat. In the case of a liquid circulating through an endless rubber tube, the positional co-ordinates would specify the shape and position of the tube. The positional coordinates will be distinguished from the cyclic coordinates by not being marked with a bar. The analytical conditions for a cyclic system will accordingly be, for all coordinates, either £)/TF £)'7T <'> afT° or Wrpi=Q' or if we use the Hamiltonian form of T obtained by replacing the velocities by the momenta, which we shall denote by Tp, since the non-cyclic momenta vanish (2} ^p pV = _p=0. dps dq* dqs We accordingly have for the external forces tending to in- crease the positional coordinates [see § 62, (17)], d(T-W)_d_(T,+ - "~ 67, 68] CYCLIC SYSTEMS. 133 and for the cyclic coordinates v TJ dT , . . dT / \ T» dTp , . - , applying the principle that a derivative by two variables is inde- pendent of the order of the differentiations we obtain six reciprocal theorems. We shall throughout suppose that there is no potential energy. I a. In an adiabatic motion if an increase in one positional coordinate qr causes an increase in the impressed force Ps belong- ing to another positional coordinate qs at a certain rate, then an increase in the positional coordinate qs causes an increase in the impressed force Pr at the same rate. For dqr I 6. In an isocyclic motion we have the same property as above. For (6) dqr dqrdq8 dqs ' II a. If in any motion an increase of any cyclic momentum pr, the positional coordinates being unchanged, causes an increase in a cyclic velocity qs' at a certain rate, then an increase in the momentum ps, the positional coordinates being unchanged, causes an increase in the velocity qr' at the same rate. For dpr dp$ps dps ' II 6. If in any motion an increase in any cyclic velocity qr', the positional coordinates being unchanged, causes an increase in a cyclic momentum ps, then an increase in the velocity qs' causes 68 — 70] CYCLIC SYSTEMS. 135 an increase in the momentum pr at the same rate. For dp, _ _&r_ _dp,. '~'-'- III a. If an increase in one of the cyclic momenta pr, the positional coordinates being unchanged, causes an increase in the impressed force Ps necessary to be applied to one of the positional coordinates qs (in order to prevent its changing), then an adiabatic increase of the positional coordinate qs will cause the cyclic velocity qr' to increase at the same rate. For _ dpr dprdqs dqs ' III b. If an increase in one of the cyclic velocities §/, the positional coordinates being unchanged, causes an increase in the impressed force Ps necessary to be applied to one of the positional coordinates q8 (in order to prevent its changing), then an isocyclis increase of the positional coordinate qs will cause the cyclic momentum pr to decrease at the same rate. For 70. Work done by the cyclic and positional forces. I. In an isocyclic motion, the work done by the cyclic forces is double the work done by the system against the positional forces. In such motions the energy of the system accordingly increases by one-half the work done by the cyclic forces, the other half being given out against the positional forces. For if we use the energy in the form we have in any change (1) ST= and in an isocyclic change, every 8^/ vanishing, (2) 8r=iS.5.'8p.. But since (3) ^ = P., &P, - P.$t, and since ?.'=§, 5/S* = Sqs, and the above expression for the gain of energy becomes (4) ST = ^s 136 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. But the work done by the cyclic forces is (5) SA = 28PsSqs=2ST. Hence the last part of the theorem is proved. Again, in any motion (6) sr=ss§85; + 2s|V and in an isocyclic motion (7) ' 8^|%- . ;, But since the work of the positional forces is (8) $A = 2sPs$qs = -^~Sqs = ~ ZT, uQs the first part of the proposition is also proved. II. In an adiabatic motion, the cyclic velocities will in general be changed. Then they change in such a way that the positional forces caused by the change of cyclic velocities oppose the motion, that is, do a positive amount of work. For since for any positional force P dl "0?,' the change due to the motion is dST ^ *s — ~ "^ — = — Zr — x — cqr — 2,r x — ^—JT- oqr . dq8 dqsdqr J dqsc)qr J Of this the part due to the change in the cyclic velocities is <> T-, 0. The interpretation of this theorem for electrodynamics is known as Lenz's Law*. 71. Examples of Cyclic Systems. The expression for the kinetic energy of the gyrostat worked out in § 67 shows that the system fulfils the conditions for a cyclic system if the velocity 6' is small enough to be neglected in comparison with the other velocities. The forces acting have been already found, and we can easily verify the theorems of the last two articles for this case. A very simple case of a cyclic system is that of a mass m sliding on a horizontal rod, revolving about a vertical axis. Let us consider the mass concentrated at a single point m at a distance r from the axis. Let the angle made by the rod with a fixed hori- zontal line be , then the velocity perpendicular to the rod is rty. The velocity along the rod Fm- 28- being r' ' , the kinetic energy of the body m is If we suppose the motion along the rod to be so slow that we may neglect r'2 T= Jrnry2, and the system is cyclic, r is the positional, the cyclic co- ordinate. A system containing a single cyclic coordinate is called by Helmholtz a monocyclic system. We have for the momenta 0/77 O/7T * = 8? = 0. ^=^, * These Theorems are all given by Hertz, Principien der Mechanik, §§ 568-583. 138 THEORY OF NEWTONIAN FORCES. [FT. I. CH. III. and introducing these instead of the velocities We have for the positional force ar .,2 dTp Pr = — ^- = - mr62 = -~* = dr dr This being negative denotes that a force Pr toward the axis must be impressed on the mass m in order to maintain the cyclic state. This may be accomplished by means of a geometrical constraint, or by means of a spring. The force or reaction — Pr which the mass m exerts in the direction from the axis in virtue of the rotation is called the centrifugal force. We see that if the motion is isocyclic, the positional force increases with r, while if it is adiabatic, it decreases when r increases. The verification of the theorems of § 69 is obvious. The cyclic force dt dt vanishes when the rotation is uniform, and the radius constant. If, the motion being isocyclic, that is, one of uniform angular velocity, the body moves farther from the axis, P^, the cyclic force is positive, that is, unless a positive force P$ is applied, the angular velocity will diminish. In moving out from rt to r2 work will be done against the positional force Pr of amount /***2 Cr2 -A = - Prdr = m^ rdr = Jfl J T! while the energy increases by the same amount. Thus the first theorem of § 70 is verified. If the motion is adiabatic, Pt = rar2' = c. If the body move from the axis, <£' will accordingly decrease. The change in Pr due to a displacement Sr is which, being of the same sign as Sr, does a positive amount of work in the displacement, illustrating the second theorem of §70. 71] CYCLIC SYSTEMS. 139 Dicydic Systems. The preceding example will suffice as a mechanical model to illustrate the phenomenon of self-induction of an electric current (Chapter XII). To illustrate mutual in- duction we must have at least two cyclic coordinates. Such models have been proposed by Maxwell, Lord Rayleigh*, Boltzmann, J. J. Thomson f, and the author J. In the model of J. J. Thomson, there are two carriages of mass m^ and w2 sliding on parallel rails, Fig. 29, their distances from a fixed line perpendicular to the rails being x± and a?2. Sliding in swivels on the carriages is a bar, on which is a third mass ra3. We shall suppose that this mass is movable along the bar, and is at a distance y from the line midway between the rails, y being positive when ra3 is nearer w2. Then, if d is the distance between the rails, and the kinetic energy is, if we may neglect y in comparison with #/, a?a', T = 4- The system is cyclic, y being the positional, x^ and #2 the cyclic coordinates. The positional force y i \ i r y d2+2d)~ X*Xz d2 vanishes if #/ = x*. The cyclic forces are /I y S (4 +S * PMi. Mag., July 1890, p. 30. t Elements of Mathematical Theory of Electricity and Magnetism, p. 385. I Science, Dec. 13, 1895. 140 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. Suppose that the coordinate y and the velocity a?a' are constant. If now #/ is increased, say by raj starting from rest and moving to the right by the application of a positive force PXi, then P^ is positive if y\< d/Z and m3 is within the rails, — in other words, unless a force to the right is impressed on mz also, #2' will diminish, and if a?/ was also zero, ra2 will move to the left. The force PXz must be greater the smaller \y\. This is the analogue of the induction of currents. Similar effects may be produced by moving m3 along the rod, instead of applying a force to ml or ra2. FIG. 30. Maxwell's model, which undoubtedly suggested Thomson's, •differs from it only in having motion of rotation instead of *eve- -, so that there is no limit to the possible difference in the coordinates sclt a?2. The independent masses are represented by the moments of inertia of masses m1? m2 carried by two shafts Slt $2, Fig. 30, each of which carries a bevel-gear wheel A, B at one end. Engaging these is a pair of bevel-gears C running loosely upon a third perpendicular shaft, carrying the inter- mediate mass, ra3. If all the bevel-gears are of the same diameter, and fa, 2, 3 are the angles made by the three horizontal rods with a fixed horizontal line, then it is evident, since the velocity of the centre of the wheel C is a mean between the velocities of its highest and lowest points, which have respectively the velocities of the rims of the wheels A and B, that + '• 71] CYCLIC SYSTEMS. 141 Consequently the kinetic energy of the system consisting of the three masses ml, w2, m3 at distances from the axis r1} r2, rs is T = if the velocities / can be neglected. The system is cyclic, the r's being positional, the <£'s being cyclic coordinates. In order to make the model a more complete representation of two electric currents, Boltzmann modified it so as to have between the co- ordinates rlt r2, rs the relation r? + rf=yf, r* + r/ = yf, where yly yz are two independent parameters. The two masses m1} m2 are chosen equal, being made one-fourth of m3. The expression for the energy then becomes and we may independently change either of the three co- efficients. The Pythagorean theorem suggests a geometrical means of imposing the above constraints. To each of the masses m is attached a string, which runs along the rod to the axis of ro- tation, where, after passing round a pulley it is carried vertically downward to be attached to the following device (Fig. 31). A FIG. 31. pair of rods are articulated at (7, the point of articulation being made to slide in a vertical line CO. The string from m3 is fastened to the point C. Sliding on a horizontal line AB and in slots in the rods AC, BC, are the points of attachment of the strings from m± and m2, which are then carried outward and upward 142 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. over pulleys. The lengths of the strings being chosen so that m3 is at the axis when the rods are horizontal, m^ and m2 when the rods are vertical, we must have, if AC = yl} EG = yz , For the actual construction of the model, the reader is referred to Boltzmann, Vorlesungen uber die MaxweWscTie Theorie der Electricitat und des Lichtes. By means of these models all the properties of Cyclic Systems may be illustrated, and all the phenomena of induction of currents imitated, as will be described in Chapter XII. 72. Hamilton's Principle the most general dynamical principle. We have seen in this chapter how by means of Hamilton's Principle we may deduce the general equations of motion, and from these the principle of Conservation of Energy. As Hamilton's Principle holds whether the system is conservative or not, it is more general than the principle of Conservation of Energy, which it includes. The principle of energy is not sufficient to deduce the equations of motion. If we know the Lagrangian function we can at once form the equations of motion, and without forming them we may find the energy. For we have L = T-W, E=T+W. Accordingly so that the energy is given in terms of L and its partial deri- vatives. If on the other hand the energy is given as a function of the coordinates and velocities, the Lagrangian function must be found by integrating the above partial differential equation, involving an arbitrary function. In fact if F be a homogeneous linear function of the velocities, the above equation will be satisfied not only by L but also by L + F. For, F being homogeneous, 71, 72] HAMILTON'S PRINCIPLE. 143 Consequently a knowledge of the energy is not sufficient to find the motion, while a knowledge of the Lagrangian function or kinetic potential is. In case we wish to ignore some of the coordinates we may modify the statement of Hamilton's Principle by the use of the modified Lagrangian function and put where we suppose only those coordinates which are not ignored are varied. CHAPTER IV. •s NEWTONIAN POTENTIAL FUNCTION. 73. Definition and fundamental properties of Poten- tial. We have seen in § 59, (29), (31), that if we have any number of material particles m repelling according to the New- tonian Law of the inverse square of the distance, the function TT _ g 2s MS — — 1 — — r ...... -\ r, r where r1} r2 ...... rn are the distances from the repelling points, is the force-function for all the forces acting upon the particle ms . If we put the mass ms equal to unity the function (i) is called the potential function of the field of force due to the repulsions of the particles mlt m2 ...... mn, and its negative vector parameter is the strength of the field, that is, the force experienced by unit of mass concentrated at the point in question. Since any term - - possesses the same properties as the function - , § 39, we have for every term, for points where r is not equal to zero, A f-J = 0, and consequently (2) AF-W.A (1) + m2A (I Vl/ V 2 = 0. 74. Potential of Continuous Distribution. Suppose now that the repelling masses, instead of being in discrete points, form a continuously extended body K. Let the limit of the ratio of the mass to the volume of any infinitely small part be p = lim — - , which is called the density. AT=O AT 73, 74] NEWTONIAN POTENTIAL FUNCTION. 145 Let the coordinates of a point in the repelling or attracting* body be a, b, c. The potential at any point P, x, y, z, due to the mass dm at Q, a, b, c, is FIG. 32. where r is the distance of the point x, y, z from the repelling point at a, b, c. The whole potential at x, y, z is the sum of that due to all parts of the attracting body, or the volume integral Now we have dm = pdr, or in rectangular coordinates dr = da db dc, dm = p da db dc. If the body is not homogeneous, p is different in different parts of the body K, and is a function of a, b, c, continuous or discon- tinuous (e.g. a hole would cause a discontinuity). Since (A) V = [([ — = [([ pdadbdc JjJK r JJ]K J^-a^+ty-by + ^-c)*' For every point x, y, z, V has a single, definite value. It is accordingly a uniform function of the point P, x, y, z. It may be differentiated in any direction, we may find its level surfaces, its first differential parameter, whose negative is equal to the whole action of K on a point of unit mass, and the lines of force, normal to the level, or equipotential surfaces. * In order to save words, and to conform to ordinary usage, we shall say simply attracting, for a negative repulsion is an attraction. W. E. 10 146 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. If for any point so, y, z outside K, r^ is the shortest distance to any point of K and r2 the greatest distance, we have for any point r2 > r > rlt 111 — < -<— , r2 r T! dm dm dm — < — < — ; r2 r r-j dm dm f7 dm Since ^ and r2 are constant Now since 1 1 1 dm = M} the whole mass of the body K, the above is M Jr M (6) — 2 ^2 ». 2 ' /2 T TI p / \ p p — -1-: cos (r#) > — - cos (rx) > — — cos (rx). r2z r2 n2 Multiplying and dividing the outside terms by cos .4 and integrating, (5) -^ Multiplying by jR2 and letting R increase without limit, since ,. jR2 ,. R2 ,. cos(?^) lim — = lim — = lim — ^-j-7 = 1, -R=c° _ _ 10—2 148 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. (6) li Therefore the first derivatives of V, and hence the parameter, vanish at infinity to the second order. In like manner for the second derivatives Every element in all the integrals discussed is finite, unless r = 0, hence all the integrals are finite. We might proceed in this manner, and should thus find that : At points not in the attracting masses, T^and all its derivatives are finite and (since their derivatives are finite) continuous, as well as uniform. Also since (z - cf - we have by addition .,, that is, V satisfies Laplace's equation. This is also proved by applying Gauss's theorem (§ 39 (4)) to each element — . r 76. Points in the Attracting Mass. Let us now examine the potential and its derivatives at points in the substance of the attracting mass. 75, 76] NEWTONIAN POTENTIAL FUNCTION. 149 If P is within the mass, the element - - at which the point Q, where dm is placed, coincides with P, becomes infinite. It does not therefore follow that the integral becomes infinite (§ 25). FIG. 33. Let us separate from the mass K a small sphere of radius e with the centre at P. Call the part of the body within this sphere K' and the rest K". Call the part of the integral due to K', V, and that due to K" ', F'. Now since P is not in the mass K" ', V" and its derivatives are finite at P, and we have only to examine V and its derivatives. Let us insert polar coordinates -E-rjx^" * * 27r sin 6 cos 0 | r* c* f27r I drl I I which also vanishes with e. Hence ^— is everywhere finite, and in ox dv dV like manner — , — . If we attempt this process for the second derivatives ^ , . . . it fails on account of — , which gives a logarithm becoming oo in the limit. dV We will give another proof of the finiteness of — . ox We have (3) ^=11^^ dadbdc which by Green's theorem is equal to This is however only to be applied in case the function - is everywhere finite and continuous. This ceases to be the case when P is in the attracting mass, hence we must exclude P by drawing a small sphere about it. Applying Green's theorem to the rest of the space K'1 ', we have to add to the surface-integral the integral over the surface of the small sphere. /* f ^7^f Since cos(nx) £ 1, this is not greater than p^l I — = 4<7repm, which vanishes with e. Hence the infinite element contributes nothing to the integral. dV In the same way that — was proved finite, it may be proved ox dV' dV" continuous. Dividing it into two parts ^— and ^ — , of which the dx dx 76, 77] NEWTONIAN POTENTIAL FUNCTION. 151 dV second is continuous, we may make, as shown, -= — as small as we ox please by making the sphere at P small enough. At a neighbour- ing point P! draw a small sphere, and let the corresponding parts , W{ , dVS ™ , dVS be ^— - and -—= . Then we can make -r— ^ as small as we please, ox ox ox •~\ ~rrr o -rr / and hence also the difference -= --- ^-. Hence by taking P and Pj near enough together, we can make the increment of - - as ox ' dv. small as we please, or ^r— is continuous. dx 77. Poisson's Equation. By Gauss's theorem (§ 39 (5)), we have when r is drawn from 0, a point within 8. Multiplying by m, a mass concentrated at 0, (i) j)-2cos(nr)dS = The integral I I x7O / / D / D \ ^7C* — II T — a& = — 1 1 JL cos (.1/2^ c&o J J Ufl J J is the surface integral of the outward normal component of the parameter P, or of the inward component of the force. The surface integral of the normal component of force in the inward direction through $ is called the flux of force into S, and we see that it is equal to — 4?r times the element of mass within S. Masses without contribute nothing to the integral. Every mass dm situated within S contributes — to the potential at any point and — kirdm to the flux through the surface 8. Hence the whole mass, when potential is V—\\\ — , contributes to the flux pdr, and 152 THEORY OF NEWTONIAN FORCES. [FT. I. CH. IV. Now the surface integral is, by the divergence theorem, equal to (3) The surface S may be drawn inside the attracting mass, providing that we take for the potential only that due to matter in the space r within S. Accordingly for r we may take any part whatever of the attracting mass, and (4) JjJ(AF+ As the above theorem applies to any field of integration what- ever, we must have everywhere (by § 23) (5) AF+4wy> = 0. This is Poisson's extension of Laplace's equation, and says that at any point the second differential parameter of V is equal to — 4-7T times the density at that point. Outside the attracting bodies, where p = 0, this becomes Laplace's equation. In our nomenclature, the concentration of the potential at any point is proportional to the density at that point. A more elementary proof of the same theorem may be given as follows. At a point a?, y, z construct a small rectangular parallelepiped whose faces have the coordinates and find the flux of force through its six faces. At the face normal to the X-axis whose x coordinate is x let the mean value of the force be — -x— = — Px- ox The area of the face is 97 f, so that this face contributes to the integral — 1 1 P cos (Pri) dS the amount — ^— 17 f. At the opposite face, since — is continuous, we have for its ox value dV .9 /9F\ ,_. , , . ,. - — h cs-'i ^— + terms of higher order in f, 77, 78] NEWTONIAN POTENTIAL FUNCTION. 153 and hence, the normal being directed the other way, this side con- tributes to the integral the amount dV 3 idV\ . and the two together * + terms of higher order. 32F Similarly the faces perpendicular to F-axis contribute £77 f ~— , and the others £tf?-;rr . d^" Hence the surface integral is and by Gauss's theorem this is equal to — 4?rm = — where p is the mean density in the parallelepiped. Now making the parallelepiped infinitely small, and dividing by £77 f, we get AF=-47T/!>. 78. Abbreviations for Operators. If p is any point function, the potential function at any point due to a distribu- tion through all space of matter whose density at any point is p has been denoted by Gibbs and Heaviside by the abbreviation Pot p} standing for the definite integral **,-/// P-dr. (The suffix oo denotes integration through all space.) We may thus abbreviate Poisson's equation - A Pot p = V2 Pot p = 4-777) , so that the operation Pot followed by the operation — A = V2, performed on any scalar function, has the effect only of multiply- ing it by 47T, or the operations A and — -.- are the inverses of each other. 154 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 79. Characteristics of Potential Function. We have now found the following properties of the potential function. 1st. It is everywhere holomorphic, that is, uniform, finite, continuous. 2nd. Its first partial derivatives are everywhere holomorphic. 3rd. Its second derivatives are finite. 4th. V vanishes at infinity to the first order, lim(JKF)«Jf; .8=00 -*-,... vanish to second order, das 1 • I T"fcf> " " \ hm LR83--) = - R=oo V OX 5th. V satisfies everywhere Poisson's differential equation 82F 82F ++ = ~ p) and outside of attracting matter, Laplace's equation | 82F | = dec2 dy2 dz° Any function having all these properties is a Newtonian potential function. The force X, F, Z is a solenoidal vector at all points outside of the attracting bodies, and hence if we construct tubes of force, the flax of force is constant through any cross-section of a given tube. A tube for which the flux is unity will be called a unit tube. The conception of lines of force and of the solenoidal property is due to Faraday. Since V is a harmonic function outside of the attracting bodies, it has neither maximum nor minimum in free space, but its maximum and minimum must lie within the attracting bodies or at infinity. In the attracting bodies the equation — AF=477y> says that the concentration of the potential at, or the divergence of the force from any point is proportional to the density at that point. 79, 80] NEWTONIAN POTENTIAL FUNCTION. 155 SO. Examples. Potential of a homogeneous Sphere. Let the radius of the sphere be JK, h the distance of P from its center, r-JT/J*. • . Let us put s instead of r, using the latter symbol for the polar coordinate, Now s2 = A2 4- r2 — 2hr cos 6. FIG. 34. Differentiating, keeping r constant, sds = hr sin 0d0, and introducing s as variable instead of 0, We must integrate first with respect to s from h — r to h + r, if P is external ; h-r M 3h ~ h' Hence the attraction of a sphere upon an external point is the same as if the whole mass were concentrated at the oenter. A body having the property that the line of direction of its resultant attraction on a point passes always through a fixed point in the body is called centrobaric. If instead of a whole sphere we consider a spherical shell of internal radius R^ and outer R2, the limits for r being Rlf R2, 156 THEORY OF NEWTONIAN FORCES. [PT. I. OH. IV. dV M Wehave ~dh=-¥> _ dh* ~ hs ' If, on the other hand, P is in the spherical cavity, h the limits for s are r — h, r+h _ V = . , rdrds f ^ = 4?rp I JE, rdr which is independent of h, that is, is constant in the whole cavity. Hence ^j- — Q> and we get the theorem that a homogeneous spherical shell exercises no force on a body within. (On account of symmetry the force can be only radial.) If P is in the substance of the shell, we divide the shell into two by a spherical surface passing through P, find the potential due to the part within P, and add it to that without, getting (Bf - dV_4nrp{Rf } dh " 3 U2 J ' d*V_ iirppRf . ,| dh,* ~ ' 3 1 A» j ' Tabulating these results dV dh d^V dh* 2,7rp (R* - 0 0 c"r / T> 3 7? 3\ OL V-"^ ~ -"1 / 3_ 80, 81] NEWTONIAN POTENTIAL FUNCTION. 157 Plotting the above results (Fig. 35) shows the continuity of FIG. 35. V and its first derivatives and the discontinuity of the second derivatives at the surfaces of the attracting mass. We see that the attraction of a solid sphere at a point within it is proportional to the distance from the center, for if R^ — 0, d V 4f7rph dh= ~3~' and is independent of the radius of the sphere. Hence experi- ments on the decrease of the force of gravity in mines at known depths might give us the dimensions of the earth. 81. Disc, Cylinder, Cone. Let us find the attraction of a circular disc of infinitesimal thickness at a point on a line normal FIG. 36. 158 THEORY OF NEWTONIAN FORCES. [FT. I. CH. IV. to the disc at its center. Let the radius be Rlt thickness 6, distance of P from the center h erdrd(f) o Jo Jo Jo f* rdr rn 1 , = 2?rep V h2 + r2 J o V h2 4- r2 _ _ = 2-Trep •! .. — 1 [• . r + .R2 Attraction of circular cylinder on point in its aods. Let the length be I and let the point be external, at a distance h from the center. By the above, a disc of thickness dx at a distance x from the center produces a potential at P - xf -(h- x)}. Hence the whole is fl J - * \ 27T/3 -i^-a- ^2 + (^ - «)2 + y log (» - Circular cone on point in axis. Let R be the radius of base, a the altitude, h the height of P above the vertex. A disc at distance x below vertex and radius r causes potential at P, dV= Zirpdx {(£ + xj + r2 - (h 81, 82] and NEWTONIAN POTENTIAL FUNCTION. 159 R - • a R r = — a If we have a conical mountain of uniform density on the earth, and determine the force of gravity at its summit and at the sea level, this gives us the ratio of the attraction of the sphere and cone to that of the sphere alone, and from this we get the ratio of the mass of the earth to the mass of the mountain. Such a deter- mination was carried out by Mendenhall, on Fujiyama, Japan, in 1880, giving 5'77 for the earth's density. FIG. 37. Circular disc on point not on axis. Let the coordinates of P with respect to the center be a, 6, 0. Then =/T J 0 J Q erdrd(t> o Va2 + (6 — r cos $)2 + r2 sin2 <£ ' an elliptic integral. The development in an infinite series will be given in § 102. 82. Surface Distributions. In the case of the circular disc of thickness e, ep is the amount of matter per unit of surface of the disc. It is often convenient to consider distributions of matter over surfaces, in such a manner that though e be considered infinitesimal p increases so that the product ep remains finite. The product ep — o- is called the surface density, and the distribu- tion is called a surface distribution. We have dS 160 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. In the case of the disc, we had 8F ^ | h_ ) When h = 0 we have '8F\ The repulsion of a disc upon a particle in contact with it at its center is independent of the radius of the disc, and is equal to 2?r times the surface density. Ff FIG. 38. If the force on a particle in contact on the right be called F2, positive if to the right, we have ^2^4- 2-7TO-. By symmetry, the force on a particle at the left in contact with the disc is F! = - 27TC7, F^-FZ = - 47T<7. Now if x denote the direction of the normal to the right, and we see that on passing through the surface there is a dis- continuity in the value of ^— of the magnitude 47ror. ox Consider a thin spherical shell. We have for an external point V = i£ (JV - Sf) * 7ro: 83. Greenes formulae. Let us apply Green's theorem to two functions, of which one, F, is the potential function due to any distribution of matter, and the other, U = - , where r is the r distance from a fixed point P, lying in the space r over which we take the integral. Let the space r concerned be that bounded by a closed surface S, a small sphere 2 of radius e about P, and, if P is wi£hsttt 8, a sphere of infinite radius with center P. FIG. 40. Now the theorem was stated in § 33 (2) for the normal drawn in toward r, which means outward from S and S, and inward from the infinite sphere, as and since 82, 83] NEWTONIAN POTENTIAL FUNCTION. 163 in the whole space r, so that (i) becomes The surface integrals are to be taken over S, over the small sphere, and over the infinite sphere. For a sphere with center at P, a! a1 the upper or lower sign being taken according as the sphere is the inner or outer boundary of T ; and for r = V vanishes, hence this integral vanishes. Also rriaF^e Vr18F^ /raF,7 (3) — 1 1 - -5- dtf = — I - ^- f2aft> = - r K- c?o>. Jj r 3n JJ r 3?^ JJ 9n Now at infinity, — is of order — , and being multiplied by r, on T still vanishes. Accordingly the infinite sphere contributes nothing. For the small sphere the case is different. The first integral becomes, as the radius e of the sphere diminishes, (4) -FPjTdft> = -47rFp. The second part [[dV -ell^-do), JJ dn however, since 5— is finite in the sphere, vanishes with e. Hence on there remain on the left side of the equation only — 4-TrFp and the integral over $. We obtain therefore II 11—2 164 THEOEY OF NEWTONIAN FORCES. [PT. I. CH. IV. the normal being drawn outward from $. This formula is due to Green. Hence we see that any function which is uniform and con- tinuous everywhere outside of a certain closed surface, vanishes at infinity to the first order, and whose parameter vanishes at infinity to the second order, is determined at every point of space considered if we know at every point of that space the value of the second differential parameter, and in addition the values on the surface S of the function and its vector parameter resolved in the direction of the outer normal. In particular, if V is harmonic in all the space considered, we have (6) and a harmonic function is determined everywhere by its values and those of its normal component of parameter at all points of the surface S. Since •I r 1 or dn r'2 dn If x dr .dr , , dr} cos(nr) = - — jcos (nx) -z- + cos (ny) — 4- cos (nz) g^ f = -£ — - , we may write (6) p 4ir JJ V^2 r dn) Consequently, we may produce at all points outside of a closed surface 8 the same field of force as is produced by any distribution of masses lying inside of S, whose potential is F", if we distribute over the surface S a surface distribution of surface-density, In the general expression (5), the surface integral representing the potential due to the masses tuithin S, the volume integral AF, - dr 83, 84] NEWTONIAN POTENTIAL FUNCTION. 165 represents, since everywhere J_ ~4^ that is, the potential due to all the masses in the region r, viz., outside S. 84. Equipotential Layers. As a still more particular case of (7), if the surface S is taken as one of the equipotential surfaces of the internal distribution, we have all over the surface V= Vs = const., and the constant may be taken out from the first integral, r 47rJJ r Now by Gauss's theorem 1 1 - -^ — • d$ = 0 ; accordingly, r n r so that VP is represented as the potential of a surface distribution of surface-density 4?r dn 4?r The whole mass of the equivalent surface distribution is ***— *8 a which, being the flux of force outward from S, is by Gauss's theorem, § 77 (i), equal to M, the mass within £ Accordingly we may enunciate the theorem, due to Chasles and Gauss* : — We may produce outside any equipotential surface of a dis- tribution M the same effect as the distribution itself produces, by * Chasles, " Sur 1' attraction d'une couche ellipsoidale infiniment mince." Journ. EC. Polytec., Cahier 25, p. 266, 1837 ; Gauss, Allgemeine Lehrsatze, § 36. 166 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. distributing over that surface a layer of surface-density equal to -: times the outward force at every point of the surface. The mass of the whole layer will be precisely that of the original internal distribution. Such a layer is called an equipotential layer. (Definition — A superficial layer which coincides with one of its own equipotential surfaces.) Reversing the sign of this density will give us a layer which will, outside, neutralize the effect of the bodies within. Let us now suppose the point P is within S. In this case, we apply Green's theorem to the space within S, and we do not have the integrals over the infinite sphere. The normal in the above formulae is now drawn inward from S, or if we still wish to use the outward normal, we change the sign of the surface integral in (5), 1 ff I irdr 1 9M j'0 1 fffAP, (12) K = --T— V F- --- ^— JdS --:—-- dr, 4<7rJJ s \ dne r dne/ 4>7rJJJ r (P inside S). Note that both formulae (5) and (12) are identical if the normal is drawn into the space in which P lies. Hence within a closed surface a holomorphic function is determined at every point solely by its values and those of its normally resolved parameter at all points of the surface, and by the values of its second parameter at all points in the space within the surface. A harmonic function may be represented by a potential func- tion of a surface distribution. Now if the surface S is equipotential, the function V cannot be harmonic everywhere within unless it is constant. For since two equipotential surfaces cannot cut each other, the potential being a one-valued function, successive equipotential surfaces must surround each other, and the innermost one, which is reduced to a point, will be a point of maximum or minimum. But we have seen (§ 34) that this is impossible for a harmonic function. Accordingly a function constant on a closed surface and harmonic within must be a constant. If however there be matter within and without S, the volume integral, as before, denotes the potential due to the matter in the 84, 85] NEWTONIAN POTENTIAL FUNCTION. 167 space r (within S), and the surface integral that due to the matter without. If the surface is equipotential, the surface integral is 19F The first integral is now equal to 4?r, so that Vs being constant contributes nothing to the derivatives of V, so that the outside bodies may be replaced by a surface layer of density 04) o- = ~~-- . ~ 4?r The mass of the surface distribution ne being the outward normal, is the inward flux of force through S, which is equal to minus the mass of the interior matter, and is not, as in the former case, equal to the mass which it replaces. 85. Potential completely determined by its charac- teristic properties. We have proved that the potential function due to any volume distribution has the following properties : 1. It is, together with its first differential parameter, uniform, finite, and continuous. 2. It vanishes to the first order at oo , and its parameter to the second order. 3. It is harmonic outside the attracting bodies, and in them satisfies The preceding investigation shows that a function having these properties is a potential function, and is completely de- termined. 168 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. For we may apply the above formula (5) to all space, and then the only surface integral being that due to the infinite sphere, which vanishes, we have If however, the above conditions are fulfilled by a function V, except that at certain surfaces S its first parameter is discon- tinuous, let us draw on each side of the surface 8 surfaces at distances equal to e from 8, and exclude that portion of space lying between these, which we will call Si and $2. If the normals are drawn into r we have The surface integrals are to be taken over both surfaces Si and $2 and the volume integrals over all space except the thin layer between Si and $2. This is the only region where there is discon- tinuity, hence in r the theorem applies, and (i/) i dV rr i -^-dSi— I r on-, JJ&r ^. vvrw*) s,r 3n2 Now let us make e infinitesimal, then the surfaces Si, S2 approach each other and S . V is continuous at S, that is, is the same on both sides, hence, since 0 (-) = — ^— (-) , in the limit oni \rj dn2 \rj the first two terms destroy each other. This is not so for the next two, for ^ is not equal to — because of the discontinuity. 85, 86] NEWTONIAN POTENTIAL FUNCTION. 169 In the limit, then (I8) 7,--J-ff!(|r+!V-fff -*• 47T JJ r \gfii dnj JJJ^ r The volume integral, as before, denotes the potential j M - dr due to the volume distribution, while the surface integral denotes the potential of a surface distribution 1 1 - - , where dv - Hence we get a new proof of Poisson's surface condition, §82. 86. Kelvin and Dirichlet's Principle. We shall now consider a question known on the continent of Europe as Dirichlet's Problem. Given the values of a function at all points of a closed surface S — is it possible to find a function which, assuming these values on the surface, is, with its parameters, uniform, finite, continuous, and is itself harmonic at all points within 8 ? This is the internal problem — the external may be stated in like manner, specifying the conditions as to vanishing at infinity. Consider the integral of a function u throughout the space r within 8. J must be positive, for every element is a sum of squares. It cannot vanish, unless everywhere ^- = r- = ^- = 0, that is dx dy dz u = constant. But since u is continuous, unless it is constant on S, this will not be the case. Accordingly J(u) > 0. Now of the infinite variety of functions u there must be, according to Dirichlet, at least one which makes J less than for any of the others. Call this function v, and call the difference between this and any other hs, so that u — v + hs, h being constant. 170 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. The condition for a minimum is that J(v)< J(v + hs\ for all values of h. -nj du dv , ds Now — = — + h x— , etc. dx dx ox «*.•:.. (/a»y. AV . dy 9 ds dv ds dv ds [das dx dy dy dz dz Integrating / \ ?•/ \ r/ \ z.o r/ \ OL [[[ fdvds dv ds dv ds (3) J(u) = J(v) + h2J(s) + 2h U- 5- JJJ \9^9^ Now in order that J"(v) may be a minimum, we must have j,2 r/ \ oi, rrr^ ds , ^ a5 8v^^ ^ (4) h*J(s) + 2h L ^-+^-^-+^ 5- dr>0, JJJ \9^ 8^c 9/ d dz dzj for aW values of h, positive or negative. But as 5 is as yet un- limited, we may take h so small that the absolute value of the term in h is greater than that of the term in h2, and if we choose the sign of h opposite to that of the integral, making the product negative, the whole will be negative. The only way to leave the sum always positive is to have the integral vanish. (It will be observed that the above is exactly the process of the calculus of variations. We might put Sv instead of hs.) The condition for a minimum is then dvds dvds dv ds , N (5) o-5-5~5- JJJ(9#9# dydy But by Green's theorem, this is equal to Now at the surface the function is given, hence u and v must have the same values, and 5 = 0. 86, 87] NEWTONIAN POTENTIAL FUNCTION. 171 Consequently the surface integral vanishes, and /// But since 5 is arbitrary, the integral can vanish only if every- where in r, Av = 0, v is therefore the function which solves the problem. The proof that there is such a function depends on the assumption that there is a function which makes the integral <7 a minimum. This assumption has been declared by Weierstrass, Kronecker, and others, to be faulty. The principle of Lord Kelvin and Dirichlet, which declares that there is a function v, has been rigidly proved for a number of special cases, but the above general proof is no longer admitted. It is given here on account of its historical interest*. We can however prove that if there is a function v, satisfying the conditions, it is unique. For, if there is another, v, put u = v — v. On the surface, since v = v', u = 0. In r, since A# and A?/ are zero, Aw = 0. Accordingly J (u) = 0. But, as we have shown, this can only be if u = const. But on 8, u = 0, hence, throughout T, u = 0 and v = v'. 87. Green's Theorem in Curvilinear Orthogonal Co- ordinates. We shall now consider Green's theorem in terms of any orthogonal coordinates, limiting ourselves to the special case U = const., or the divergence theorem, § 35, where ns is the outward normal to 8. * Thomson, "Theorems with reference to the solution of certain Partial Differential Equations," Cambridge and Dublin Math. Journ., Jan. 1878; Keprint of "Papers in Electrostatics and Magnetism," xm. The name Dirichlet's Princip was given by Eiemann ( WerJce, p. 90). For a historical and critical discussion of this matter the student may consult Bacharach, Abriss der Geschichte der Potential- theorie, as well as Harkness and Morley, Theory of Functions, Chap, ix., Picard, Traite tf Analyse, Tom. u., p. 38. 172 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. Let the coordinates be qlt q2, q3. The parameter P is the resultant of the derivatives of Fin any three perpendicular directions. Let these be in the directions of the normals to the level surfaces qly qz, qs. Then, calling these Pft, P,2, PQa. (1) = = Pqi cos (n&s) + Pqz cos (nzns) + Pqa cos (nsns). Now Pgit the partial parameter with respect to ql} is (§ 16) k£. 9^i Hence 9F , dV , 8F , . , 9F (2) -— = A! — COS («!»!») + Aa ^— COS (w2fts) + A3 g~ COS (ft^l,). If we divide the volume r up into elementary curved prisms bounded by level surfaces of q2 and q3, as in the case of rectangular FIG. 42. coordinates, we have, at each case of cutting into or out of S, ± dScos (nsnl) = dS1, where dSl is the area of the part cut by the prism from the level surface qt. By § 20, _ dq, dqs ** -]t V accordingly 87, 88] NEWTONIAN POTENTIAL FUNCTION. 173 Transforming the other two integrals in like manner, ,dV , v VdF + h2 — cos (n2ns) + hs ^— cos (n3ns) j- ct>S L (A. a.^ + 91 ^ |Z) + ^. (A|Z)1 ^5l Now this is equal to 1 1 1 A F«c?r. But cZr = » Multiplying and dividing in the last integral of (4) by kjiji3, we find that, since the integrals are equal for any volume, the integrands must be equal, or ( \ This result was given by Lame, by means of a laborious direct transformation. The method here used is similar to one used by Jacobi, and is given by Somoff *. 88. Laplace's Equation in Spherical and Cylindrical Coordinates. Applying this to spherical coordinates hr = 1, h0 = - , /,9F\ 8/1 9F\) 'w).+ = 28F 1^8^F 1 8F 1 92F 8r2 + r 8r + r2 8(92 + r2 CC 8^ + r2 sin2 6> 8<£2 ' We may apply this equation to determine the attraction of a sphere. For external points AF=0, and since by symmetry Fis independent of 6 and 0, * Lam6, Journal de VEcole Polyte.chnique, Cahier 23, p. 215, 1833 ; Lemons sur les Coordonnees curvilignes, n. Jacobi, "Ueber eine particulars Losung der par- tiellen Differentialgleichung AF=0," Crelle's Journal, Bd. 36, p. 113. Somoff, Theoretische Mechanik, n. Theil, § 51 — 2. 174 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. (7) _0 . dr2 r dr dr \ dr J _ _^ dr ''} dr ~r2' V=-- + c'. r But since Lim (rV) = M, r=oo Lim [— c + c'r] = M, T= 00 we must have c' = 0, — c = M. Apply the above transformation to cylindrical coordinates /L = 1, h0 = 1, h.,. = - , p _ idv " 8? + dp2 + p dp + p"2 9»2 ' If we apply this to calculate the potential due to a cylindrical homogeneous body with generators parallel to the axis of z and of infinite length, the potential is independent of z and satisfies at external points, - __ 18F 1 8^F dp2 p dp p2 do)- ' If the cylinder is circular, V is independent of &>, and we have the ordinary differential equation 1 dV +- j— =0, a/52 p dp dp p' d (. dV\ 1 — 'l°g-^T =-- log -7- = - log p + const. dp 88, 89] NEWTONIAN POTENTIAL FUNCTION. 175 e The force in the direction of p is inversely proportiona first power of p. We may verify this by direct calculation. Let us consider the FIG. 43. cylinder as infinitely thin, with cross-section «r. We will find the component of force in the direction of p. The action of dm at z on P at distance p (Fig. 43) is dm dm The component parallel to p is dm Now since, calling the density 8, dm = Sixdz, we have for the total force in direction p F=^r ^™Pdz Jo Put z = p tan 0, z = /osec2 Qd6. o /> C — -- = — as before. P P 89. Logarithmic Potential. We may state the above result in terms of the following two-dimensional problem. Sup- pose that on a plane there be distributed a layer of mass in such a way that a point of mass m repels a point of unit mass in 176 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. the plane with a force — where r is their distance apart. The potential due to m is V= — m logr and it satisfies the differential equation ~ x df ~ Similarly, in the case of any mass distributed in the plane, with surface-density /*, an element dm = pdS produces the po- tential — dm log r, and the whole the potential V = - II dm log r = - M//, log rdS, where r is the distance from the repelling dm at a, b to the repelled point a, y, so that r2 = (x - a)2 -h (y - b)2. We may verify by direct differentiation that, at external points, this V satisfies W dy* 8 a? -a) , ,, dadb = ~ ~ ~ ' This potential is called the . logarithmic potential and is of great importance in the theory of functions of a complex variable. 90. Green's Theorem for a Plane. In exactly the same manner that we proved Green's Theorem for three dimensions, we may prove it when the integral is the double integral in a plane ff (dUdV and sin co, and consequently on the circum- ference of a circle about the origin is simply a trigonometric function of the angular coordinate o>. Any homogeneous function Vn of degree n satisfies the differential equation wn zvn (i) x^ + yw» = nVn, so that a circular harmonic is a solution of this and Laplace's Equation simultaneously. The homogeneous function of degree n anxn + an^xn-* y + ...... a^y"-1 + a0yn contains n + 1 terms, the sum of its second derivatives is a homo- geneous function of degree n — 2 containing n — 1 terms, and if this is to vanish identically each of its n — 1 coefficients must vanish, consequently there are n — 1 relations between the n + 1 co- efficients of Vn, or only two are arbitrary. Accordingly all har- * See also Harnack, Die Grundlagen der Theorie des logarithmischen Poten- tiates ; Picard, Traite d' Analyse, torn. n. 12—2 180 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. monies of degree n can be expressed in terms of two independent ones. We have found in § 44 that the real and imaginary parts of the function (x-\-iy)n are harmonic functions of x, y, being respectively equal to pn cos nco and pn sin no). Accordingly the general harmonic of degree n is (2) Vn = pn [An cos nco + Bn sin nco} = pnTn. We may call the trigonometric factor Tn, which is the value of the harmonic on the circumference of a circle of radius unity, the peripheral harmonic of degree n. If a function which is harmonic in a circular area can be developed in an infinite trigonometric series W=°° oo (3) V (x> y) = % {An cos nco + Bn sin nco} = ZTn «=o o on the circumference of the circle of radius R, the solution of Dirichlet's Problem for the interior of the circle is given by the series (4) r-T.+%Tl+£Tt+.... For every term is harmonic, and therefore the series, if con- vergent, is harmonic. At the circumference p = E, and the series takes the given values of V. The absolute value of every term is less than the absolute value of the corresponding term in the series (3), in virtue of the factor pn/fin, therefore if the series (3) converges, the series (4) does as well. Since the series fulfils all conditions, by Dirichlet's principle it is the only function satis- fying them. We may fulfil the outer problem by means of harmonics of negative degree. Taking n negative, the series (5) F=r0+|r1 + ^r2+... ' .-' is convergent, takes the required values on the circumference, and vanishes at infinity except the constant term. For a ring-shaped area between two concentric circles, we may satisfy the conditions by a series in both positive and negative harmonics, 00 (6) F= 2pn {An cos nco + Bn sin nco} o 00 H- 2p~n {Anf cos nco + Bn' sin nco}. i 93, 94] NEWTONIAN POTENTIAL FUNCTION. 181 94. Development in Circular Harmonics. We may use the formula (12), § 92, to obtain the develop- ment of a function in a trigonometric series on the circumference of a circle. Let the polar co- ordinates of a point on the circumference of the circle be R, « and of a point P within the circum- ference p, - <£)]*. Removing the factor R2, inserting for cos (o> — p) its value in exponentials, and separating into factors we obtain (7) Taking the logarithm we may develop log (i- 1 and log (l-^e-* <<•>-*) by Taylor's Theorem, obtaining (8) log r = log R - - 2 - jL 0™ <«-*) + -<*» ) QO _n = log R — ^L t~n cos w (a) — <(>). This series is convergent if p < R, and also if p = R, unless Inserting this value of logr in (12), differentiation with re- spect to the normal being according to — R, we have (Q\ 182 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. Expanding the cosines, we may take out from each term of the integral, except the first, a factor pn cos n or pn sin nfy, so that VP is developed as a function of its coordinates p, , in an infinite series of circular harmonics, the coefficients of which are definite integrals around the circumference, involving the peripheral values of V and dV/dn. This does not establish the convergence of the series on the circumference. Admitting the possibility of the development, we may proceed to find it in a more convenient form. In order to do this let us apply the last equation to a function Vm, which is a circular harmonic of degree ra. Then at the circumference we have and 1 n=» /n) \ /*2ir - 2 nRm-n(- + 1 Tmcosn(a>-6)da>. \n /Jo l The expression on the right is an infinite series in powers of /o, while Fm(P) is simply pmTm. As this equality must hold for all values of p less than R, the coefficient of every power of p except the mth must vanish, and we have the important equations fftr (11) I Tm cos n (w — <£) da) = 0, ra =*= n, J o i f27r (12) Tm (<£)=- I Tm(co)cosm(ct)— ) da, for all values of n, and for all values of m except 0. Since T0 is a constant, we evidently have T0=*rT0dco. Z7T./0 These two important results can be very simply deduced by direct integration, inserting the value of Tm (o>), but we have preferred to deduce them as a consequence of Green's formula (12), § 92, in order to show the analogy with Spherical Harmonics. Let us now suppose that the function F(o>) can be developed in the convergent infinite trigonometric series 00 00 F(o>) = 2 (An cos no) + Bn sin nw) = 2Tn (o>). 0 0 94, 95] NEWTONIAN POTENTIAL FUNCTION. 183 Multiply both sides by cos m (ay — <£) dco and integrate from 0 to 2?r. /•27T oo r-Zn (r3) F(&>)cosra(ft> — $)d« = 2 Tn Jo o Jo cos m o> — Every term on the right vanishes except the rath which is equal to 7rTm ((/>). Accordingly we find for the circular harmonic Tm the definite integral (14) Tm($) = i For m = 0, we must divide by 2. Writing for Tm () its value Am cos m -f 5m sin ra$, expanding the cosine in the integral, and writing the two terms separately, we obtain the coefficients 1 /*2jr 1 f27r (15) J.0 = ^— I F(ft>)c?a>, Am = — V(a>)cosmct)dco, ^TTJo 7TJO This form for the coefficients was given by Fourier*, who assuming that the development was possible, was able to determine the coefficients. The question of proving that the development thus found actually represents the function, and the determination of the conditions that the development shall be possible, formed one of the most important mathematical questions of this century, which was first satisfactorily treated by Dirichlet"!'. For the full and rigid treatment of this important subject, the student should consult Riemann, Partielle Differentialgleichungen; Picard, Traite d' Analyse, torn. 1, chap. ix.J 95. Spherical Harmonics. A Spherical Harmonic of degree n is defined as a homogeneous harmonic function of the coordinates x, yy z of a point in space, that is as a solution of the simultaneous equations * Fourier, Theorie analytique de la Chaleur, Chap, ix., 1822. t Dirichlet, " Sur la Convergence des Series Trigonometriques, " Grelle's Journal, Bd. 4, 1829. I A resum6 of the literature is given by Sachse, Bulletin des Sciences Mathe- matiques, 1880. 184 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. vv vv vv_ 8*2 + 8^+ S?"!"- ar ar 8F (2) fl?-5- + y^-+s-5-=wF. 3# " 9y 9.3 The general homogeneous function of degree n + a0 02/n contains 1 + 2 + 3 +rc + l = (™+l)(n + 2)/2 terms. The sum of its second derivatives is a homogeneous function of degree n — 2 and accordingly contains (n — 1) n/2 terms. If the function is to vanish identically, these (n— 1) n/2 coefficients must all vanish, so that there are (n — l)w/2 relations among the (n + l)(n + 2)/2 coefficients of a harmonic of the nth degree, leaving 2^ + 1 arbitrary coefficients. The general harmonic of degree n can accordingly be expressed as a linear function of 2^ + 1 inde- pendent harmonics. EXAMPLES. Differentiating the arbitrary homogeneous function, and determining the coefficients, we find for ?i = 0, 1, 2, 3, the following independent harmonics : n = 0 constant n = l x, y, z n=2 #2- 2/2, 2/2-22, xy> yz> zx n = 3 3^y-2/3, 3ate- zs, %fx - a?, %fz - z\ 3z*x — a?, 3z*y — y3, xyz. If we insert spherical coordinates r, 0, , x=r sin 6 cos 0, y = r sin 6 sin , 0 = r cos 0 the harmonic Vn becomes 95] NEWTONIAN POTENTIAL FUNCTION. 185 where Yn is a homogeneous function of the trigonometric functions cos 6, sin 6 cos <£, and sin 0 sin . Tn being the value of Vn on the surface of a sphere of unit radius, is called a surface harmonic. The equation Yn = 0 represents a cone of order n, whose inter- section with the sphere gives a geometrical representation of the harmonic Vn. If u and v be any two continuous functions of x, y, z, a2 (uv) = u^v , 2 — — + — 9#2 da? dx dx da? ' / \ A / \ A A , o fiu dy du dv du dv (3) A (uv) = u&v + v&u + 2^-3-+ — — + — 5- \dx dx dy dy dz dz Put u = rm, and since _vU __ mrm-i ^ ox ox 02 (rm\ } we get (4) A (rm) = 3rarm~2 + m (m - 2) rm = m (m + 1) rm~2. If Vn is a harmonic of degree n, (5) A (rmVn) = rmA Fn + m (m + 1) rm~2Fn = [m (m + 1) + 2m^] rm~2 Vn, by virtue of equations (i) and (2). Consequently if m = — (2w + l), the product rmFw is a harmonic. Since Vn is of degree w, and r is of degree unity in the coordinates, r-(2n+i) Yn is of degree — (n + 1). Accordingly to any spherical harmonic Vn = rnYn of degree n there corresponds another, v Vn -(n+i) — ^+i> of degree — (n + 1). Compare this with the corresponding property of circular harmonics, where the degrees of the two corresponding harmonics are equal and opposite. 186 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 96. Dirichlet's Problem for Sphere. By means of these harmonics we may solve Dirichlet's problem for the sphere. If a function harmonic within a sphere of radius R can be developed at the surface in an infinite series of surface harmonics, (6) F=F, + F1+F2 , the internal problem is, solved by the series For each term is harmonic, and therefore the series (7), if con- vergent, is harmonic. At the surface the series takes the given values of V. Every term of the series (7) is less than the corre- sponding term of the series (6) in virtue of the factor rnjRn, therefore if the series (6) converges, the series (7) does as well. Since the series fulfils all the conditions it is the only solution. We may in like manner fulfil the outer problem by the series of harmonics of negative degree, which vanish at infinity. 7? 7?2 7?3 (8) v = ^Y0 + ^T1 + ^T,+ .... For the space bounded by two concentric spheres, we must use the series in positive and negative degrees, as will be illustrated by an example in § 198. 97. Forms of Spherical Harmonics. Before considering the question of development in spherical harmonics, we will briefly consider some convenient forms. Since if we have and any derivative of a harmonic is itself a harmonic, so that is a harmonic of degree n — (a + /3 + 7). Since to F0 = c corre- sponds the harmonic F_x = c/r, we have V9) o~i r\n,Q 27 96, 97] NEWTONIAN POTENTIAL FUNCTION. 187 If ^! be any constant direction whose direction cosines are £) /i \ and 37- ( - ) is a harmonic of degree — 2, and to it corresponds the Ofli \T / harmonic, <"» *- which is of the first degree. Since Zja + mf + n^ = 1, the harmonic contains two arbitrary constants, and multiplying by a third, A, we have the general harmonic of degree 1, in the form If in like manner h2) h3 ...... hny denote vectors with direction cosines 12) ra2, n2 ...... Intent nn> _a_ _a_ _a /i\ a/ij dh2 " "dhn \r) is a spherical harmonic of degree — (n + 1) and to it corresponds (12\ V -r"*1— — — ^ dh.dh," ~dhn(r)> a harmonic of degree n, and since every h introduces two arbitrary constants, multiplying by another, A, gives us 2n 4- 1, and we have the general harmonic of degree n in the form, Jr . a a a /i\ (I3) ...... The directions A1? /i2, ...... hn are called the axes of the har- monic. To illustrate the method of deriving the harmonics we shall find the first two. „ / lx = A^(~^ 89 /7 a a a\ /, a a a\ /i f tj^- -l-.wii ^- + *h 5- U2 5- + ^2 5- +^25- - \ 3* ay fa) \ ox oy dz/ \r 188 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. = _ AT* fo 1 + 3 + Ml |. ) f^ \ 8# 9?/ 8,37 V \ ) r3 r5 r5 r* /m2 _ 3m2y2 3l2xy mi ~ ~~ **- F2 = A {- ( + 3 ( + (m^ + m^j) 2/5 + n,2 + ?i2 ^a?. The coefficients are of course subject to the relations 98. Zonal Harmonics. If all the axes of the harmonic coincide, we may conveniently take the axis for one of the coor- dinate axes, and write (14) Vn It is evident that this will contain only powers of z and r, so that the surface harmonic will be simply a polynomial in zfr = cos (rz). The equation Tn (cos (rz)) = 0 may be shown to have n real roots lying between 1 and — 1, and hence represents n circular cones of angles whose cosines are these roots, intersecting the surface of a sphere in n parallels of latitude which divide the surface into zones. The harmonics are therefore called Zonal Harmonics. The polynomial in cos (rz) which constitutes the zonal surface harmonic, when the value of the constant A is determined in the manner to be shortly given, is called a Legendre's Polynomial, and denoted by 99. Harmonics in Spherical Coordinates. We have transformed Laplace's Operator into spherical coordinates in § 88, and A V = 0 becomes ,8 / 9F\ d . ,aF\ 9 C5) 97 — 100] NEWTONIAN POTENTIAL FUNCTION. 189 If we put in this Vn = rn Yn we obtain as the differential equation satisfied by a surface harmonic Yn (@, <£)• This is the form of Laplace's equation originally given by Laplace*. If Tn is the zonal harmonic Pn, which is independent of 2 + r'* ~ 2rr>]~^ = [>2 + f + (z - r')2]~ I Considering this as a function of z let us develop by Taylor's Theorem, f?r\\ — — f(v v'\ — -o; d-f(*- -j v/ - v~ / j ^/r/=o - 21 * Laplace, "Theorie des attractions des spheroides et de la figure des planetes." Mem. de VAcad. de Paris. Annee 1782 (pub. 1785). 190 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. andsmceforr=0, ^ = -, ^--, Now multiplying and dividing each term by rn+l, we find (22) a-£ {^+£^+5^+ •••+£*»+•••• where This is the determination of the constant A adopted by Legendre, for the reason that, since by the binomial theorem, for r r we find In order to find Pn as a polynomial in //, we may write r/c2 as and develop by the binomial theorem. Developing the last factor 100, 101] NEWTONIAN POTENTIAL FUNCTION. 191 Picking out all the terms for which s -f r = n we get for the ir'\n coefficient of f -J n _ *(*-*) nl 1.2(271-1)^ n The first polynomials have the values P.* I, P3 (/.) = ! (5^-3^), 101. Development in Spherical Harmonics. We may use the formula (6) § 83 for an internal point, to obtain the development of a function of 0, , on the surface of a sphere in the same manner as in § 94 for the case of a circle. Since the polynomials in the development of the reciprocal distance involve only the cosine of the angle between the radii to the fixed and variable points, we have if r' < r, (22) i = i and differentiating this with respect to — r, the internal normal, <-> Inserting these values in (6), § 83, namely eft' and applying it to the case that F is a spherical harmonic Vm we obtain, since (29) o + mrm~* Tm I ( -Y Ps O)j r2 sin ed0d o \*V 192 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. If the coordinates of P be r', #', <£' we have, while on the right we have an infinite series in powers of /, with definite integrals as coefficients. Since the equality must hold for all values of r' less than r, we must have, collecting in terms in r's Ym (6, 6) Ps O) sin 0d0dd>=0, s^ m, o o (30) m 4- '). By means of the above integral expressions (30) and (31) we may find the development of a function of 0, $, assuming that the development is possible. Suppose we are to find the development (32) Multiply both sides by Pn(/i,)sin#c£0d<£, and integrate over the surface of the sphere, and since every term vanishes except the nth we obtain (33) /(*• *> P" 0*) si __ (34) FM (0', f ) = /(tf, 0) Pn Ot) sin Accordingly to find the value of any term Tn at any point P, (0', <£') we find the zonal surface harmonic whose axis passes through the point P, multiply its value at every point of the sphere by the value of / for that point, and integrate the product over the surface. It remains to show that the development is possible, that is that the sum of the series ~ 2 (2» + l)j"J*Jf(0, <£) Pn Gt) sin 6d6 d$, actually represents the function f(0', $'). This theorem was demonstrated by Laplace, but without sufficient rigor, afterwards 101, 102] NEWTONIAN POTENTIAL FUNCTION. 193 by Poisson, and finally in a rigorous manner by Dirichlet. A proof due to Darboux is given by Jordan, Traite d' Analyse, Tom. II. p. 249 (2me ed.). 102. Potential of Circular Disc at points not on axis. We have found in § 81 the potential of a disc of surface density R, (2) 7= 1 r" 1 1 r4 1.1. 3 r6 ) <> r qsin the directions of the coordinate axes at the point, R1} R2) R3, which by § 16 are 7? ft dV K /, 8F 7? I, W B,.-A,g-, £.= -4,^, ft.-A,—, If now ds be an element of a line of force, its projections on the three axes being fa _ dfc fa - <%' fa _ <%* **-.-£-, dS*'^> °^-I7> we have (2) dsl : ds2 : ds3 = Rl : R2 : R3, or dq: : dq2 : dq3 = h^ : h2R2 : h3R3) so that the differential equations of the line of force are (3) 4«*'*-vg:Vg:.vg. or, dividing by hjiji3, h, dV h2 dV hs dV n n n (4) *-*s*-«ifesia«s^.5"*^^sft while we have by Laplace's equation the relation, (§ 87 (5)) dq3 \h-Ji2 dq. that is dql dq2 dqs We may now use the principle of the last multiplier demon- strated in § 37, replacing x, y, z, by ql} q2, q3 and X, F, Z by Qi, Qz, Qs- That is to say, if we have found an integral ^ (2i> #2> #3) = const., we may obtain the other at once by a quadrature as (6) 103] NEWTONIAN POTENTIAL FUNCTION. 195 and inserting the values of Q2, Q1} where of course all the functions under the integral are to be expressed in terms of ql) q2, X. This principle will be made use of in the treatment of the flow of electric currents in thin curved surfaces. The theorem becomes very simple in two particular appli- cations. First let qlt q2, qs be rectangular coordinates x, y, z, and let V be independent of z, that is, the problem is uniplanar, or the lines of force lie in planes all parallel to the Z- plane. Then X = z — const, is one integral and the other is From this we obtain , a/*, dp, dV, dV . da = f dx + — ay = -^— dx — — - du, dx dy ' dy dx djt^dV d^ = _dV dx dy ' dy~ dx ' and the function //, is the function conjugate to the potential function V, as found in § 42. Since by § 36 the flux of the vector R across any cross-section of a vector tube defined by four surfaces X, X + d\, //,, //, + dfju is d\dp, the function //, represents the flux through a tube bounded by two parallel planes z = Q, z= 1, by the surface /* = 0, and by the surface p = const. If the vector R represent the velocity of a fluid motion, //, is called Earnshaw's current function, and the amount of fluid crossing unit height perpendicular to the ^-plane of any cylindrical surface projected into a curve on the ^-plane is given by the difference in the values of JJL at the two ends of the curve. We may call the function p for any vector the flux-function. In the second case let q1, q2, q3 be cylindrical coordinates p, &, z, and let V be independent of o>, so that the lines of force are in planes intersecting in the ^-axis. The figure is then symmetrical around this axis, and we have a problem of revolution. We then have an integral X = , or du + idv = where <£ and ty are independent of the differentials dq1} dq2. If we put that is + idq2 198 THEORY OF NEWTONIAN FORCES. [FT. I. CH. IV. equating the coefficients of dql} dq2, we obtain du .dv 6 O -- I"* 0~ =7 > (12) % fyi «i 9w . 9v . _ I n _ __ n r o * ^ — T~ • oq2 cq2 hz Now eliminating <£, , N , \du .dv} ., (du . dv} (13) Mo- +*5rr!BS«*iio" +*V-M and equating the real parts on each side, and the imaginary parts in like manner we obtain / \ z, du i dv i dv i 3u (14) h>2^- = — ni^, h2^-=hl — . 2dq2 ldqS *dq2 l dq, Solving for the derivatives of v , . _dv _h2du dv _hldu ~^~h^ 872~r2g^' differentiating respectively by q2 and qlt and adding ' 104] NEWTONIAN POTENTIAL FUNCTION. 199 which is the condition, § 42, that the complex variable u1 + iv' is an analytic function of u + iv. Thus from the solution of one problem for the surface qs may be deduced the solution of any number of others for the same surface. If now the quantities u, v be taken as rectangular coordinates in a plane, the arc of any curve is expressed in the form, do-2 = du* + dvz. To any point u, v in the plane corresponds a point with the same values of u, v, on the surface qs. In virtue of the relation da2 = Mds2 between corresponding arcs on the plane and on the surface, we see, as in § 43, that corresponding infinitesimal triangles are similar, or the surface qs is conformally represented upon the plane. If the Z7F-plane is conformally transformed to another plane XY, we have seen that we have u + iv an analytic function of the complex variable x + iy and the real functions u, v are potential and flux-functions in the JTF-plane. As we have just proved that they retain this property on the surface q3, we see that the method of the functions of a complex variable will give us the solution of any number of cases upon a surface, and that the surface may be conformally repre- sented on the plane in an infinite number of ways. Such a representation of a surface on a plane constitutes a map. Surfaces which may be conformally represented on a plane may be conformally represented on each other. The theory of such transformations is the subject of an important memoir by Gauss*. The method here given is due to Beltrami^, and may be applied even when the coordinates qlt q2 are not orthogonal. The method is particularly applicable to the case of electrical currents flowing in thin conducting surfaces, and the conformal transformations may be found by experiment. A thin space bounded by two surfaces qs in which is distributed a solenoidal vector which may be represented by a potential or by a flux-function as here de- scribed, is termed a vector-sheet. * Gauss, " Allgemeine Auflosung der Aufgabe die Theile einer gegebenen Flache auf einer anderen gegebenen Flache so abzubilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen ahnlich wird. " WerJce, Bd. iv., p. 189. + Beltrami, " Delle variabili complesse sopra una superficie qualunque. " Annali di Matematica, ser. 2, t. i., p. 329. 200 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 105. Example. Conformal Representation of Sphere on Plane. Let the surface q3 be a sphere of radius R, and take for the coordinates q1 and qz, the co-latitude 6 and the longitude . Then by § 17, we have hl=R> hz ds* = R2 (dd* + sin2 6d, du = -; — ^ . sin 0 Integrating we obtain 0 v = $, u = log tan ^ . If now we take u and v for rectangular coordinates in a plane, the surface of the sphere is conformally represented upon the plane by means of the above transformation. This particular representation is known as Mercator's Projection. The meridians

between the limits 0, 2-7T, the projection on the plane has the finite width 2?r, but the length of the projection is infinite, the poles 9 = 0, 0= TT corresponding to u = — oo , u = oo . If we make a conformal trans- formation of the £7F-plane by means of the function u + iv = log (x + iy\ we obtain the formulae, u = log r = log Jx2 + y*, v = tan"1 - , CO = r = tan ^ , $ = tan~x - -j X * For an example see Fig. 71, § 177. 105, 106] NEWTONIAN POTENTIAL FUNCTION. 201 which give a new conformal representation of the sphere on the plane, the meridians corresponding to radial lines yjx = tan (£, and the parallels to concentric circles. This is the stereographic projection, obtained by projecting points on the sphere upon a plane tangent at one pole from the other pole as a center of projection. Figure 23 projected upon the sphere by this transformation is shown in perspective in Fig. 47. FIG. 47. 106. Diagrams. If we have a diagram representing a plane section of a set of equipotential surfaces, corresponding to equal increments of potential, and we superpose upon this a second diagram representing a second set of equipotential surfaces, drawn for the same differences of potential, we may draw the curves representing the equipotentials due to a distribution which is the sum or difference of the other two by simply drawing lines con- necting opposite corners of the curvilinear quadrilaterals into which the diagram is divided by the two equipotential systems. For as we go from vertex to vertex, the increase of potential due to one system is just counterbalanced by the decrease due to the other. Fig. 49 represents a combination of Fig. 23 with a straight field in this manner. 202 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. In like manner if we have diagrams of the flux-function of Q any two systems superposed, we may draw diagrams of the flux-function of the sum or difference of the two systems, for if we con- sider two flux tubes bounded by the lines AB, CD, and A'K, C'D', Fig. 48, the line PQ has the flux A//, through it in opposite direc- tions from the two systems, so that the total flux through it is zero, or it is a flux-line. In this manner the Figures 49*, 72, 74, 75, 76, 77, 78 have been drawn. FIG. 48. Fm. 49. * Fig. 49 is to be considered a diagram of lines of flow or of equipotentials according as the directions of the component vectors at the origin are the same or opposite. The analogous cases of the rotational problem are represented in Figs. 74, 75. CHAPTER V. ATTRACTION OF ELLIPSOIDS. ENERGY. POLARIZED DISTRIBUTIONS. 107. Ellipsoidal Homoeoids. Newton's Theorem. If we transform Laplace's equation to elliptic coordinates and attempt to apply the methods of § 88 to the problem of finding the potential of a homogeneous ellipsoid, we are at once con- fronted with a difficulty. It is not evident, nor is it true, that the potential is independent of two of the coordinates, and that the equipotential surfaces are ellipsoids. The following theorem was proved geometrically by Newton. A shell of homogeneous matter bounded by two similar and similarly placed ellipsoids exerts no force on a point placed anywhere within the cavity. Such a shell will be called an ellipsoidal homoeoid. FIG. 49 a. Let P, Fig. 49 a, be the attracted point inside. Since the attraction of a cone of solid angle da) on a point of unit mass at 204 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. its vertex is dm we have for an element of the homoeoid the attraction d(BD-AC). Draw a plane through ABO, and let ON be the chord of the elliptical section conjugate to AB. Since the ellipsoids are similar and similarly placed, the same diameter is conjugate to the chord CD in both. But CD and AB being bisected in the same point, AC = BD, and the attraction of every part is counterbalanced by that of the opposite part. 108. Condition for Infinite Family of Equipotentials. Although the equipotentials of an ellipsoid are not in general ellipsoids, we may inquire whether there is any distribution of mass that will have ellipsoids as equipotential surfaces. Let us examine, in general, whether any singly infinite system of surfaces F(at!,ytz,q) = 0 can be equipotential surfaces. If so, for any particular value of the parameter q, V must be constant, in other words V—f(q). If x, y, z are given, q is found from F (x, y, z, q) = 0 and from that V from the preceding equation. Now in free space, V satisfies the equation AF=0. But, since F is a function of q only, dx dq dx ' d*V ~* da? dx 107, 108] ATTRACTION OF ELLIPSOIDS. 205 In like manner 2 dq dy2 \dy> dq2 ' dq dz2 \dz) dq2 ' Accordingly ^F d dq Now since F is a function of q only, the expression on the right must be a function of q only, say (q). Consequently, that may represent a set of equipotential surfaces, the parameter q must be such that the ratio of its second to the square of its first differ- ential parameter is a function only of q, If this is satisfied, we have d , dV dq There must be one value q such that the level surface is a sphere of infinite radius, and for this V must vanish. 206 THEORY OF NEWTONIAN FORCES. [FT. I. CH. V. These conditions are satisfied by the polar coordinate r for by §95, (4) For r = oo , we must have V=0, accordingly we must put B = 0. We may get a convenient expression for •— by transforming hq Ag into terms of three orthogonal coordinates, of which it is itself one. Put q-qi, and since it is independent of q2 and qs i_2s h, s 109. Application to Elliptic Coordinates. Applying this to elliptic coordinates gives I I ,(a ° 2 c2+X 108, 109] ATTRACTION OF ELLIPSOIDS. 207 which is independent of /JL and v, and hence the system of ellipsoids X can represent a family of equipotential surfaces. We have (8) = log V(a2 (9) 7 = 4 [7= -- — =+5. JVa2 + The constant B must be such that for X = oo , which gives the infinite sphere, F=fO. This is obtained by taking the definite integral between X and oo , (10) " - r ds X being taken for the lower limit, so that A may be positive, making V decrease as X increases. V is an elliptic integral in terms of X, or X is an elliptic function of V. For dV A 0 1) ^ ) = (a2 + X) (62 + X) (c2 + X), a differential equation which is satisfied by an elliptic function. We may determine the constant A by the property that \i-rn (rV) = M, x=» or that lim (V2 -5— ] = - M cos (rx\ ,.=« \ 9# / We have (a2 + X) (a2 + X) (62 + X) (c2 + X) , i OF THF ' y > UNIVERSITY 208 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. From the geometrical definition of X, Now consider, for simplicity, a point on the X-axis, where SA=#=r. The denominator becomes infinite in X^, that is, r5, and so does the numerator. Hence lim so that (12) - -' & 110. Chasles's Theorem. We have now found the potential due to a mass M of such nature that its equipotential surfaces are confocal ellipsoids, but it remains to determine the nature of the mass. This may be varied in an infinite number of ways ; we will attempt to find an equipotential surface layer. By Chasles's theorem, § 84 (n), this will have the same mass as that of a body within it which would have the same potentials outside. If we find the required layer on an equipotential surface 8t since the potential is constant on 8, it must be constant at all points within, or the layer does not affect internal bodies. The surface density must be given by § 84 (10), 1 8F °" — ~~ ~r~ o — > where n^ is the outward normal to X, 4?r dnx and a_F==^F_ax==/, dv Now since h^ — 28x, .—.a'**" Since F is a function of X alone, the same is true of -=— , which for a constant value of X is constant. Hence cr varies on the ellipsoid 8 as S\. Therefore if we distribute on the given ellipsoid 8 a surface layer with surface density proportional at every point to the perpendicular from the origin on the tangent plane at the point, this layer is equipotential, and all its equipotential surfaces 109 — 111] ATTRACTION OF ELLIPSOIDS. 209 are ellipsoids confocal with it. Consequently if we distribute on any one of a set of confocal ellipsoids a layer of given mass whose surface density is proportional to 8, the attraction of such various layers at given external points is the same, or if the masses differ, is proportional simply to the masses of the layers. For it depends only on X, which depends only on the position of the point where we calculate the potential. Since by the definition of a homceoid, the normal thickness of an infinitely thin homceoid is proportional at any point to the perpendicular on the tangent plane, we may replace the words surface layer, etc., above by the words homogeneous infinitely thin homoeoid. The theorem was given in this form by Chasles.* 111. Maclaurin's Theorem. Consider two confocal ellip- soids, 1, Fig. 50, with semi-axes «i, &, 71, and 2, with semi-axes FIG. 50. a2> /52, 72- The condition of confocality is « - = 2 - 2 = 2 - = 72 - 7l = s, say. If we now construct two ellipsoids 3 and 4 similar respectively to 1 and 2, and whose axes are in the same ratio 0 to those of 1 and 3, these two ellipsoids 3 and 4 are confocal (with each other, though not with 1 and 2). For the semi-axes of 3 are 0alt 0/31} 6ylf and of 4 are #«2, #/32, #72, and hence the condition of confocality, #V - 0V = 02/322 - 6%2 = 6V - 0V = fr s is satisfied. Now if on 3 we distribute one infinitely thin homoeoidal layer between 3 and another ellipsoid for which 0 is increased by dO, and on 4 a homoeoidal layer given by the same values of 6 and d&, and furthermore choose the densities such that these two homceoidal layers have the same mass, then (since these homceoids are confocal) their attractions at external points will be identical. * Chasles, "Nouvelle solution du probl^me de 1'attraction d'un ellipsoide h6tdrog£ne sur un point ext£rieur." Journal de Liouville, t. v. 1840. W. E. 14 21 0 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. Now the volume of an ellipsoid with axes a, I, c, is f Trabc, that of the inner ellipsoid of the shell 3 is accordingly and that of the shell is the increment of this on increasing 6 by dO, or (vol. 3) = Similarly (vol. 4) = 47r<92d0a2/32y2. Consequently, if we suppose the ellipsoids 1 and 2 filled with matter of uniform density p± and p2 the condition of equal masses of the thin layers 3 and 4, is simply that is, equality of masses of the two ellipsoids. And since for any two corresponding homceoids such as 3 and 4 (0 and 0 + d6) the attraction on outside points is the same, the attraction of the entire ellipsoids on external points is the same. This is Maclaurin's celebrated theorem : Confocal homogeneous solid ellipsoids of equal masses attract external points identically, or the attractions of confocal homogeneous ellipsoids at external points are proportional to their masses.* 112. Potential of Ellipsoid. The potential due to any homo3oidal layer of semi-axes a, ft, 7, is to be found from our preceding expression for F, § 109 (12), ds where X is the greatest root of 2 _^_ *2 Now if the semi-axes of the solid ellipsoid are a, b, c, those of the shell a = da, /3 = Ob, 7 = 6c, we have M = 4>7r02d0abc, if the density is unity, and (i) * Maclaurin, A Treatise on Fluxions, 1742. Ill, 112] ATTRACTION OF ELLIPSOIDS. 211 where X is defined by /y»2 nft nZ _i " To get the potential of the whole ellipsoid, we must integrate for all the shells, and (3) V= 2>rrabc (I Jo o For every value of 6 there is one value of X, given by the cubic (2), we may say X = (6). Let us now change the variable s to t, where, 6 being constant, s = 0% ds = d*dt ; and put X = fru. Then (4) V= 27rabc P 0d0 T -p Jo Ju v/(a2+ where u is defined by * si Since 0* is thus given as a uniform function of M, we will now change the variable from 0 to u. Differentiating (5) by 0, When 0 = 0, u = oo , and when 0 = I,u has a value which we will call cr, defined by a? y2 £2 _ + + Accordingly, changing the variable, (8) F=TO6o/J(^2+(^+(^}^/ --= The three double integrals above are of the form where /" («) = , . v/(a2 + 0(&2 + 0(c2 + 0 14—2 212 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. This may be integrated by parts. Call I J - Now ^(oo)- f(i)dt = 0, (since /( oo ) = 0), J 00 4 <•)-/-/($•* ./ is an elliptic integral independent of a?, y, z, and so are its derivatives with respect to a2, 62, c2. Calling these L M N respectively — -r , — -7- , — -r , we nave a symmetrical function of the second order, and since L, M, N are of the same sign, the equipotential surfaces are ellipsoids, similar to each other. Their relation to the given ellipsoid is how- ever transcendental, their semi-axes being 7 IV IV 9$ • Vo ** ' V21* ' ~V) a(62) 8(c2) We have for the force 37 ,, dV ^— = My, - 5- = Nz. dy dz Hence, since for two points on the same radius-vector, #2 2/2 ^2 r2 X* F2 Z2 T2 - = ^* = -J = -• we have ^-2 = -/=-=; = — . #1 yj *i n J?! F! ^ n • The forces are parallel and proportional to the distance from the center. 114. Verification by Differentiation. For an outside point, we have 6 ° \_ u c2 + u V(a2 + u) (62 + u) (c2 + u) 113, 114] ATTRACTION OF ELLIPSOIDS. 215 dV r , JT a2 + wVa2 d*V r du (a2 + u) V(a2 + u) (62 + u) (c2 + u) fol 1 (a2 + cr) (a2 -f o-) (62 4- a) (c2 + t>2 Co the point as — , jf jn, £ - Cli PI Ci lies on the ellipsoid These will be called corresponding points. We shall now assume that these two ellipsoids are COD focal, and (2) the smaller. Then The action of (2) on the external point x, y, z is I"*5 du X, = - &ra,ta0 J ^ (aj, + B) ^-—^-j-^, + where a;2 «2 ^ and since _ + |L+_ = 1, we must have a = X. If now we substitute a22 = w' + cr, /.'T^ Now the attraction of the ellipsoid (i) on the interior point a2 &a c2 . a? — - , y-r, z- is «b • fc ci TT o L «a A j = — ZTraACi a? — (a? + u) v(a!2 + u) (b-? + u] (d2 + w) 114 — 116] ATTRACTION OF ELLIPSOIDS. 217 The definite integrals being the same in both cases, we have c2a2 This is Ivory's theorem : Two con focal ellipsoids of equal density each act on corresponding points on the other with forces whose components are proportional to the areas of their principal sections normal to the components.* 116. Ellipsoids of Revolution. For an ellipsoid of revolu- tion, the elliptic integrals reduce to inverse circular functions. Put b = c, a being the axis of revolution, r du (i) F=7rao2 . — UXx + Yy), J^ = ]*™~l 4/iq^J - 1 (** + ry} , F-9 ^ f_ 2(62-a2)g2 53c?5 4 - 2woWa?J - ^TI^(^I^ ^ /" s2c?5 s ] (i=j- VfTJ- ds so that (7) z=(S5{v/fe?-sin"1V// (8) F = /6^-a2 V 62^ Now so that r S2^s _ I = J (sin"1 s — s v 1 — s2}, , N ^ 27ra62y f . 62 - a2 (62 - a2)(a2 + o-)) (9) F=(62^)i{sm-1VFT,-V- -FT^ TP . /62-a2 ., /62 - a2 For sin"1 A / -7 — - we may write tan"1 A / - — , V o2 + a, they introduce imaginaries, from which they may be cleared as follows. 116, 117] Call then therefore ATTRACTION OF ELLIPSOIDS. sin"1 (iu) — 0, iu = sin 0, Vl + w2 = cos 0, = cos 0 — i sin 0 = Vl -f w2 + w, — 1'0 = log {Vl + u2 + u], sin-1 (iu) = 0 = i log { ViTw2 + w}. /a2 - 62 Va2 + (7 + Va2-62) 219 2 - 2 Hence , In all these formulae, 222 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. y + 8y, z + $z. The amount of matter in the fixed infinitesimal parallelepiped dxdydz, dm = pdxdydz is thereby changed, and the work necessary is the same as that required to bring the mass Sdm from infinity to the point x, y, zy where the potential is V, namely, SW = V8dm. We have found in § 38, Consequently the whole increase of energy is (it) JJJ ( dx Integrating by parts - dxdydz = -VpSxdydz + p&v dxdydz, the integral being over all space, and the surface integrals vanishing at infinity. But since p — — -r- A V, 47T this becomes . a* "9* ty ty *dz dz a so that For a third deduction, since in moving a mass dm a distance whose components are Sx, By, $z the energy lost is equal to the work done by the system - S JF= dm {XSx + YBy + ZSz] , ttVK 9F, 3V (14) = - dm ] ^ 8x + d- oy + r— 118,119] -ENERGY. 223 The whole variation of the energy is , asin(i2). Applying Gauss's Theorem to the mutual energy of two distri- butions, one of which has density p, producing the potential V, the other the density Bp, producing potential SV, we have jjj and W = pVdr gives in agreement therewith (15) The integrals may be now restricted to the space occupied by matter. 119. Maximum theorem for Energy. By making use of the two different expressions for the energy we can deduce an important theorem relating to the energy of a distribution. We may use the form, § 118, (8), (i) W* = *Vd8+ PVdT> which is distinguished by the suffix d to denote that the densities occur explicitly. This form, by the definition of the potential, holds for any law of force, whether the Newtonian or not.* On the other hand we may use the form, § 118, (10), to which we give the suffix / in order to denote that it is ex- pressed only in terms of the field at all points, and does not * By this we mean any conservative law in which the action is proportional to the product of the masses, and to some function of their relative position. 224 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. explicitly contain the densities. This expression holds good only for a distribution acting according to the Newtonian Law. As these two expressions must be equal for all distributions, we may write (3) W aFV V aF\2 ff ir^o fff f Tr 1 = o-VdS + pF- JJ JJJn L & If in this latter expression for W we make an arbitrary varia- tion in the form of the function V, we obtain for the varied value of W an integral containing the variations of V, a, and p. If we suppose that the new distribution also acts according to the Newtonian law, in virtue of Poisson's equation there will be relations between SF, So-, 8p. We shall however remove this restriction, and consider V, cr, p as perfectly independent functions, which can be varied inde- pendently. We shall choose Bo- and Bp as zero, in other words we shall suppose V to be varied from the values that it actually has for the original Newtonian distribution, the variation being entirely arbitrary, while the densities are unchanged. Calling the varia- tion under these circumstances 8TF, (4) W+&rW=e) = — ge COS (geWe) = ge COS (ggW*). The vector g being everywhere solenoidal, its surface integral over any closed surface vanishes, so that as many unit tubes enter as leave the surface. Tubes leave the polarized body where a- is positive, and enter it where a- is negative. They form closed tubes, every one of which passes through the body. The vector g is called by Maxwell the induction, and is characterized by the solenoidal property. The line separating the region of positive a from those of negative is linked with all the tubes of induction belonging to the body. The induction is not in the same direction as the force F unless the polarization / is. We obtain another physical conception of the induction by considering the force in a cavity in the conductor. By hollowing out a space in the body we remove a portion of the volume distri- bution, but give rise to a new surface distribution. We shall suppose the cavity so small that the volume-density of the part removed may be considered constant. Now if we consider the forces at corresponding points of geometrically similar distributions of constant densities, we have for the action of the volume-density, fdr •ffi and if we increase the dimensions in the ratio n, the element of volume and the potential at a corresponding point are dr = nsdr, and the force _ ds' d (ns) ds 121, 122] POLARIZED DISTRIBUTIONS. 231 while for surface distributions r,_Tf/*S' ff»1 D 3 n d A^^'B=tt'G=Tc- Outside the polarized body, since / = 0, must be constant, and accordingly discontinuous at the surface. Inserting this in the value of V the potential becomes Applying Green's theorem we obtain (io) F= But since 1/r is harmonic except for r = 0, if the attracted point is outside of the polarized body, V is given by the surface integral, (ii) F = - 236 THEORY OF NEWTONIAN FORCES. [PT I. CH. V. Accordingly the potential at points outside of a lamellarly polarized body depends only on its form and position, and on the values of the potential of polarization at the surface. If the attracted point is within the substance of the polarized body, we may integrate (9) in the other manner, interchanging <£ and 1/r, obtaining which, by the theorem of § 83 (5) or § 84 (12) applied to , becomes 03) = In the case of lamellar polarization the induction becomes so that the induction, being the parameter of the function — F+ 47T(£, is also lamellar. For both inside and outside points, this function is equal, except for a constant, to the surface integral Os) vn as we see from (13) and (n), together with the fact that outside is constant. 125. Polarized Shells. The characteristic of a lamellar polarization is that if we construct two infinitely near equi- potential surfaces of polarization <£> = fa and = fa, the polariza- tion is normal to them at all points, inversely proportional to the distance between them, and in the direction from the smaller to the larger value of fa The portion of matter included between the two surfaces, which need not be closed, is called a simple polarized shell. If we consider the infinitesimal portion of the potential due to such an unclosed shell, the surface integral (i i) is taken over both sides of the shell, the portion over the edge vanishing, since the width of the edge is infinitesimal. Conse- quently, replacing n, the internal normal, by n-^ and n2) away from 124, 125] the shell, POLARIZED DISTRIBUTIONS. (16) V= sub- tended at P by the surface 8, if n^ points toward the side of 8 on which P lies. Con- sequently the potential at any point P due to the shell is equal to the product of the difference of potential of polarization on the two sides of the shell by the solid angle subtended by the shell at P, the potential being positive if P is on the positive side of the shell, that is, the side toward which the polarization is directed. Now we have seen in § 39 (5) that the solid angle integral is equal to — 4?r for a point inside a closed surface, and to zero for an outside point, that is, it experiences a dis- continuity of 4-7r as P crosses the surface. When the surface is not closed the same thing takes place. For the integral FIG. 54. a) = — is a continuous function of P so long as r is not zero, that is, so long as P does not lie on the surface. If P lies on the surface, the in- tegral has an infinite element. We remove this by cutting out a small area around P. If now o>' be that part of the integral due to the remainder of the surface, o>' is finite and continuous even when P passes through the surface. As P approaches the surface the solid angle subtended by the small area cut out, which may be treated as plane, approaches 2vr, so that at the surface on the side 1, G)1 = ft)/+27r. At an infinitely near point on the side 2, how- ever, the cosine in the numerator has changed sign, for the small area, so that the solid angle subtended by the latter is to have the negative sign. Accordingly on the side 2, o>2 = &>' — 2?r, and ac- cordingly, ft)j — ft>2 = 4-7T, 238 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. and the potential V experiences a discontinuity of — 4?r (fa — fa) in passing through the shell from the positive to the negative side. The discontinuity may be also explained by considering the solid angles subtended at points 1 and 2 approaching a point on the surface from opposite sides. If the solid angles have different signs on opposite sides, as the points come together the sum of the absolute values of the two angles approaches 4-Tr, so that at the surface If the thickness of the shell is e, the polarization is (fa — fa)/6, and the moment of the equal and opposite charges on the element of surface dS on the opposite sides of the shell is, since the volume of the element is edS, equal to (fa — fa) dS. Thus the surface density times the thickness, or the moment of polarization per unit of surface of a simple polarized shell, is constant. The value of the constant = fa — fa is called the strength of the shell, and it is this strength that is multiplied by the solid angle in the expression for the potential.* Suppose now that the intensity of polarization increases without limit, so that the strength of the shell fa — fa is finite, instead of infinitesimal. Then the difference of potential on the two sides of the shell is finite, or the potential is discontinuous in crossing the shell, by the amount The derivative, dV/dn, is however continuous. We may prove the converse of this proposition. If a function satisfies Laplace's equation, vanishes at infinity, and is continuous everywhere ex- cept at a certain surface, its first derivatives being everywhere continuous, the function represents the potential of a double distribution on the surface of discontinuity. If the function were uniform and continuous, it must, by Dirichlet's principle, vanish everywhere. The demonstration will be given in § 210. * Gauss. " Allgemeine Theorie des Erdmagnetismus," § 38, 1839. Werke, Bd. v., p. 119. 125, 126] POLARIZED DISTRIBUTIONS. 239 126. Energy of Polarized Distributions. If a polarized distribution is placed in a field of which the potential is F, then- mutual energy is, by § 117, which, by § 120, is equal to ( i ) W= - (JV {A cos (nx) + B cos (ny) + C cos (nz)} dS i dB +^ Integrating by Green's theorem, this becomes " The integrand is the negative of the geometric product of the polarization and the force of the field. This result may be ob- tained directly for a doublet as we obtained the potential in §122. If the polarization is lamellar, the energy of the distribu- tion is dV For a polarized shell the volume integral disappears, and the surface integral becomes Accordingly the energy of a polarized shell is equal to the product of its strength by the flux of force through it in the direction opposite to the polarization. If we wish to find the energy of the polarized distribution itself, we must put for V in the above formulae the potential due to the distribution itself, and multiply by the factor one- half, as in § 117. It is important to notice that the energy of polarized distributions is defined as the work that they are capable of doing if every particle is allowed to retire to infinity carrying 240 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. its own charge. But if the distribution should be cut up into small parts, new surface densities would appear on each part. To prevent this the distribution must be supposed split up into infinitely thin shreds along the lines of polarization — on separating these from each other no new surface densities would be formed, so that the energy as calculated would be the work obtained by letting these shreds be bodily removed to infinite distances from each other. Similarly polarized shreds side by side of course repel each other, so that this energy is positive. If we should further break up each shred into infinitely short lengths, and separate these from each other, we should have to do positive work to pull them apart, and if we should remove all the parts to infinite distances from each other, it has been shown by Lord Kelvin* that we should have to do exactly as much work as was before obtained by separating the shreds. Consequently the energy must be defined by the first operation alone. 127. Development of Potential of Polarized Body in Spherical Harmonics. We have seen in § 123 (7) that the potential due to a doublet placed at the origin is a spherical harmonic. We may develop the potential due to any polarized distribution in a series of spherical harmonics. If we call r and r' the distances from the origin of the attracted point x, y, z, and the point of integration a, b, c, so that 7-2 _ tf + y2 + ^ /2 _ a2 + fc3 _|_ c2j we have for the distance between the two points, by § 100 (22), if r < r, where //-, the cosine of the angle between r, r, is (ax + by + cz)/rr'. Inserting this value and those of P0, Plt P2, § 100, we have 1 _ 1 asc + by 4- cz 1 3 (ax + by + czf — r2/2 = + ~ ~~ ~ + ~ ~~ +•••• * Thomson. "On the Mechanical Values of Distributions of Matter, and of Magnets." Papers on Electrostatics and Magnetism, p. 437. 126, 127] POLARIZED DISTRIBUTIONS. 241 Now inserting this value of \/d in the expression for the potential in § 122 (3), in which 1/r is to be replaced by 1/d, or performing the differentiations, x, y, z, r being constant, and collecting in powers of x, y, z, which can be taken out from under the integral signs, we get the development in spherical harmonics F=F_2+F_3+F_4+..., where the coefficients are the definite integrals In like manner the coefficients in the harmonics of higher orders are definite integrals throughout the polarized body of the components of polarization multiplied by powers of the co- ordinates of the point of integration. By a change of the origin the integrals L, M, N may be made to vanish. For putting a = a0 + a', b = b0 + V, c = c0 + c', W. E. 16 242 THEORY OF NEWTONIAN FORCES. [FT. I. CH. V. we have L = a, Adr + flAafdr = M + L't M = bojjlBdT+jjJBb'dr = b0B + M', Xr = c0 (ffcdr + [((Cc'dr = c0<7 + N'. JJJ JJJ and if we choose the integrals L'y M'} Nr vanish, and F_3 reduces to three terms. The values of the integrals A, B, C, are not changed by this change of origin, but those of all the others are. The new origin is called the center of the polarized distribu- tion. If the polarization is uniform, it is the center of gravity of the body. If we find a vector M whose components are A, B, C, we have .„, _ M (cos (Mx) cos (rx) + cos (My) cos (ry) + cos (Mz) cos (rz)} _ M cos (Mr) But this is equal to the potential due to a doublet of moment M situated at the center. M is called the moment of the polarized body, and since at great distances the first terms are relatively the most important, we see that at great distances the body acts as if concentrated at its center. The line through the center having the direction of M is called the axis of the distribution. :TY PAET II. ELECTROSTATICS, ELECTROKINETICS AND MAGNETISM. CHAPTER VI. ELECTRICAL PHENOMENA. SYSTEMS OF CONDUCTORS. 128. Fundamental Experiments. We shall begin the treatment of Electricity by the description of a number of simple experiments, for the most part due to Faraday and described by Maxwell, the explanation of which will devolve upon the theory, when mathematically established. EXPERIMENT I. Let a piece of glass and a piece of resin, neither of which exhibits properties different from those of ordinary bodies, be hung up near each other by silk threads. They do not affect each other, and the threads hang vertically. Let the glass and the resin be rubbed together, and left in contact. They still exhibit no peculiar properties. Let them now be separated. They attract each other, and the strings take an inclined position. The system composed of the glass and resin has now acquired energy, which has enabled it to do work against the force of gravity in lifting the two bodies through a certain distance. Let a second piece of glass be rubbed with a second piece of resin, and be similarly suspended. Then it may be observed that the two pieces of glass repel each other, and have therefore acquired energy, which is evinced by their overcoming gravity in lifting themselves. 16—2 244 ELECTROSTATICS. [PT. II. CH. VI. The two pieces of resin in like manner repel each other. Each piece of glass attracts each piece of resin. All of these phenomena, each of which indicates the acquisition of a positive amount of potential energy, are known as Electrical phenomena, and the bodies exhibiting them are said to be electrified, or charged with Electricity. The properties of the two pieces of glass are similar, but opposite to those of the resin. What the glass attracts the resin repels, and vice versa. Bodies repelled by the glass and attracted by the resin are said to be vitreously, those attracted by the glass and repelled by the resin, resinomly electrified. By general convention we say positive, instead of vitreous, negative for resinous. EXPERIMENT II. Let a hollow metal vessel be hung up by silk threads, and let a lid completely closing it be also so hung, so that it may be removed and replaced without touching it. Then if the electrified glass be hung inside the vessel without touching it, and the lid placed on, the outside of the vessel will be found vitreously electrified, and the manner of the electrification will be exactly the same in whatever part of the interior the glass may be. That is to say, if we place successively at different points of the external space the same small electrified body, it will be acted upon at each point by a certain force. The direction and magni- tude of this force determine a vector called the strength of the electrical field of force. The field may be geometrically repre- sented by lines of force in the usual manner. The electric field is the tangible evidence of the electrification, and the measurement of a force is the means of its measurement. We may therefore describe the above experiment by saying that the field external to the closed metal vessel is independent of the position of the charged body within. If the glass be removed without touching the vessel, the electrification of the glass will be unchanged, and that of the vessel will have disappeared. If resin be substituted for glass the outside of the vessel will be negatively electrified. Such electri- fication, which depends on the proximity of electrified bodies, is called electrification by influence, or induction. In this manner a body may acquire energy without contact with other bodies, and it is natural to suppose that the energy has passed through the intervening medium from the electrified body. Such a medium, 128] ELECTRICAL PHENOMENA. 245 which allows electrical influences to pass through it, is called a dielectric, as was proposed by Faraday*. EXPERIMENT III. Let the vessel be positively electrified by induction as before, let a second vessel be suspended by silk threads, and let a metallic wire, similarly suspended, be made to touch both simultaneously. The second will be found to be positively electrified, and the positive electrification of the first is lessened. EXPERIMENT IV. If instead of a metal wire we had used a rod of glass, sealing wax, or hard rubber, no such effect would have been produced. Bodies may accordingly be divided into two classes, 1°, those which, like metals, allow a transference of electri- fication from place to place. These are called conductors. The second body above is said to be electrified by conduction : 2°, those which do not allow such transfer. These are called non-conductors or insulators. The dividing line cannot be drawn with perfect sharpness, since no bodies have been found to be absolutely non- conducting. All insulators are dielectrics, but not all dielectrics are necessarily insulators. EXPERIMENT V. In Experiment II it was shown that the external electrification of the vessel due to the introduction of the electrified glass was independent of the position of the latter in the vessel. If we now introduce the piece of glass together with the piece of resin with which it was rubbed, without touching the vessel, the electrification of the latter disappears. We therefore conclude that the electrification of the glass and resin, which are able to counteract each other's effects, are equal in amount. By putting in a number of bodies, and examining the external field, we may show that the induced electrification is proportional to their algebraic sum. We thus have an experimental method of adding the effects of several electrifications without altering the electrifications. EXPERIMENT VI. Let there be two insulated metallic vessels, A and B, and let the glass be introduced into A and the resin into B, and let them be connected by a wire. All electrification disappears, as was to be expected. Now let the wire be removed, and then let the glass and resin be taken out. It will be found * Exp. Res., § 1168. 246 ELECTROSTATICS. [PT. II. CH. VI. that A is now negatively electrified, and B positively. By intro- ducing A and the glass together into a larger metal vessel C, its outside will be found to have no charge, consequently the induced charge on A is equal and opposite to that of the glass. In like manner the charge of B may be shown to be equal and opposite to that of the resin. The charge of A, which is not apparent as long as the glass is within, is said to be bound by the inducing charge of the glass, and resides on the inside of A. By the withdrawal of the glass it becomes free, and appears on the outside of A. We have thus a method of charging a vessel with an electrification equal in amount and opposite in kind to that of a given electrified body without changing its electrification. EXPERIMENT VII. Let the vessel B, charged with a quantity of positive electricity, which we shall take for a provisional unit, be introduced into the vessel C without touching it. C will be found charged on the outside with a unit of positive electricity. Now let B touch the inside of C. The external electrification is unchanged. If B be now removed from C without touching it, and taken to a distance, the field external to C is still unchanged, that is, C is charged with a unit of electricity, but B is completely discharged. If B be now recharged with a unit of positive electricity, and again introduced and made to touch (7, on removal it will again be found to be completely discharged, and the charge of G will be increased by one unit. This may be repeated indefinitely, and no matter how highly C may become charged, it will be found that B is always completely discharged. This is a cardinal point in the theory of electricity. Since when in contact B virtually forms a part of the conductor (7, we may state that there is no electrifica- tion on the inside of a charged conductor left to itself. We now have a means of charging a body with any number of units of electricity. A machine for the purpose of generating electricity on this principle is Kelvin's Replenisher, whose theory will be considered later. The last experiment may be modified by examining the field of force within a hollow charged conductor. This cannot be done by introducing anything through a hole, but was accomplished by Faraday by building a closed conductor large enough for a person to remain inside. Even when the outside was so highly electrified that large sparks were flying off from it, the strength of the field at points within was absolutely zero. 128] ELECTRICAL PHENOMENA. 247 EXPERIMENT VIII. Suppose that while the pieces of electrified glass are suspended as in Experiment I, we surround them with a dielectric fluid insulator, such as turpentine, kerosene, or melted paraffin. It will be found that, if the buoyancy of the liquid be just counterbalanced by weights, the threads will now hang more nearly vertically, showing that the repulsion is less. The energy of the system is consequently less. We see then that the energy of a system of electrified bodies depends not only on their charges and positions, but on the nature of the dielectric medium in which they are placed. The consideration of the part played by the medium is now one of the principal parts of electrical theory. If any of the other experiments be repeated with the closed vessel filled with any dielectric fluid, the results will be unchanged, showing that the values of charges induced on a closed conductor by charges within are unaffected by the dielectric. We will now briefly recapitulate the results of our experi- ments. We may examine the nature and the magnitude of the charge of any electrified body without altering it, by placing it within an insulated hollow conductor without touching it, and examining the charge induced on the outside of the latter. The amount of electricity on a body remains unchanged, unless it be put in conducting communication with another body. When a body electrifies another by conduction, the quantity of electricity on the two remains unchanged. When electricity is produced by friction (or otherwise, as we shall find) equal quantities of positive and negative electricity are produced. When electrification is caused by induction from a body surrounded by a conductor, the amount of electricity on the inside of the conductor is equal in quantity and opposite in sign to the charge of the inducing body. There is no electricity on the inside surface of a closed hollow conductor, charged but under the action of no internal bodies. The forces between charged bodies, and their electrical energy, depend on the dielectric medium in which they are placed. The charges induced on closed conductors by charges within do not. 248 ELECTROSTATICS. [PT. II. CH. VI. 129. Mathematical Conclusions. Law of Force. We have used the words electrification, electricity, and charge to denote a measurable quantity, which possesses the property of conservation, that is of remaining unchanged in amount. For if electricity disappears, it is by the disappearance of two equal quantities of opposite sign, whose algebraic sum was zero. We need define these terms no further than by their properties, and for the present, the single property of exerting force is sufficient. We may speak of electrification occupying definite portions of space, for the field of force is such that lines of force issue from positive electrifications and run into negative electrifications. Electrifications being always examined by examining their fields of force, we may consider the field of force as specifying the electrification. Certain writers have gone farther, and insisted that electricity does not exist, but that lines of force and electrical energy are the only real entities. Such a question is purely metaphysical, and of no importance to the physicist. It is obviously of no importance whether we define electricity as that which exists where lines of force converge, or say that the electricity exerts force upon other electrifications. If we wish to use the term "electrical fluid" or "matter" we may do so, provided we use "fluid" or "matter" simply as convenient terms, without attributing to electricity any of the properties of ordinary fluids or of matter. It has, so far as we know, no inertia, the fundamental property of matter, nor is it incompressible. We may then define a charge of electricity as a "something," "fluid," or "matter," which possesses the unique property of repelling or attracting other charges of electricity, according to the signs of the two charges. By definition the force is proportional to the charge, and it is natural to suppose that the force between two electrified elements will be in the line joining them, and proportional to some function of the distance. Experiment VIII shows that the force depends on something beside the distance, but if we suppose all space to be filled with the same dielectric medium, such as air, the assumption is justified by experiment. This supposition will accordingly be made for the present. We shall also suppose all conductors to be made of a single material. We shall now deduce the law of the force from the result of Experiment VII, — that there is no force within a hollow conductor. Let the conductor be in the form of a sphere. On account of 129] LAW OF FORCE. 249 symmetry the charge is so distributed that equal areas possess equal charges. Let the charge per unit area be , the co-latitude 0 being measured from the radius OQ. Let the angle PQO be 8. Then the whole force at Q resolved along the radius OQ is proportional to ( I ) R = (ja-dS .f(r) cos 8 = cr f* (2*f(r) cos «5 . a2 sin Od6d. FIG. 56. We may at once integrate with respect to (2) R = cos £ sin OdO. Now OQ is the sum of the projections of OP and PQ on the radius r cos 8 4- a cos 0 = 6, (3) ~ cos o = b — a cos 6 From the relation between the sides of the triangle POQ, (4) r2 = a2 + 62 - 2a6 cos 0, we get on partial differentiation with respect to b, (5) r ~ — b — a cos 0, <7C> dr b — a cos 0 = cos o. Substituting this value of cos 8 in the integral, R = 27ra27 (r}. 250 ELECTROSTATICS. [PT. II. CH. VI. so that we have (6) R = 2-TraV J- f " (r) sin do J o We may now change the variable from 6 to r by differentiating the relation r2 = a2 + 62 - 2a6 cos 0, 7-^r = ab sin #c£#, rdr (7) s For 0 = 0, r = a — 6, and for 0 = TT, r = a + b, so that . b ab Calling we have ra+ .(8) ^ a- and (9) R = ^aa- 1 JJ (^(a + 6) - ¥ (a - 6))J . By the conditions of the problem this must vanish, so that we have the differential equation for "¥", (10) which being integrated gives (IT) i ¥ (a + 6) -¥(«-&) =(76, a functional equation to determine ¥. Differentiating twice with respect to b, (12) 129, 130] LAW OF FORCE. 251 This equation holding for all values of a and 6, since a + b and a — b are entirely independent variables, M*" must have the same value for all arguments. Accordingly, putting r for the argument, (13) V"(r)=A, (14) V'(r) = Ar + B = r<$> (r), T> (is) *(>0 = ^ + -, (16) ' *Xr) -/ in which the volume and surface densities are p and r'i//J. 3s Suppose now that if we change p in the conductors to and o- to 0, (§ 58). Making the above changes in the integral (i), we have (4) and subtracting W, we get 131] ELECTRICAL EQUILIBRIUM. 255 Now F, F, SV are potentials due respectively to distributions of densities p, a- in the space K for F, p in the space D for V, and S/o, &r in the space K for SF, and accordingly by Gauss's theorem of mutual potential energy, § 117 (5), (6) jj 7rJJ 4-rr or inserting the above notation for the integrals, (3) es = q^ 4- q2sV2 . . . + qnsVn + Q.. 135, 136] SYSTEMS OF CONDUCTORS. 265 There is one such equation for each conductor. These n equa- tions determine the charges in terms of the potentials, and if the potentials of some of the conductors are given, and the charges of the rest, all the remaining charges and potentials are deter- mined. Qs is the charge of the conductor KB by induction from D when all the conductors (including Ks) are connected to earth, and consequently F, = F2=...= Fn = 0. 136. Coefficients of Induction. Reciprocal Relation. We shall now suppose the system of conductors to be under the action only of their own field, so that Q = 0. Then we have 0i = tfn "Pi + 721F2 ... -f qmVn, i , e2 = q12V1 + q^Vz • • • + qwVn, (4) The constants qrs are called coefficients of induction, and any qrs is defined as the charge induced on the conductor Ks when it and all the others are earthed, except Kr which is brought to potential 1. Any coefficient with double suffix qss is the charge of K8 when it is at potential unity, and all the other conductors are earthed. It is called the capacity of the con- ductor Ks. The dimensions of the q's are Tr = [L]. We shall now show that the order of the suffixes in qrs is immaterial. Applying Green's theorem in the second form to the functions vg and vr) we have IL (*%, - * The volume integral being taken throughout the space ex- ternal to the conductors where vr and v8 are both harmonic, vanishes, and since vr vanishes on all conductors except Kr where it is constant and equal to unity, and vs vanishes on all conductors except Ks where it is equal to unity, (5) becomes (6) --Lff ^.Iff p. *Tr]] Kgc>ne torJJfrjfa* 266 ELECTROSTATICS. [PT. II. CH. VI. We may accordingly state the reciprocal theorem : The quantity of electricity induced upon a conductor A of a system when another conductor E is brought to a given po- tential V and all the others including A are earthed, is the same as the quantity induced on B when it and the others are earthed, except A, which is brought to the same potential V. 137. Energy of System. The energy of the system is (7) Vnen), or introducing the values of es and bearing in mind the relation (8) That is, the energy is a homogeneous quadratic function of the potentials of the n conductors, the coefficients being the coefficients of capacity and induction. The energy expressed in this form will be denoted by Wv. We have (9) ~-ff = qurl + qmVt + +9n,FM = es> or the charge of any conductor is obtained by differentiating the energy-function expressed in terms of the potentials partially with respect to the potential of the given conductor. 138. Coefficients of Potential. Solving the linear equa- tions (4) we get n equations do £:£ . ' n =Pin @i T PMI &2 ~\~pnn &n > where any coefficient prs is the minor of qrs divided by A in the determinant ri2 q™~> q™ 136 — 138] SYSTEMS OF CONDUCTORS. 267 The coefficients p are called coefficients of potential Their dimensions are — = \ -f- \ • Since the determinant A is un- L«J LAJ changed by interchanging columns and rows, the determinant of the p's must have the same property, or prs = p8r. We may prove this directly as we did for the q's. Let V be the value of the potential when Kr has the charge e and all other conductors charge 0. Let V be the value of the potential when Ks has charge e and the others charge 0. Then as in (5) and since on any conductor Ki, V and V ' ' are constant and re- spectively equal to Viy V{ ', Now since the potential V is due to a distribution in which only K8 is charged, all the integrals ^n C vanish except for i = s, for which the integral is — 4?re, likewise all the integrals vanish except for i = r, when the value is — 4tire. Consequently we have (13) Now from the equations (10), putting er = e, the other es zero, Vs=prse. Again putting es = e, the others zero, Vr'=psre. Whence 04) Prs=Psr, and we have the reciprocal theorem : If a conductor A receive a certain charge, e, all the other con- ductors of the system being uncharged, the potential of any other conductor B is the same as would be attained by A if E should receive the charge e, all the other conductors being uncharged. 268 ELECTROSTATICS. [PT. II. CH. VI. Making use of the equations (10) and the condition prs=psr, the energy W=^8esV8 becomes (15) W = %pueS or W is a homogeneous quadratic function of the charges of the n conductors, the coefficients being the coefficients of potential p. This form will be denoted by We. If we differentiate partially by any charge es we get dW or the potential of any conductor is the partial derivative of the energy of the system as a quadratic function of the charges, by the corresponding charge. 139. Properties of the Coefficients. As the energy of an electrified system is intrinsically positive, the values of the co- efficients q and p must be such that the functions Wv and We shall be positive for all possible values of the Vs and e's. We may deduce certain properties of the coefficients from the elementary properties of the tubes of force and equipotential surfaces. Let one conductor Ks receive a positive unit of charge, all the others being uncharged, its potential is then_pss, and the energy We = ^psse2 = ^pss, and since this must be positive pss is positive, or: Any coefficient of potential with double suffix is positive. Any conductor Kr completely enclosed by Ks has the same potential, so that for these two prs = pss. Any conductor JTr out- side of Ks has a potential of the same sign but of less absolute value. For the charge of a conductor is proportional to the excess of unit tubes issuing from it over that entering. An uncharged conductor accordingly has as many leaving as entering. Accordingly all tubes have one end on K8 and the other at infinity. (Fig. 57), and the potential of Kr> prs is consequently intermediate between that of Ks and that at infinity, /Y) > v\ >• 0 Pss — Prs ^ v> All coefficients of potential are positive, and those with double suffixes are not greater than those with single suffixes. 138 — 140] SYSTEMS OF CONDUCTORS. 269 Secondly, let all the conductors be at potential 0, except KS) which is at potential unity. The energy is so that the capacity of any conductor is positive. The number of unit tubes of force issuing from K8 is proportional to the charge qss. Some of these extend to infinity, while others end on the other conductors. At the latter the charges will be negative, but the sum of all such charges is not as great as qss. Accordingly, every q with double suffixes is negative, and 07) qss > - (?if + ?«...+ qs-i, s + qs+i,S'" + qns)' If, however, Ks is completely surrounded by a conductor Kr, If a new Conductor be introduced into the field, the coefficient of potential with the double suffix for any conductor is diminished. For if any portion of the field be made suddenly conducting, elec- tricity will move in it so as to make the energy less than before. If Ks was the only charged body, the energy ^pssez must be diminished, but as the charge e has not changed, pss must be diminished. Introducing a new body into the field increases the capacity of any conductor, and diminishes the absolute value of every co- efficient of induction. For if the new conductor and all the others be at potential 0, while Ks is at potential unity, some of the tubes of force which before ended on the other conductors, now end on the new conductor, which receives a negative charge. This in- duces a positive charge on Ks, increasing its charge qss, and positive charges on the other conductors Kr) diminishing their negative charges qrs. 140. Work done during displacement of conductors. Suppose that we deform or displace the conductors of the system, thus changing the geometrical coefficients p and q. Suppose the configuration of the system is specified by m parameters so that if the conductors are displaced as rigid bodies m = 6n. Let the mechanical forces due to the electrification be denoted by <£ so that the force tending to change the parameter <^ is fy. 270 ELECTROSTATICS. [FT. II. CH VI. Then the work done in a displacement S^, ... Sm is (i) ^Sfc + <£,$& ...... +««*m, and if no energy is furnished to the system this work must be done at the expense of the electrical energy W and (2) In the differential STF we may use according to circumstances either of the three forms TT.r = iS.«.Fw We or Wv, which are of course identical, though expressed in terms of different variables. If we choose Wev the total differential (3) does not contain the S<£'s explicitly. For neither the coefficients p nor q appear in WeV. However, the Se's and SF's are not inde- pendent, being connected by either set of equivalent linear relations (4) or (10) above, which in the coefficients q or p involve the para- meters <£, consequently we may eliminate either the SF's or Se's, and replace them by 8'a. Now we see by the relations dWv = e§ that the functions We and Wv are reciprocal functions (§ 63) with respect to either set of independent variables el}...en, or Fj, ... Vn, containing also the independent variables , corre- sponding to the variables z of § 63. Accordingly by the last of equations (5), § 63, (4) e ~ ~ • If the conductors are insulated, so that all the charges are constant, we use the form We, so that any force 4>s has the value, from (2), (t\ ct> = ds ' The system tends to move so as to diminish the energy. If on the other hand the potentials are maintained constant we must use the form Wv. In this case we must supply energy from 140] SYSTEMS OF CONDUCTORS. 271 outside and the equation (5) can no longer be used, but in place of it we have, by (4), *-£-• The system now tends to move so as to increase the energy, and the increase of energy is exactly equal to the work done by the electrical forces. For (7) We accordingly see that the system is analogous to a cyclic system. The forces <£>s correspond to the negative values of the positional forces Ps, for the latter are denned as the forces that must be applied from outside in order to equilibrate the reactions of the system. Comparing equations (5) and (6) with (i) and (3) of § 69, we obtain the analogous results P- i _<£- fy. ' 8*. ' The electrical energy W plays the role of the kinetic energy T in the cyclic system. In order to determine which of the variables e or V are to be assimilated to velocities and which to momenta, we must recall that in an adiabatic motion work is done through the positional coordinates at the expense of the energy T. This corresponds to the case of constant charges (2). The charges are accordingly the analogues of the momenta, and the potentials of the velocities. Accordingly to an isocyclic motion will correspond a motion in which the potentials are maintained constant. We have already seen that in this case electrical energy must be supplied from without, and since this must not only do work but also increase the energy of the system by an equal amount we have the analogue of the Theorem I of § 70 : — In any motion of a system of conductors in which the potentials of the conductors are maintained constant, an amount of electrical energy must be supplied from without equal to twice the amount of work done by the electrical forces during the motion. The equations for the cyclic forces Ps — ~ are here not cat applicable. 272 ELECTROSTATICS. [PT. II. CH. VI. Example. In the case of a single conductor, the coefficients q and p reduce in each case to a single one, the capacity and its reciprocal respectively, and e = qV, V=pe, If the conductor is a sphere of radius r, we have The only geometrical parameter is here r, and since We tends to decrease, Wv to increase, r tends to increase. If the sphere is elastic, as in the case of a soap-bubble, the electrical forces tending to enlarge the sphere may be held in equilibrium by a greater pressure of the air on the outside than on the inside, or by the surface tension of the film. If P denote this excess of pressure, that is, the force acting normally on a unit of surface, the work done by the whole surface 8 in increasing the radius by dr is PSdr. If T is the surface tension of the film, or the elastic force tangential to the surface exerted normally across a curve on the surface per unit of length, in increasing the surface by dS we must do work TdS. Hence we have = PSdr + TdS = -dWe = dWV) S = *irr\ dS = Sirrdr, We = 47r {Pr2 + 22V} dr = | ^ dr = \ e* = 8-Trr3 {Pr + 2T\, If the soap-bubble be blown on a tube connected with a mano- meter, the difference in pressure P may be observed. T may be determined by an observation when the bubble is unelectrified. Calling r0 the radius under these circumstances, P0 the pressure, T = -- p , Po being negative, 140, 141] ELLIPSOID. 273 and using this value of T, Accordingly a potential may be measured in this simple manner by a measurement of P and r, P0 and r0 having been observed. If the tube on which the bubble is blown is open to the air, P = 0, and 141. Distribution on an Ellipsoid. We have found the potential clue to an equipotential layer of amount e distributed on an ellipsoid of semi-axes a, b, c to be, § 109 (12), v=e~ r ds 2Jx V(tt2 + 8) (62 + *) (C2 + S) ' where X is the greatest root of the cubic x2 * z At the surface of the ellipsoid X = 0 ds so that the capacity q is the elliptic integral 2 ds The surface density of the charge is given by 1_ dV_ JL_ fdV ~- which by § 110 gives e Ellipsoids of Rotation. If a = 6, the ellipsoid is one of rotation, arid the elliptic integral simplifies into r ds 2 . Vo^c2 , I — ,— — =- = .-- sin"1 - — — , if a > c, W. E. 18 274 ELECTROSTATICS. [PT. II. CH. VI. i.e., when the ellipsoid is oblate, and f ^ =^=10 Jo (a2 + s) V c2 + s Vc2-a2 when the ellipsoid is prolate. The capacity is — — 1 1 ir -< 'rl Va2 - c2 ' sin"1 a Vc2 - a2 ir ., , prolate ellipsoid. For a very long prolate ellipsoid, neglecting f-J , c «=— • so that as a approaches zero, the capacity approaches zero, but more slowly, viz., logarithmically*. In the limiting case of an oblate ellipsoid, for c = 0, we have a circular disc, whose capacity is If in the expression for the surface density we eliminate c by the equation of the ellipsoid, a2 b* /*£*(- a? f* V a4+64+7A a2"^ 141, 142] ELLIPSOID. CONCENTRIC SPHERES. 275 If we now make z — 0, we get the density on an elliptical disc e and if a — b, 4-rra Va2 - r2 ' for a circular disc of radius a. At the edge of the disc, and the density is infinite, so that this case is not physically possible. It is however of considerable theoretical importance. For the case of the circular disc the potential at any point becomes Tr e f°° ds e (TT . Vx) e , a V = ^ — ._ = - \- - tan"1 — \ = - tan"1 -= , 2 J \ (a2 + s) v s & (2 a j a Vx where X is the greatest root of the quadratic ^ "*" ^2 4. z- — i a2 + X X 142. Concentric Spheres. Suppose we have a sphere of radius J^x, surrounded by a concentric spherical shell of radii R% and Rs. In the space between the conductors and outside of the outer, V satisfies the equation, § 88 (7), whose integral is dVA . r If Fj is the potential of the inner sphere, F2 of the shell, 18—2 276 ELECTROSTATICS. [PT. II. CH. VI. which determines A : A - ^^2 (V V\ Rz-Ryi~ The surface density on- the inner sphere is The charge on the sphere is In like manner, at the surface r = R^ differentiating by — r, we get for the charge e2', To find the charge at the outer surface r = R3, we must redetermine the constants A and B. Since V = 0 r" V, and the charge at the outer surface The whole charge of the conductor 2 is e, = el + e2" = -J^- ( V, - F.) + FA- We accordingly have for the coefficients q 142, 143] CONDENSERS. 277 143. Condensers. The capacity qn decreases as Rz in- creases, becoming equal to ^ when R.2 = oo . Accordingly by the presence of the envelope the capacity of the sphere is increased T> in the ratio 2 which may be made very large. Such an Jl>2 — KT, arrangement of two conductors, by which the presence of the second largely increases the capacity of the first, is called a con- denser, for by it a larger quantity of electricity is condensed on the first by raising it to a given potential, the second being to earth. The coefficient qu, or — ) F2 F3 , giving ~F7"2 l/iT7'2_4_jL//> /^M^2 />T7"T7"_i_rl7"T/r ^ 2^1rl 2L/2'2i2 V^l °t/ '3 Klrlr*iVB'i'8« If the electrometer is correctly constructed and adjusted Cj = c2 and The needle is usually suspended either by a torsion fibre, or by a bifilar suspension, so that the force of restitution is proportional to the deflexion, the factor of proportionality being denoted by A. In the usual method of use, the potential F3 of the needle is made large in comparison with V1 and F2. We may then neglect the second term in the brackets, and the deflection is proportional to F2 — Fj. This is called the heterostatic method of use, the needle being charged by an extraneous source of potential. In the idiostatic method, the needle is put into connection with one pair of quadrants, which are put at the potential to be measured, the second pair of quadrants being to earth. Then 148] ELECTROMETERS. 285 F2=F3 = F, F1 = 0, <£ = -F2 2 V ' and the deflection is proportional to the square, instead of to the first power of the potential. This method does not show the sign of V like the former. If the electrometer is not in perfect adjustment, we use the more general form <£> = A0 = ic^2 - ic2F22 + i(d - c2)F32 - c1F1F3 + c2F2F3. In order to be able to adjust cx and c2 to equality, two of the quadrants should be capable of motion toward or from the center, one roughly, the other micrometrically, so that the amount of surface of the needle covered by them may be varied. In order to make the adjustment, we may first put both pairs of quadrants to earth and observe the deflection when the needle, originally to earth, is charged. Calling this 0Q, we have which shows whether d or c2 is greater. We may then adjust until there is no deflection, however the needle is charged. If a high potential is not available for F3 , we may conveniently proceed as follows : By means of a voltaic battery and two commutators, we may charge either of the quadrant-pairs to a given potential F either positive or negative, the other quadrant-pair being to earth. We thus have four combinations, as follows : e v, F2 reverse commutator A -V Oh D reverse „ B _v et o Fjreveree The deflections are given by i - c2)F32 + ClFF3, l - c2)F32 - c2FF3, I - c2)F32 + c2FF3; 286 ELECTROSTATICS. [PT. II. CH. VI. Taking differences we obtain In this manner we can accurately bring the ratio — to unity. C2 Whether the adjustment be made or not, and without the necessity of making F3 large, if we can reverse the sign of V we may by observing 01 and 02 get the correct value, since so that V is directly proportional to the difference of the two deflections, or to the arithmetical mean of their absolute values. 149. Induction Electrical Machines. As a further example of induction in 'a system of conductors, we shall con- sider the action of a class of electrical machines typified by Lord Kelvin's Replenisher. This consists essentially of two semi-cylindrical conductors A and B called the inductors, and two smaller conductors G and D called the carriers, which may be rotated as a rigid system about the axis of symmetry. If Vl be the (positive) potential of A at any time, F2 that of B, supposed negative, then if C and D be put in conduct- ing communication with the earth while in the position shown, G will have a negative, and D a positive charge induced upon it. Now on insulating G and D, and turning them until G Tji nn is opposite B and D opposite A, if C be put into communication with B, being nearly surrounded by B, it will give up its charge, thereby increasing the absolute value of the negative potential of B. D being put into communication with A gives up its positive charge, and increases the positive potential of A. The connections of G and D with each other and with A and B are made automatically by contact springs once in each half revolution. If Ft(n) and F2(7l) are the potentials of A and B after n half- revolutions, KI and Kz the capacities of A and B and whatever 148, 149] INDUCTION MACHINES. 287 conductors they are respectively connected with, their charges are p (n) _ IT Y (»> f> M — K V 7rJJs A function satisfying these conditions is called Green's function for the space r and pole P. The problem is unique, if it has a solution. For if there are two solutions GI and 6r2, by 3°, so that by subtraction for any harmonic function V. But by 2°, ^_1 and G2-- r r are harmonic, so that their difference GI — 6r2 is also harmonic. Applying the above result to the harmonic function Gl - Gz, (s) * Green, Essay, § 5. The name Green's Function is due to C. Neumann, who applies it, however, as does Maxwell, to the function G - - . 151] GREEN'S FUNCTION. 291 But by Green's theorem this is equal to the volume integral which, as in § 86 can vanish only if GI — 6r2 = const. That is, with the exception of a constant, Green's function is unique. But as in the employment of the function only its derivative is used, the constant makes no difference. Since the function G — is harmonic, we have by § 33 (2) or transposing, cni r d' , by § 83 (6). If on the surface G = 0 we obtain Consequently if we can solve Dirichlet's problem for the given space, obtaining a harmonic function F which takes at the surface S the values then the function G = F 4- - r solves Green's problem. Conversely if we can solve Green's problem for the space and for any pole P, the equation (1) enables us to find any harmonic function V from its values at the surface, solving Dirichlet's problem. The problems of Green and Dirichlet are thus exactly equivalent. In physical language, Green's function is the potential due to a positive unit of electricity placed at the pole P together with that of the charge which it induces on the surface 8 made con- 19—2 292 ELECTROSTATICS. [PT. II. CH. VII. ducting and connected to earth. If o-G is the density of the induced charge, 1 ~ and (1) is Fp = -|l VtrcdS. Suppose that G is Green's function for a certain space, with the pole P, whose co-ordinates are a, b, c, and that G' is Green's function for the same space, but a different pole P' whose co- ordinates are a', &', c. Then there exists the reciprocal relation that the values of either function at the pole of the other are equal. For ' where the suffixes indicate from what point the distance is measured. Now since F and F' are harmonic, by the property of the two Green's functions G and G ', (8) so that The last integral but one vanishes because F and F' are harmonic functions, while on account of the surface values of F and Fr, the last becomes dn rp Since both the functions l/r> and !/?>• are harmonic except at their poles P and P', by constructing small spheres about the points P and P' and proceeding as in § 83, we find that the two parts of the last integral destroy each other (each being equal to 151, 152] GREEN'S FUNCTION. 293 , so that T? — T'P. Accordingly we have for the two points P and F, (10) GV = IV + - In order to show the dependence of the function G on the co- ordinates of its pole P let us write it (! !) Q(x,y,*) = 9 O, y, z> a, b, c), and G' (x, y,z) = g (x, y, z, a', &', c'). Then by the above theorem G(a',b',c')=G'(a,b,c), (12) g (of, b', c', a, b,c) = g (a, b, c, of, 6', c'), or Green's function is a symmetric function of its variables a, b, c and a', 6', c'. 152. Examples of Green's Function. Plane. Let us seek Green's function for all that portion of space lying on one side of a given plane. Let A be the given pole, at a distance a from the r. A/ *rl B Fio. 62 a. plane, on the left, and let B be its geometrical image in the plane. Let the distances of any point at the left of the plane from A and B be r and r respectively. Now for every point at the left of the plane the function -, is harmonic, and for points on the plane, where r = r, it assumes the value — . It is therefore the function F of the preceding article. We have then - "r 7' dG _ cos (rii r) cos fa r') _ 2 cos 0 dn~i~ ~r*~ r'2 r2 ' where 6 is the acute angle included between the radius r and the normal to the plane. Consequently, the equation 294 ELECTROSTATICS. [PT. II. CH. VII. solves Dirichlet's problem for the left-hand side of the plane. If we suppose a charge of a positive unit placed at A, and a negative unit at B, the plane of symmetry will be an equipotential surface of zero potential, and we may apply the theorem of equipotential layers. If the plane of symmetry is made conducting, and the charge B removed, the conducting plane receives a charge — 1 which screens the space on the right from the action of A. The surface density on the plane is OG = — cos 0/2-Trr2, so that the whole charge on the plane is, applying Gauss's theorem, — 1 /7cos0 , This is an example of the second theorem of § 150, the space on the right being considered internal. The charge — 1 at B is said to be the electrical image in the plane of the charge + 1 at A. Two point-charges A and B are said to be electrical images of each other in a certain closed surface separating them if either one, gay B, produces in the portion of space in which the other, A, lies, ;he same effect as would be produced there by the charge induced n the surface made conducting and connected to earth, by the joint A alone, the image B, being removed. 153. Planes intersecting in a sub-multiple of two right angles. Let us seek Green's function for a portion of space lying in the acute angle between two planes intersecting in an angle which is equal to two right angles divided by an integer. Let the planes be denoted by 1 and 2, let the pole be P, and let Pl be the geometrical image of P in 1, P2 that of Px in 2, P3 that of P2 in 1, and so on alternately in the two planes. Let Qi be the image of P in 2, Qz that of Ql in 1, Q3 that of Q2 in 2, and so on. Since the angle is a submultiple of TT it is easily seen that the series of images will be finite, the Q's and P's finally coinciding. Let the distance of any point from P be denoted by r, from any Ps by rs, and from any Qs by rsf. Then the reciprocal of any distance rs or rs' is a harmonic function in the space between the planes since none of the images lie in that space. Also for all points lying on the plane 1, i_I = o l-l=o I-i=o l-l=o r r, r,' r,' r, rz r,' rt' 152 — 155] GKEEN'S FUNCTION. 295 and for all points lying on the plane 2, r r n r2 Consequently the function vanishes for points on either plane, and being harmonic except at P, is Green's function. 154. Two parallel Planes. If P lie in the space between two parallel planes the successive images will all lie in a straight line, and will be infinite in number. Using the same notation as in the last example, we have the same equations, and the same form of Green's function, except that we shall have an infinite series. 155. Sphere. Let A be the given pole, at a distance a from the center of the sphere of radius R. Take a point B lying on the same radius as A, at a distance from the center 6 such that ab=Rz. Then A and B are said to be inverse points with respect to the sphere. If M be any point on the surface of the sphere, the triangles OMB and MAO are similar, for they have a common angle at 0, and the sides including it are proportional, for by hypothesis, FIG. 63. Accordingly, for points on the surface r' r . I Rl <2> E=aand7==a/ 296 ELECTROSTATICS. [PT. II. CH. VII. r and r being the distances of any point from A and B respec- tively. Therefore, since I// is harmonic in the space containing A, Green's function is for that space n l R l (3) **r~ir" dG _ 1 dr R 1 dr' cos (^r) R cos ^^2 + ~rz~~ ~~~ f (5) so that the density of an equipotential layer induced by a unit charge at A on the sphere made conducting is 1 [COS (UjT) R COS («&' ~ ~~~ Now in the triangles OMB and MAO we have a- = R2 + r2 - 2jRr cos - cos so that cos (np) _ E cos (mrf) _ a2- (R2 + r2) R (&*_- (R* + r'2)) ^ "^7 ^/2 9 P/*^ *" ^ which by ( i ) and (2) gives p , and The whole induced charge is 47rJJ V ^ a r2 and if J. is an outside point by Gauss's theorem so that 155] GREEN'S FUNCTION. 297 If we should place a charge e at the outside point A, and a charge e at the inside B, the potential on the surface of the e e' sphere would be V— - + —. which, if we make e = — Re/ a becomes r r zero. The action of the charge e at B in portions of space outside of the sphere may thus be exactly replaced by making the sphere conducting and removing the charge B. Accordingly the charges at A and B are electrical images of each other in the sphere. Suppose now that the sphere, instead of being connected to earth, is insulated and charged to a potential V, then beside the induced charge it will have a uniformly distributed charge VR of density V/4i7rR, so that the whole charge of the sphere is now (I3) Jf The surface density 4-7T (R ' Rr* vanishes along the circle Fr3 = a2 — R2, which divides the surface into two parts oppositely electrified. If however or (a - R)* the surface density is of the same sign all over the sphere. Since the action of the induced charge on external points is the same as would be that of a charge e' at B, and the action of the uniform charge is the same as that of a charge VR at the center, the repulsion of the whole charge of the sphere on the charge e at A is FZte e'e ^(V ea (15) -TiT-+7^ i^ = e' = — \& — e This is negative, so that there is an attraction, when V= 0, or E = 0, or a — R is small; that is if the sphere is connected to earth, if it is insulated without charge, or in any case if the charged point A is very near to the sphere. On the other hand, by making V or E of the same sign as e and great enough in absolute 298 ELECTROSTATICS. [PT. II. CH. VII. value, we have a repulsion, when or \E\> ear 156. Electrical Images in a Sphere. Points which are electrical images of each other, besides having the properties connected with equipotential layers described above, possess peculiar reciprocal properties with respect to the portions of space in which they are respectively situated. There thus arises a method of finding from the known solutions of electrostatic problems a new class of problems whose solutions can be found. This method of electrical images was discovered by Lord Kelvin in 1848*. Suppose as before that A and B are inverse points with reference to the sphere of radius R, A being outside. Let M and M1 (M outside) be two other inverse points situated at distances I and I' from the center, and at distances r and r respectively from A and B. Then the triangles 0AM and OM'B are similar, since ab— II' = R*. Suppose a charge e placed at A, and a charge e = — eR/a placed at B. If we call V the potential at M due to the charge e, and F' the potential at M' due to the charge e', we have V _e r_J r__Rl_ _l^_ _R T~~r'' ~e~ e ' r'~ a b~ R~ I' ' If then we have any number of electrified points such as A, and find their images B, and if V be the potential of the system A at any external point, M, then RV IV & • r- -*• will be the potential at M ' the inverse point to M, of the system B which is the electrical image of the system A. We shall give an analytical proof of the same proposition, based on the method of curvilinear coordinates. If x, y, z are the coordinates of the point M, of, y'y /, those of the point M', we have x jx — y'\y — z'jz and since IV = R2, -_ „ '~W~* y z I'2 = a/2 + 2 4- * Papers on Electrostatics and Magnetism, p. 144. 155, 156] ELECTRICAL IMAGES. If x, y, z are given, we know #, y, z and the position of M, so that we may consider x', y', z given by the above equations, as curvilinear coordinates of the point M, disregarding for the present their relation as rectangular coordinates of the inverse point M '. Forming their differential parameters, a# 2 (i _ 2^ dx ~ I/2 I4 (4) ^r = -2^2ir> to It is easily seen that the surfaces, x' — const., y' — const., / = const., cut each other orthogonally, for example the cosine of the angle between the normals of x' = const, and y' = const, is proportional to ty Md£ dtfty_ p4 [4 (x*y + xy* + xyz*) 4ney] _ x dx + dy dy ^ 3z dz " \ Is ' I* J = We have then by § 87, (5) a /iar\ a and performing the differentiations ^ M ' e Now we have ic' 3iE' , „ , ' ''+ ' ' " 300 ELECTROSTATICS. [PT. II. CH. VII. Forming the derivatives for y and z, writing 92 a2 a2 and comparing with the expression for A V above, we get But I / 1' is harmonic except at 0, and therefore (8) L If we put (9) e/ ' and we get the proposition that if V is a harmonic function of the point M, then V is a harmonic function of the corresponding point M'. If the distribution causing F is distributed continu- ously in three dimensions, the density is p — — A F/4?r and in the image the density is p = — A7 F'/47r so that «°> *•• £-*-*• " If ds and ds be corresponding infinitesimal arcs, expressing ds in terms of the curvilinear coordinates x, y', z dx'z dv'2 dzf* I4 , l*ds* so that we have for the ratios of corresponding infinitesimal arcs, surfaces, and volumes ^_E2_Z^ dS'_&_l^ dT^_It?=l^ Ts~l*~R*' dS~l*~R*' dr~l«~W The ratio of charges of corresponding infinitesimal volumes is de' _p'dr' _R_lf de~ pdr == I ~ R" and of the surface densities v__ Z) ~ y ' If the values of the functions for points internal and external to the sphere S be distinguished by the suffixes i and e, on the surface S, since x = x'} y = y', z = z, (3) Wt'=Wt, W.'-Wt. W' (x, y, z) vanishes for / = oc and is finite for I = 0 since (4) W (x, y,z) = j^W (V, y', z'} and lim Wf=~ lim I' W (of, y', /) = const. Let us now put (5) V(x, y, z) = W(x, y, z) + W (x, y, z), and we shall show that the function W may be so defined that V will be the potential of an equilibrium distribution on the spherical segment. We have seen in § 141 that the potential at any point due to an equilibrium distribution on a circular disc of radius a is e (TT J\] -15-— tan"1 — }• , a (2 a } where X is the greater root of the quadratic The derivative of this function according to the normal of course has a discontinuity by changing sign on crossing the disc. If we consider the disc placed in the mouth of the bowl, on 304 ELECTROSTATICS. [FT. II. CH. VII. account of the change of ^-coordinate we must take X as the root of the equation and if X' be the same function of x', y', z that X is of x, y, z we must have v ~£4' We will now define our function W by two different analytic expressions. In the space T we take and in the space T' and T" This makes TF continuous at 2 as required, since on the disc X = 0, and the change of sign in the second term makes the normal derivative continuous in crossing the disc 2. By the definition of W we have in T and T" (since the inverse of T is T', and of T" is itself) and in T' Accordingly we have for the values of F in (9) r, F'=|-tan-.f H-fgn-tan-f), 157] ELECTRICAL IMAGES. 305 The function V is everywhere continuous, for W and W are continuous except at S and there, by (3), (10) Vi=Wt+ W{ = We'+ We=Ve, so that V is continuous. We have already seen that the derivative of W is continuous in crossing 2, and accordingly that of W is continuous in crossing 2'. Now the derivative of W is continuous in crossing 2', since W is defined by the same continuous analytic expression in T' and T", and the derivative of W is continuous in crossing 2, since W is defined by the same continuous analytic expression in T and T". Accordingly the derivatives of V as well as V itself satisfy the required conditions of continuity, and on S, since x — x', y = y, z = zf, I = R, X' = x, we have F=TT and F, the function assumed, is therefore the potential of an equilibrium distribution on the bowl. At the center of the bowl we have on the one hand while in order to employ the formula (9) we have x = y — z — ^, X = c2. But when I is infinitely small, X' must be infinite of the second order, as we see by making I infinitesimal in the equation <*> The terms of the lowest order are V/4 ^+^+^=^. Hence approximately x'-*4 I* ' and TF'(0) = limf (f -tan-'f ) = Ihn £ tan-| = |. I=Q i \4 iaj I=Q l iff A Therefore we have finally ? + tan->%| = -l, (13) e = W. E. 20 306 ELECTROSTATICS. [PT. II. CH. VII. If we call 7 the angular half-opening of the bowl, a/R = sin 7, a/c — tan 7 and the charge of the bowl is (14) e = R (IT — 7) + R sin 7, giving as its capacity To complete the problem we have to find the surface density. We find 3F = 1 1 8X R I _JL_SX/ dnt ~ X ' 2a \/X fa* I -, V a2 a2 Rdl fir l Vx7\ __ _ _ ton — * _ 1 • aF7= 1 1 8X jR 1 1 d\' dne~ l \ ' 2a VX 9we ^ x V a2 a2 R dl /TT - i; o~ ^ t2 9we \2 Now on the surface S v/ _ av_ ax 7-7? ^_ _8[_ :A" 97i""8^' ^ 8r2 _t- tft -L- fz\ rv\ -4- 9/^2 \J/ t 2 /^T7\2 /32 T7\ 2 /3 T7\ 2 /3 l/\ 2 7> 2 ' fly [0 V \ (Or\ (u v \ 10 V \ fly or the ratio of the second differential parameter of any function to the square of the first is unchanged by a conformal transformation. We may call such a quantity an invariant of the transformation. 310 ELECTROSTATICS. [PT. II. CH. VII. We now require the condition that an equation (#, y)=G represents an equipotential family of curves. In this case we shall have for the potential function F, F=/(^>) and as in § 108, (2) (6) so that /'()' The right-hand member depends on alone, consequently the left hand must also. Consequently in order that <£ (a?, y)= C shall represent an equipotential family the ratio of the second to the square of the first differential parameter of 0 must be a function of alone. Let now c/> (x, y)=C represent an equipotential family, and let (u,v) = C' be the transformed family. Since by (5) V2 ~ V ' and since A<£//^2 depends only on cf>, A'/^/2 will depend only on , for and are constant together. Accordingly a conformal transformation leaves every equipo- tential family equipotential. It is upon this property that the application to electrostatical and other physical problems depends. If we integrate the second parameter of F over a portion of the JTF-plane where it does not vanish, using the element of area in curvilinear coordinates dudv / Q\ I I I v r . *•* ' X 7 i I I T „ / I/ r . V f \ CtUCtV (°) and now considering the second integral to refer to the trans- formed plane, and e and e' to be charges of corresponding regions, (9) e=jj p dxdy=-^H&Vdxdy = -^ ff XV dudv = II p dudv — e', or corresponding regions in the two planes have equal charges (the densities being different). 158,159] CONFORMAL REPRESENTATION. 311 If "SP" be the conjugate function to V, we have for the charge upon any conductor V=C between points A and B, e = I ads = — -j— I ;r — ds - J A 4f7rJ Adne or since cos (nx) ds = dy, cos (ny) ds = — dx, i (B fiv , dv, \ e=—.-~ \-5-dy — -^- da) 4f7rJ A \dx ' d J so that the flux of force of any tube of force is measured by 1/4-n- times the difference of the values at its two sides of the conjugate function to the potential, as in § 103. 159. Examples. Eccentric Cylinders. Let us transform by means of the function w = log z giving (§ 45), 1 i ) u = log r = log A pair of parallel planes u=V1)u=VZt transforms into a pair of concentric circular cylinders r = rl} r=rz. To the potential function V— u we have the conjugate function 'W = v so that for the charge of the cylinders we have between <£ = 0 and 2ir, (2) e = -:— {^27r - ^0} = T'- . 2?r=4, 4?r l 4?r 2 and the capacity is (3) K = + V6_ v } = — Z— = ± 1/2 log ^ , as in § 144. ±logn The use of the fractional linear function w = (2 + «)/(•& — »), gives us an important new result. Replacing i in (4) » + by — i, as a reference to the theory of the complex variable shows is always possible, gives ,. . x — iy+a (5) u-iv= ¥- , x — l — a 312 ELECTROSTATICS. [PT. II. CH. VII. and multiplying together, ~~ (x — a)2 + 2/2 ' We will now make use of the results of the preceding example, denoting, however, by our present u and v what were there denoted by x and y. The cylinders (as + a)2 + transform into (7) Vu, — u/y -r y \aJ- which on clearing of fractions, O2 + 2/2) (1 - rf) 4- 2a# (1 + (8) + a2(l-n2)=0, + a2(l-r22)=0, are seen to be eccentric circular cylinders. Their trace on the new XF-plane is shown in Fig. 65, which represents the transformation of the right-hand part of Fig. 24 by means of the function a w log — a If we denote for either cylinder the radius by R and the distance of the center from the origin by d, since we may write (8) FIG. 65. 158] CONFORMAL REPRESENTATION. 313 1 4- r2 /I l r2\ 2 (9) * + f + 2a* -j + a? {^J — 1 > = we have r2 + 1 r + l/r „ ± 2ar ± 2a a = a — — 7 = & -- ST-j ^~~y - 7 = -- 7T~- r-2 — 1 r— 1/r r2 — 1 r — 1/r from which < + c2 - 2 <"> r= -±^" Since ^2 and r must be positive, we take the upper signs in (10) and (i i) for r > 1, which makes c£ > 0, and gives the circles on the right, the lower for r < 1, which makes d < 0, and gives the circles on the left. Now making use of the results of the last example, we have for the functions V and SP V= log r = log (w2 + t;2), so that in the transformation, for r = i\, r = r2, T. , , c^ F! = log n = log - (I3) di F2=logr2 = log- and the capacity of the pair of eccentric cylinders is ±1 * (14) In case r2=l, c?2 and R2 become infinite, and we have for the capacity of a single cylinder in presence of the infinite conducting plane x — 0, (15) K= The formula given above for the capacity of a pair of cylinders of which one is internal to the other is not convenient in practice, since we are given not the distances dlt d,2, but only their difference 314 ELECTROSTATICS. [FT. II. CH. VII. d = d2 — d1} the distance apart of the lines of centers, together with the radii Rl} R2. We must therefore solve the equations (I6) 4 = ag±l, (17) d, = a'?±±, (18) ^ = 20-, (19) a, = 2a, (20) d.2 — dl = d, so as to obtain r1? r2, a, dl} d2) in terms of R1} ^2, d We need for use only the ratio 7*1/7*2 . Eliminating c?1} c?2 from (20) by (16) and (1 8), (17) and (19), and a by (18), (19), (22) jR»%- — RiTl^ 'Zr2 LT^ Taking the sum and difference of these two equations we obtain (23) Rzr2 - R^ = d, / \ -*-*'9 -*-*/l 7 (24) — - — = d- Multiplying these equations together or a quadratic for rjr^ or ra/r^ Solving we obtain ± It is easily seen that taking the square root with one sign makes the whole expression the reciprocal of its value with the other sign. Consequently we use the upper or lower sign according as r2 is greater or less than ?v The capacity is accordingly (27) K= — 0 , log "~ 2R& } ' I which for d = 0 becomes ± 1/2 log(^2/-R,) as in § 144. 159, 160] CONFORM AL REPRESENTATION. 315 If the two cylinders are external to each other, we must insert the minus sign on the right of (18), so that the equations are the same as before, with S^ replaced by its negative. Accordingly we obtain - Rf - d% 2 log The formulae (15), (27) and (29) are important in calculating the capacities of telegraph wires. 160. Elliptic and Hyperbolic Cylinders. In the pre- ceding examples we were given a function of a complex variable, and from that obtained a conformal representation. We will now consider a case in which we are given a set of orthogonal curves, and we shall seek a function of a complex variable which will make them the conformal representation of orthogonal straight lines. The functions X and ^ defined by the equations * * ' a2 + X b2 + \ ' are a pair of orthogonal coordinates. Solving for x and y we obtain X = A / — , (2) y = V'~b-cf + ^' and differentiating logarithmically cfo? 1 / d\ x dy _ 316 ELECTROSTATICS. [PT. II. CH. VII. so that _lj / ^T7^ 7 / 0?+^ ) (4) ) + (a2 - 62) e~w (cos v — i sin v)] = i. [e«+*> + (a2 _ #» which gives the form of the function sought, (13) * = i{^+(a2-62)e-w}, or (14) w = log {z ± *!& - (a2 - ft2)}. We may now conveniently change our unit so that the focal distance Va2 — b2 shall equal unity. Then the function z becomes the hyperbolic cosine of w. A table of comparison of the principal properties of the hyperbolic and circular functions is appended*. * 160 A. Hyperbolic and Circular Functions. coahx=%(ex + e-x) si: rihx = %(ex-e-x) cosh"1 x = log (a; ± */a;2 - 1) si] ah"1 a; = log (x ± >/a;2 + 1) tanh x = sinh a;/cosh x (I) <*) sinh ( - a;) = - sinh x. cosh ( - a;) = cosh x. M sin ( - a;) — - sin x. cos ( - a:) = cos x. (3) cosh2 x - sinh2 x = 1. (30 cos2 x + sin2 x = 1. - (4) 1 - tanh2 x = sech2 x. (40 l + tan2a; = sec2a:. (5) sinh (a: ± ?/) = sinh x cosh ?/ (50 sin (a; ± ?/) = sin x cos ?/ ± cosh a; sinh y. ± cos a; sin y. (6) cosh (a; ± y) = cosh a: cosh y (60 cos (a; =t ?y) = cos a; cos y ± sinh a; sinh y. =Fsina: sin?/. (7) tanh a? ± tanh y (70 ta.n(x±v) tan-r±tan2/ • y' 1 ± tanh x tanh y' v y' 1 T tan a? tan y /Q\ . cosh2a;-l /o'\ 1 - cos 2a; \°t 2 (o ) 2 (9) cosh 2a; + 1 (90 l + cos2# cos a? ^ 10) — sinh x = cosh a:. dx (ioO (cos v' ~t~ * s^n v') ~ cosn x cos y + * sinh x sin y, so that (4) eu' cos v = cosh x cos y, eu> sin v' — sinh x sin y, from which, in the same manner as above, cos2 v' sin2 v 1 cos2 v' sin2 v 1 cosh2 x + sinh2 x~~^" cos2 y ~ sin2 y = ~^' ' and taking logarithms, , 1 , f cos2 ?/ sin2 v' 2 ° (cosh2 a? sinh2 a , 1 , f cos2 v' sin2 v' nt' _ __ IfiCT •< • — 2 5 (cos2 y sin2 y From these equations the curves corresponding to x = const, and y = const, may be immediately plotted by the aid of tables of logarithms and hyperbolic functions. They are shown in Fig. 67. It is at once seen that u' is a periodic function of v', the period being TT. The figure is the same for negative x and y as for positive. In order to represent the whole of the £7F-plane corresponding to the half strip in the X F-plane, we must however let v' vary from 0 to 2-7T. The curves x = const, are sinuous curves, u' having maxima for v = 0, TT, 2-7T, . . . and minima for v = Tr/2, 3?r/2, .... The maxima u = log cosh x and minima u' = log sinh x differ but little for large values of x, since then approximately cosh x = sinh x = e?/2 so that we may then take out this factor from u', obtaining u —x — log 2 for all values of v, so that the curves x = const, are nearly straight lines. As x diminishes the maxima and minima both diminish, but get farther apart, the maxima being always positive, while the 161] CONFORMAL REPRESENTATION. 323 minima eventually become negative. The curves all cut the axis of u' to the right of the origin, but stretch out farther and farther 27T FIG. G7. U' F'-plane. toward the left, so that for # = 0 the curve reaches from 0 to minus infinity, coinciding with the left-hand half of the u' axis. 21—2 324 ELECTROSTATICS. [PT. II. CH. VII. In fact we see from the equation that for any finite u' , v must be zero when x is zero. The curves y = const, are different in appearance, on account of the minus sign, u has minima for v'=Q, TT, 2-Tr... having the values u = log cos y which are all negative, and decrease more and more rapidly as y increases to Tr/2. The maxima of u' are how- ever infinite. In fact while u increases continuously as v varies from 0 to 7T/2, as soon as v > y the parenthesis becomes negative and u is imaginary. The curves y — const, accordingly approach horizontal asymptotes v' = y. These curves correspond to the hyperbolas of the last figure, the sinuous curves corresponding to the ellipses. Corresponding regions in the three figures are simi- larly shaded. The circle in Fig. 66 corresponds to the vertical V- axis in Fig. 67. If we choose for the functions V and ^ the values V—y, ty = x, and consider the strip between v' — Tr/2 and v = — w/2, we have the case of the electrification of an infinite plane with a free edge, lying between two infinite planes parallel with it at distances 7T/2 from it, and extending to infinity on all sides. Since at a distance from the edge x is equal to u + log 2, the field is straight, but the charge from the edge to the point u' is greater by (log 2)/47r than if the plate extended to infinity instead of stop- ping at the edge. Thus the edge increases the capacity of the upper or lower side of a portion of the plate of any width by the amount K = (log 2)/47r ( F2 - Fa) = (log 2)/27r2. This result may be used to find the capacity of a circular plate between two infinite parallel plates at a distance from it d so small that the edge of the circular disc may be considered straight *. The effect of the edge is the same as that of increasing the radius by (log 2) d/(ir/2), so that the capacity would be, counting both sides, d (lo£ 2V7r _ & 2 * Maxwell, Treatise, Vol. i. Art. 196. CHAPTER VIII. ELECTROKINETICS. STEADY FLOW IN CONDUCTORS. 162. Ohm's Law. The condition of equilibrium of electri- city in homogeneous conductors has been found to be that in each conductor the potential has a constant value. If this condition is not fulfilled in any conductor, the electrification changes with the time, if the conductor be left to itself, or in ordinary terms elec- tricity moves from one place to another in the conductor. The laws of this flow of electricity were enunciated in 1827 by Georg Simon Ohm*, although the notion of the potential was unknown to him. If at any point in the conductor we construct an element of surface dS, the quantity of electricity q crossing the surface in the unit of time per unit of area will vary according to the direc- tion of the normal to dS at the point. That direction of normal for which the quantity per unit of time is greatest is called the direction of the current at the point, and the quantity q is called the current density. The current density is a vector quantity, and its components according to the axes will be denoted by u, v, w. If the quantities u, v, w are independent of the time, we call the state of the conductor a state of steady flow. We shall now consider the properties of the steady state. If we consider any portion of a conductor in which there is no electricity created nor destroyed, as much electricity must enter the space during any interval as leaves it, or the whole flow resolved along the inward normal must be zero. Accordingly ( I ) 0 = 1 1 q cos (qn) dS = II {u cos (nx) + v cos (ny) + w cos (nz)} dS Mdu dv dw\ ~- -f ~- + ~- }dr) dx dy fa) * G. S. Ohm, Die galvanische Kette mathematisch bearbeitet. Berlin, 1827. 326 ELECTROKINETICS. [PT. II. CH. VIII. and as this must be true for any portion of space fulfilling the above conditions, we must have everywhere in such regions du dv dw . (2) r- + 5- + 3- = 0. dx dy dz This is called the equation of continuity, and shows that the current density is a solenoidal vector. Current lines and tubes accordingly possess properties similar to those of tubes of force in the case of equilibrium. The law of Ohm is identical with that stated by Fourier* for the conduction of heat, and connects the current density with the potential, the corresponding quantity for heat being the tempera- ture. If the conductor be isotropic, that is if its properties are at each point the same for all directions, the direction of the current is the same as that of the electrostatic field, and their magnitudes are proportional, the factor of proportionality depending on the physical properties of the conductor at each point. If n is the normal to an equipotential surface at the point in question drawn in the direction of the current, we have (3) ff-vr~x£ ; as the mathematical statement of Ohm's Law. The factor of pro- portionality X is called the conductivity, and its reciprocal the specific resistance, or resistivity of the conductor. If \ is the same at all points of the conductor, the conductor is said to be homo- geneous, if X is variable, the conductor is heterogeneous. The above equation is equivalent to the three 3F SF 3F (4) u= -\^~ , v — — X ^— , w = — \^-. dx dy dz Inserting these in the equation of continuity, (5) fa V/v fa) " dy V'v dy If the conductor is homogeneous, this becomes A F= 0. Hence the density p is zero, or there is no free electricity in any portion of a homogeneous conductor in the state of steady flowf. * Fourier, Theorie analytique de la chaleur, 1822. t Kirchhoff. "Ueber eine Ableitung der Ohm'schen Gesetze, welche sich an die Theorie der Elektrostatik anschliesst." Pogg. Ann., Bd. 78, 1849. Ges. Abh., p. 49. 162—164] STEADY FLOW IN CONDUCTORS. 327 163. Boundary Condition. Refraction of Lines of Flow. In passing from one homogeneous conductor to another, A may be discontinuous, and since the current must be continuous, we must have at the surface ql cos fan*) + q2 cos (qzn2) = 0, (6) A^ cos (F^) + \2F2 cos (Fzn^ = 0, or The boundary condition (6) has a simple geometrical meaning. Since the derivatives of V are discontinuous only on crossing the surface, we have the derivatives in any direction t tangent to the surface, dV/dt the same on both sides of the surface. If 0l be the acute angle made by the current line with the normal on one side of the surface, 02 the acute angle on the other, resolving along the normal, (7) A! J^ cos 0! = \2F2 cos 02. Resolving along the tangent plane, since this component is continuous, (8) Fl sin 0! = F2 sin <92. Dividing the second of these equations by the first, we obtain tan 0i tan 09 (9) X, or the line of flow is refracted on passing the surface, so that the tangents of the angles of incidence and refraction are in the ratio \J\a dependent only on the media. The law of refraction is different from the optical law, in which we have the sine instead of the tangent, and in the case of the tangent law we do not have the phe- nomenon of total reflection, since the tangent takes all values from zero to infinity. 164. Systems of Conductors. All the statements heretofore made are true for the flow of heat, if V represent the temperature, but whereas in the case of heat in passing from one conductor to another the temperature FIG. 67 a. 328 ELECTROKINETICS. [PT. II. CH. VIII. is continuous, in the case of electricity, in passing from the con- ductor 1 to the conductor 2, we have at the surface of separation (10) F2-F1=:#12, where E^ is a quantity depending on the nature of the two conducting substances. In the theory of heat, if we have a chain of conductors in contact with each other, surrounded by a non-conductor, we may have equilibrium, but in the case of electricity this is the case only if the sum of the discontinuities of potential is zero, (11) E=Eu + Em + ...+Em = 0. Conductors may be divided into two classes. Those of such a nature that for any number of them an equation of this sort holds constitute the first class. To it belong all metals (their temperatures being the same). To the second class, for which in general such equations do not hold, belong solutions of salts and dilute acids. If we have a set of conductors of either class, the constants Erj r+l being given, and also the conductivity X as a point- function, we shall show that the problem of flow is determined as soon as we are given any two equipotential surfaces. Let VA be the potential at one of the surfaces A, VB that at the other B. Let be a function holomorphic in the whole space occupied by the conductors, satisfying the differential equation and the boundary condition at surfaces of separation of two conductors, taking the value unity for all points of the surface A, and the value zero for all points of the surface B, while d<&/dn = 0 for all points of surfaces separating the conductors from the surrounding insulators, or at infinity, if the conductors reach so far. Then if ^ be the potential function in the conductor 1 (in which lies the surface A), v2 that in the conductor 2, . . . vn that in the conductor n (in which 164] STEADY FLOW IN CONDUCTORS. 329 lies the surface B\ we may show that in the different conductors the potential is given by the functions V = v3 (H) For since the function satisfies the differential equation that is satisfied by the potential, any vgt which is a linear function of , must also satisfy the same equation. Also at any surface separating the conductors r and r + 1, and H dnr + dnr+l from the definition of the function . At the insulating boun- dary of any conductor that is, there is no flow across the boundary. The function vl takes at the surface A the value VA, and the function vn at the surface B the value VB. But these are all the conditions satisfied by the potential function. It remains to show that the function <3> is uniquely determined by the conditions that have been imposed upon it. The problem of finding the function is of the same nature as Dirichlet's problem, differing from it in that while the values of are given over part of the bounding surface^ over the remainder instead of <1> the values of d&/dn are given. Suppose that there are two functions <3> both satisfying the conditions of definition. Let them be denoted by <£>! and 2« Then let us form the integral taken throughout the conductors considered , f f ii] 330 ELECTROKINETICS. [PT. II. CH. VIII. By Green's theorem this is equal to -m- The surface integral is taken over the surfaces A and B, and the surfaces bounding the composite conductor, the integrals over the surfaces separating two conductors vanishing in virtue of (13). But at the surface A, X and 2 are both equal to 1, hence 3>i-<*>2 = 0, and at the surface B, ! and O2 are equal to 0, while on the remaining surfaces d^/dn — 92/9?i = 0. Consequently the surface integrals vanish. But the integrand in the volume integral vanishes in virtue of the differential equation satisfied by both functions. Consequently the integral J vanishes, but as in Dirichlet's demon- stration this can only be if j — 2 is constant. But since 4^ and 2 are equal on the surfaces A and B, they must be every- where equal. Consequently the solution is unique. 165. Properties of Vectors obeying Fourier-Ohm Law. The vectors F, the electrostatic force, and q the electric current- density are typical of a class of pairs of vector-functions of frequent occurrence in all parts of mathematical physics, distinguished by the following properties. The first vector is lamellar, the second is solenoidal. In isotropic bodies the vectors have the same direction, and their ratio depends only on the physical nature of the body at each point. When two vector-functions have these properties we shall say that they satisfy the law of Fourier- Ohm. The study of the properties of such vectors is of great importance. We shall in general call the solenoidal vector the yte-density, and the surface integral of its normal component over any surface the flux through that surface. It is remarkable that the characteristic properties of such vector-functions are embodied in the single statement that if V, the potential function of the lamellar vector, is uniform, finite, and continuous, in a certain region r, its first derivatives possessing 164, 165] STEADY FLOW IN CONDUCTORS. 331 the same properties with the possible exception of certain sur- faces 5 at which they may be discontinuous, then if the values of V are given on parts of the surface S bounding the region T, and the value of dV/dn is zero on the remainder, the integral J throughout the region T, is a minimum* for that function V which makes the vector q solenoidal, where 3F ^dV (2) u=qcos(qa}) = \, v = q cos (qy) = X -- , w = q cos — . For if we change the form of the function V by the arbitrary amount SF, 3(F+SF) T (3) = /T JJJ , dr. to \ dy J \ dz The integral with the coefficient 2 is equal, by Green's theorem, ff[ "III JJJ dy \ dy J dz\ dz where n^ and n2 are the normals on opposite sides of a surface S of discontinuity of the derivatives. On those portions of the bounding surfaces for which F is given 8F=0, and for the re- * Kirchhoff, Ges. Abh. p. 44. 332 ELECTKOKINETICS. [PT. II. CH. VIII. maining parts X3F/3ft = 0. Consequently the integrals over the bounding surfaces disappear, and we have 9 A, 9FM J s (x TZ )\ dT asF /9SF In order for J(V) to be a minimum, this must be positive for all possible choices of the arbitrary function SV. This can be true only if we have everywhere in the region r and at every surface of discontinuity S, Consequently the statement that J is a minimum is equiva- lent to stating that q is solenoidal. 166. Integral form of Ohm's Law. We have seen in § 35 that the solenoidal condition signifies that the flux, /= q cos (qn) dS across any surface bounded by the sides of a vector tube is the same for all parts of the tube. In the case of electrical flow, the flux is called the current (current-strength, or intensity) in the tube. Although V has discontinuities, the function has not. Since we have between any equipotejitial surfaces A and B, the ratio of the flux to E+VA — VB, the difference of potential plus the sum of the sudden rises of potential as we go in the direction of flow, thus depends only on the function <3>, which depends only on the configuration of the space r, and the values of the function X. That is, the ratio of the flux in any tube of the 165—16*7] STEADY FLOW IN CONDUCTORS. vector qtoE plus the difference of potential between two equi- potential surfaces depends only on the physical properties of the substance in the tube. This is the usual form of the statement of Ohm's Law, and is the integral form, whereas our previous state- ment was the differential form. In the case of electrical flow, the difference of potential VA - VB is called the external or electrostatic electromotive force from A to B, and it is evidently the line integral of electrostatic force along any line from A to B. E is called the impressed, intrinsic, or internal electromotive force. The ratio C of current to total electromotive force is called the conductance of the tube. Its reciprocal R is called the resistance of the tube. If we consider a closed tube of flow, the two surfaces A and B will coincide, and we shall have the ordinary expression of Ohm's Law, or: — For any closed tube of flow, the current is equal to the impressed electromotive force divided by the resistance of the tube. 167. Heat developed in Conductors. We shall now con- sider the physical meaning of the integral J in the case of elec- trical flow. In passing from a point where the potential is VA to one where it is VB a unit of electricity does VA — VB units of work, and that quantity of electrostatic energy thus disappears. Also at every surface of discontinuity, Er r+l units of work must be done upon it. But if we consider heat as a form of energy, if mechanical energy disappears, an equivalent amount of heat must make its appearance. If accordingly we find energy appearing in no other form, the electrostatic energy W that disappears, to- gether with the work done by the impressed electromotive forces, must be converted into heat. In the case of steady flow we find this to be the case. In unit time the quantity I = llq cos (qn) dS crosses any section of a tube of flow, so that considering that part of the conductor between the equipotential surfaces A and B we 334 ELECTROKINETICS. [PT. II. CH. VIII. have /units entering at potential VA and emerging at potential VB. The energy converted into heat in that portion of the conductor will accordingly be (i) - But transforming the integral, § 165 (i), by Green's theorem, and taking the normal at A, B, and the surfaces of discontinuity always in the direction of the current, J— /*/*/* C ^ / O T7*> / / / __ I n / Or - I Ma- X-5-) + :T lA'^-) + 5- JJJ (9#V oai J ty\3yj fa The volume integral vanishes by the equation § 162 (5), and in virtue of the surface conditions § 164 (10) and (13), (3) J=-(E + The integral which is a minimum in the actual distribution of current accordingly represents the heat generated in the conductor in the unit of time. The equation (i), written EI= n This and the equations (2) for the junctions are generally referred to as the equations of Kirchhoff s two Laws. Maxwellf treats the problem in the following more symmetrical form. 171. MaxwelPs treatment of Networks. Consider n points of junction, each of which, in the most general case, is connected with each of the others by a conductor. The number of conductors in this case is n(n — 1)/2. If some of the conductors are lacking this will be expressed by putting the conductivities * Kirchhoff. "Ueber die Auflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Strome gefiihrt wird." Fogg. Ann., Bd. 72, 1847. Ges. Abh., p. 22. t Maxwell, Treatise, § 280. 22—2 340 ELECTROKINETICS. [PT. II. CH. VIII. between the corresponding points equal to zero. Let the current from the point p to the point q be Ipq, and let the conductivity of the conductor pq be Cpq, the impressed electromotive force Epq. Then evidently The equation ( I ) may be written (4) IM=CM(Epq+Vp-Vq). Substituting the values of the currents in the equation of continuity (2) for the point p, (5) Cpl(Ep,+ Vp- Vl)+CP»(Eia + F,- F.) ...... Let us introduce a symbol Cpp, denned by the equation form (6) s=l The equations (5) may then be written in the symmetrical n + . . . + C If we add these equations for all the points of junction, the result will be an identity, so that the equations are not all independent. The equations therefore suffice to determine the differences of potential between the junctions, but not the potentials themselves. Since in the equations (4) only the differences of the potentials appear, it is evident that we may choose one of the potentials arbitrarily. Let us therefore put Vn equal to zero, and use the first n — 1 of the equations (6), which are independent, to deter- mine the potentials Vlt V2 ...... V^^. Calling A the determinant of the coefficients of conductivity, V21 » ^1 l, n—i a 2, n—i i, n— i 171, 172] STEADY FLOW IN CONDUCTORS. 841 and Arg the minor of Crs, we have A a symmetrical determinant, and Ars = Asr, since Cpq — Gqp. The solutions of the equations (6) are of the form (7) A . Vt = Alt (CUEU + C12#12 + ...... + ClnEm) + A2i (G21E21 + 632^22 + ...... 4- @znEm) ...... 4- An_lf « (Cn-i, i J&ti-i, i + @n-i, 2 ^w-i, 2 ...... Inserting these values of the potentials in the equations (4), we obtain the currents in all the branches as linear functions of the impressed electromotive forces in the branches. Picking out the terms containing Ers or its negative Esr in the current Ipq we obtain /ox Hpq In like manner the coefficient of Epq^ in Irs is (9) But since Ars = A^, etc., this is equal to n Consequently the current produced in a branch pq as a result of introducing an electromotive force E in a branch rs is the same as the current produced in the branch rs on introducing an equal electromotive force into the branch pq. This theorem is analogous to the reciprocal property of electrified conductors given in § 136. If ( 10) &pr + A9S = &qr + A^s, an electromotive force applied in one branch produces no current in the other, and the conductors are said to be conjugate. 172. Heat developed in the System. If we denote the coefficient GpqCrs (A^ 4- A9S - A9r - A^s) by Cpqrs> we have r=l g= 342 ELECTEOKINETICS. [PT. II. CH. VIII. Now the activity of the electromotive force Epq is Epqlpq. Forming the products for all the branches and summing, bearing in mind that each branch appears twice, we obtain for the total activity p=nq=n 2 2 p q r s But since there is supposed to be no electrostatic energy, this must be the heat developed in the system in unit time. The heat is accordingly a homogeneous quadratic function of the impressed electromotive forces. If we should solve the equations (n) we should obtain the electromotive forces as linear functions of the currents. Then forming the expression for the activity we should obtain a homogeneous quadratic function of the currents, and by our general theorem for the heating this must be equal to This might be obtained from the equations above by the aid of certain properties of determinants. 173. Wheatstone's Bridge. As an example of the above principles let us consider the case of Wheatstone's Parallelogram or Bridge- It consists of four points connected by six conductors, which may be represented by the sides and diagonals of a parallel- ogram, or more symmetrically as in Fig. 69. Suppose that the only impressed FIG. 69. electromotive force is in the branch 12, and that we require the current in the branch 34. The equations (6) are OnVi + 012VZ + C13F3 + (714F4 = C, C^V, + C*V,+ (723F3 + (724F4 = 172 — 174] STEADY FLOW IN CONDUCTORS. 343 from which, putting F4 = 0, and using the last three equations, Csl C3, 0 , we may write for the total current dV -ii' — dti = — Xw , , dn ds s being the length of the conductor measured from a certain point. Integrating with respect to s from Sj to s2 , „ v f2 Ids , [2 ds YI— F2= — =// — , J i Xw J t Xo> and the resistance is given by R = = r ds } ! Xft) ' 344 ELECTROKINETICS. [PT. II. CH. VIII. This formula is important in the case of standards of resistance formed of tubes filled with mercury, the varying diameter of the tube being determined by a calibration. If the conductor is homogeneous, X is constant, and if the cross-section is constant, Xw or the resistance of a uniform wire is proportional to its length and inversely to its cross-section. 175. Non-linear Homogeneous Conductors. In the case of homogeneous conductors, X being constant, the equation of flow, 5 162 (5), becomes a2F 92F or the potential is harmonic. Consequently every theorem on harmonic functions applies to the potential in this case, and every method of solving problems of electrostatic distribution may be applied to the solution of problems of steady flow. We must have the electrodes of the conductor given. Now by the equation of Ohm's Law it is evident that the effect of increasing the conductivity of any portion of a conductor is to make the potential vary less rapidly there, the current being given. If then a portion of the conductor be made infinitely conducting its potential will become constant throughout. Accordingly if we introduce a thin plate of infinitely conducting material, this will form an equipo- tential surface and may be taken as an electrode for the conductor. This supposition will be made in the following examples. Since in the electrostatic problem the capacity is given by T7- J " F.-F, and in the problem of flow the conductance by F2 - Fx ~ R ' we find that the conductance of a portion of a homogeneous con- ductor between two electrodes is equal to 4?rX times the capacity of a condenser whose plates have the geometrical form of the electrodes of the conductor, and whose dielectric occupies the 174, 175] STEADY FLOW IN CONDUCTOES. 345 space corresponding to that occupied by the conductor. The case of a straight field, § 145, gives v _ $ r,_\S T?_ d ~' " d ' ~\S' as in the case of the uniform wire. The case of flow radially between concentric cylindrical electrodes gives, § 144, I „ 2A/7rZ o = This formula might be used for calculating the resistance of the liquid in galvanic cells where the plates are concentric cylinders. The case of radial flow in a sphere from a spherical electrode of radius RQ (§ 142) gives, if the outer electrode is at an infinite distance, This formula may be used to find the resistance of the earth between two telegraphic earth-plates. If both earth-plates are equal spheres buried deeply in the earth at a distance apart so great that it may be considered infinite in comparison with their diameters, we may consider the resistance from one to the other as that of two conductors of the last case in series, so that If, as would more nearly represent the practical case, the con- ductors are hemispheres, with diametral planes in the surface of the earth, we may consider the space in the preceding problem split along the surface of flow formed by the plane through the centers of the spheres, and take the lower half, whose conductivity will be half of that just found, or 7T\RQ' In like manner the problem of the ellipsoid and the circular disk will give us the resistance between earth-plates in the form of circular disks laid on the surface of the earth as ?r/2 times that for a hemisphere of the same radius. It is important to notice that in any case of geometrically similar electrodes, the resistance is inversely proportional to the linear dimensions of the earth-plate, 346 ELECTKOKINETICS. [PT. II. CH. VIII. and not to its surface. This of course comes from the fact that the lines of flow diverge in all directions from the electrode instead of remaining parallel. It explains the necessity for large- sized plates for telegraphy or for the earth connection of a light- ning rod. In practice, the conductivity of the earth varying from point to point, the conductivity of the portions near the electrode plays the most important part, so that it is important that the earth:plate be buried in good-conducting material. The problem of the spherical bowl shows that if such a bowl should be made an electrode immersed in an infinite conductor, the other electrode being at a great distance, nearly all the current would flow from the outside of the bowl, the current density being greatest at the HP. The method of the conformal representation furnishes a means of solution for the case of two-dimensional problems, in particular for the flow of current in a thin plane sheet. Fig. 67 for instance shows the lines of flow in the case of a long ribbon of conductor slit along the axis of U'. 176. Correction for End of Wire. We shall conclude this subject with the consideration of the practical problem of finding the correction that must be made in the value of the resistance of a uniform wire when it ends in a conductor so large as to be capable of being considered infinite. This is of importance in the case of mercurial standards of resistance, for the tubes end in large cups of mercury. We shall consider a right circular cylindrical conductor ending in a conductor of in- FIG. 70. 175, 176] STEADY FLOW IN CONDUCTORS. 347 finite extent and bounded on one side by a plane perpendicular to the cylinder, Fig. 70. We may obtain an upper and lower limit for the desired correction by an artifice due to Lord Rayleigh *. It is evident that if we introduce anywhere a portion of conductor of greater conductivity we increase the conductance of the whole. Let us accordingly introduce in the mouth of the cylinder a plane sheet of infinite conductivity, thus rendering that circular section equi- potential. The flow in this case will resemble the actual flow in that V will be continuous in crossing the plane, while it will differ from the actual case in that dV/dn will be discontinuous, its integral over the section, or the total current being continuous. We may then use for the portion below the mouth the solution for a straight field, so that the resistance of a length I of radius a is i i Above the mouth of the cylinder we may use the formula for the flow from a circular disk of radius a to infinity, so that the resist- ance on the upper side is R- l ~i5x' Consequently the lower limit of the resistance is In a similar manner the resistance of the system will be increased if we introduce non-conducting surfaces not coincident with the walls of current tubes. Let us below the mouth of the cylinder suppose the cylinder split up into an infinite number of cylinders of infinitesimal cross-section, by means of cylindrical non-conducting surfaces introduced, and let the current density in these filaments be maintained constant, in the whole of the cylinder. Then below the mouth the equipotential surfaces will be planes, but on the upper side of the plane of the mouth the potential will not be constant, as we shall show. Consequently at the mouth of the cylinder V is discontinuous, while d Vfdz is in this case continuous. Since below the mouth = xdF dz * Rayleigh, Tlieory of Sound, Vol. i. § 305. 348 ELECTROKINETICS. [FT. II. CH. VIII. is constant by hypothesis, we have dV/dz = const., and if dV/dz is to have the same value on the upper side, V must there be the same as the potential due to a fictitious (non-equi- potential) distribution on a disk of radius a of constant density _1_ 9F_ JL^ q ~ ZTT dz~ 2?r X ' The mass of such a distribution would be a2q m = jra2 cr = — - . &\ The resistance of the upper side may be calculated by Joule's Law, TT o~ -5- r~ H JJJ( \dxj \dyj \c)z The integral in the numerator being through one-half of infinite space is 8?rX times one-half the energy of the distribution on the disk. The integral in the denominator is 4?rX times one-half the mass of the disk. Consequently W where W is the whole energy of the distribution of the disk. This energy is very easily calculated. The potential at the edge of a disk of radius p with constant surface density a- is r ffdS •HjT- If we introduce polar coordinates, the origin being the attracted point on the edge, and 0 being the angle included between r, the radius to the point of integration and the diameter through the origin, this becomes r-2 rr = 7r l^ ^ 9a? ^ 92/ v ?y / 9^ v dz If now the energy of the actual distribution of potential is to be a maximum for all possible values of 8F, the first two in- tegrals must vanish, which can be the case only if throughout space we have 1 9 / 9F\ 9 / 9F\ 9 / 9F ?=- * See Helmholtz, Wiss. Abh. Bd. i., p. 805. 23—2 356 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. and at every surface of discontinuity, 1 f 8F 8F \9) > °" = I A*l J~ /^2 "" These equations will henceforth be known as the generalized Poisson's equations. They give us the law of distribution of force in the differential form, and contain the forms heretofore used as a special case, obtained by putting //, = 1. The form of the integral expressing the energy is the same as that of the integral J of § 165 (i). All the properties of the integral J are accordingly possessed by the integral Wf. In particular it follows that if the potential is given at certain surfaces the condition that the energy shall be a minimum re- quires that in the space between (10) 1^9F) + 1^3F) + 1LLF) = 0 and on surfaces of discontinuity 8F 8F We may call the problem of finding a function that shall satisfy these differential equations, and take the required surface- values, the generalized Dirichlet's Problem. The function F may be called quasi-harmonic. It is evident, as in § 86, that the solution of the problem, if there be any, is unique. We have heretofore said nothing regarding the localization of the energy of a distribution, which we have represented either by an integral W$ throughout the acting distribution, or by an integral Wf, which is expressed in terms of the field at all points of space. Whereas both representations are equivalent mathematically, it is a fundamental point in Maxwell's theory to regard the energy as localized in the medium wherever a field exists. 181. Induction. If we define a vector g by the equations £ = g cos (ga?) = -p—t (12) |trg 8F 3 = g COS (g*) = - ft, 180, 181] DIELECTRICS AND MAGNETIZABLE BODIES. 357 the vector has, by (8), the property of being solenoidal in all parts of space where there are no charges. That is The force, or field-strength F, no longer enjoys this property in general, but does so in a homogeneous medium, for which //- comes out as a constant factor. The vector § is called the induction, and is connected with the force by the equations The induction accordingly satisfies everywhere the law of Fourier-Ohm. The surface integral over any surface of the induction resolved normally to the surface is called the total induction, or induction- flux, through the surface. The quantity //, is called the inductivity of the medium. A more usual name for it is the specific inductive capacity or dielectric constant, in the electric case, magnetic per- meability in the magnetic case. The latter name is due to Lord Kelvin, to whom the recognition of the analogy to the case of flow in electricity and heat is due.* The name permeability comes from the hydrokinetical analogy of water flowing through a porous medium. The lines of induction suffer refraction in the manner described in § 163 when passing from one medium to another. In the FIG. 71 a. * Magnetic Permeability, and Analogues in Electro-static Induction, Conduction of Heat, and Fluid Motion. Papers on Electrostatics and Magnetism, p. 487. 358 ELECTROSTATICS AND MAGNETISM. [FT. II. CH. IX. electric case, /JL does not differ widely for different media, seldom reaching ten times the value for air or empty space, and never being less than for empty space, while in the magnetic case, /A may be, for iron, several thousand times as great as for air, and in some cases is slightly less than for air. Consequently lines of force emerging from iron into air are generally nearly normal to the surface in the air unless very nearly tangential in the iron. This is exemplified in Fig. 7 la, showing the distribution of lines of force between the pole-pieces of the field magnet of a dynamo with the armature removed. In virtue of the analogy to electric conductivity, it is evident that the lines of force exhibit a tendency to crowd together into parts of the field where p is large. 182. Relation of Charge to Induction. Since the force no longer possesses the solenoidal property, except in homogeneous media, while the induction does, it is more logical to speak of tubes of induction than of tubes of force, although geometrically the two coincide. The flux of force through various cross-sections of a tube, however, varies, while the flux of induction is constant for the tube. The volume density is no longer determined by the divergence of the force, but of the induction, being equal to 1/4-Tr times the divergence of the induction, § 180 (8), dz while the surface density is l/4?r times the discontinuity of its normal component, § 180 (9), (16) = _ JL jj {£ cos (nx) + g) cos (ny) + 3 cos (nz)} dS, is equal to l/4?r times the excess of the number of unit tubes issuing from the space over the number entering. We shall call the densities thus defined, for a reason to be presently explained, the densities of true electricity or magnetism. 181 — 183] DIELECTKICS AND MAGNETIZABLE BODIES. 359 183. Apparent Charge. Since on passing from one medium to another where the inductivity is different the force is dis- continuous, the surface acts as a charged surface has previously been found to act, § 82. Also since the force is not solenoidal in a heterogeneous medium, there appear to be bodily charges. The magnitude of these apparent charges, whose densities are p, a, are given by the usual equations and comparing these with the equations for the true densities we find 1 fiVfy Wbp dVdp or in a homogeneous medium (20') .',' ,' For the surface density The potential is then determined by § 85 (18), as dS that is : A distribution of charges acting according to the Newtonian Law, of densities p and cr', would produce everywhere exactly the same field as that actually produced by the true charges p and a. The Newtonian Law thus reappears, and may be used to calculate the forces, only the true charges do not follow the law, but the apparent charges, which are known as soon as the true charges and the properties of the media are given. 360 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 184. Capacity. Reluctance. In the electric case we have for the capacity of a condenser whose dielectric is homogeneous, so that AF=0, ff 3F ^ fji^-dS ^== that is, the capacity equal to that found in Chapter VI multiplied by the inductivity of the medium. This is Faraday's capital dis- covery,* leading to the development of the theory according to which the energy resides in the medium, so that electrical actions are transmitted by means of the medium, and not by action at a distance. Faraday experimented with condensers in the form of concentric spheres, the intervening space being filled by the dielectric in question. The material of the dielectric outside the larger sphere was accordingly immaterial. Instead of capacity the term permittance has been proposed by Heaviside. In the magnetic case, the value of the quantity analogous to the capacity has been called the permeance or inductance, while its reciprocal, corresponding to the resistance in the case of electric flow, was called magnetic resistance by Bosanquet, a name which has given way to that of reluctance. 185. Induced Charge. The apparent charges defined above minus the true charges are called the induced charges due to the action of the forces of the field. If we examine the amount of the induced charge in a body r surrounded by a homogeneous medium, we shall obtain an important result. Let the constant inductivity of the external medium be fa, and let us denote the normal toward the interior of the body r by n{ and the normal toward the outside by ne. Then if we use the formula (21) for the apparent charge on the surface we find and transforming the second integral on the right by Green's theorem this becomes * Exp. Res. § 1252 seq. 184 — 186] DIELECTRICS AND MAGNETIZABLE BODIES. 361 / aF r te ~ dr- But by the definitions of true and apparent volume density this is (26) so that, transposing, (27) JJo-US + JJJVdr = I [JjVdS + ffjp dr^ , or the total apparent charge of a body surrounded by a homo- geneous medium is equal to the true charge of the body divided by the inductivity of the surrounding medium. In particular a body which has no true charge has a total apparent charge equal to zero, and since this remains true however the body may be subdivided, the body is polarized. In the magnetic case, the body is always found to be polarized, consequently we must conclude that the true magnetic charge of all bodies is zero, or in other words, true magnetism exists only as polarization. This is a second apparent difference between electricity and magnetism, but if we remember that whenever electrification is produced equal quan- tities of both signs appear, the difference disappears. 186. Polarizations. Since experiments on electrification and magnetization are almost always made on bodies surrounded by a homogeneous medium, namely air, it has become customary to regard their apparent charges as due to the polarizations of the bodies themselves, although it is evident by § 120 that the surface charges are due only to differences of polarization on the two sides of the surface. The surrounding medium might be uniformly polarized to any degree without producing any effect, consequently its absolute polarization cannot be determined, and is of no import- ance whatever. The apparent polarization of the body must produce the surface density, by § 120 (2), (28) we obtain for the volume density due to the polarization, § 120 (6) fdA dB dC\ (32) p' =__ + _ + _ \dx dy dzj a* ag) a3\ i /aar 3F and since by ( 1 8) we have we must have (34) ? + ?+f = 0- dx dy dz or the induction is solenoidal. The induction accordingly possesses the property of the vector called the induction in § 121, and by the equations (31) is equal to it if /^ is equal to unity. 187. Examples. Point-charge in Medium bounded by Plane Face. Suppose we have a point-charge e placed at P at a distance a from a plane face separating two media of inductivities /AJ, /*2, their extent being infinite. We may solve the problem of induction by the method of images as in § 152. Suppose that e lies on the left of the dividing plane, and that at its geometrical 186, 187] DIELECTRICS AND MAGNETIZABLE BODIES. 363 image P' we place a charge e'. Then we may determine the charge ef so that on the left the potential will be the same as that due to charges e and e placed at P and P' in a uniform medium, while on the right we shall have V the same as would be produced by a charge e + e placed at P. For if we put e e on the left F= - + — , r r dV e dr e' dr e- e cos#; on the right F = e 4- e r dV e + e' dr e + e cos 0. But at the surface these must satisfy the equation dV dV , , ,. ,x1 ° = ^ + ^^ = {^(e-e)-^(e + e>)} Consequently if we put FIG. 72. 364 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. the condition will be satisfied. The surface density induced on the surface of separation is 4?r l r2 2?r fa + fa r3 so that the distribution is proportional to the distribution on a conducting plane, as in § 152. The lines of force in the medium 2 are straight, and the refraction on crossing the plane is shown in the diagram, Fig. 72, in which fa/ fa = 4. The greater /x2 the less is the force in the medium 2, and if we make /*2 infinite, the force in the medium 2 vanishes, and the surface density becomes , e a " -2SH' as in the case of a conductor. 188. Slab in Uniform Field. Suppose a slab of induc- tivity /x2 with parallel faces of infinite extent is placed in a uniform field parallel to the equipotential surfaces, and let the indue tivity of the surrounding homogeneous medium be fa. Then the potential satisfies Laplace's equation in the slab as well as outside it. Ac- cordingly the solution for either of the three parts of the field, in, above or below the slab, is a linear function of the single co- ordinate perpendicular to the equipotential planes, and the force has values which are constant, but different, in the three regions. v V FIG. 73. If Fi and V2 are any two equipotential planes outside the slab at distances dt and d2 from its faces, F3 and F4 the potentials of the faces of the slab respectively facing them, and d the thickness of the slab, we have the conditions at the surfaces V3, V4 (1) faFl==faF, ^F that is (2) F^Ft-^ 187, 188] DIELECTRICS AND MAGNETIZABLE BODIES. 365 Now by § 145 we have W F>-F3 F F3-F4 y.^-y. (3) F^ -5—' f = ~^~' ~dT so that we have the equations Solving these for F3 and F4 y = VldsJy + F2 ( + from which we get for the force outside the slab In the electrical case if Fl5 F2, are the surfaces of conductors, the density on the upper plate Vl is w " and the capacity of the condenser of area S vu/ V V " \ ~ * 2 By measuring the capacity with the slab and with it removed we may determine the dielectric constant of the slab in terms of that of the surrounding medium. If d± and d2 are zero, the capacity is which is, as was stated in § 184, proportional to the dielectric constant. The apparent surface density on the upper face of the slab is so that the intensity of polarization is 47r [^ (d, which is in the direction of the force if fa > fj^. 366 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 189. Point- Charge in Sphere. Suppose we have a point- charge e at the center of a sphere of radius R of homogeneous substance of inductivity /4l5 surrounded by an infinite homogeneous medium of inductivity ytt2. Then Laplace's equation being satisfied in either medium, we may use the solution of § 142, and put F in either medium equal to a linear function of I/r The condition at the surface r = R gives, since so that rr e v e F! = — + C, F2 = , since the integral of gn = - p ^- over any surface inclosing e must be 4-7T0. The apparent surface density and charge of the sphere are '- JLJ^E1 8F4_ JL_iP_jL 4?r 1 tin* dn* \ 4?r R2 (Lin IL« The real charge e at the center acts, by § 183 (20'), like an apparent charge e/yu.l5and the apparent charge of the sphere e acts at outside points as if concentrated at the center. Accordingly the whole force in the medium 2 is efh + e' e which is the same as found from — 3 V.2/dr. 190. Unit of Electricity or Magnetism.. If the charge be situated in a medium of inductivity /u, extending to infinity, the force of the field is, by the above, equal to e/pr2 and the action of e on a charge of ^ units is ^ times as large, or e^/pr3. Now the unit charge has been defined as the charge which repels an equal charge placed at the unit of distance from itself with a unit of force. We accordingly see that the magnitude of the unit will 189 — 191] DIELECTRICS AND MAGNETIZABLE BODIES. 367 depend on the medium, and if the experiment be made in a medium of inductivity /-i the unit thus obtained will be larger in the ratio V//, than if it had been determined in a medium of unit inductivity. We also see that the dimensions of the unit involve fj,, for we must have the dimensional equation so that It is customary to choose the unit of inductivity so that the inductivity of empty space is unity, or as it is sometimes stated, the inductivity of the ether is unity. This is, as we have seen, purely arbitrary, as experiments enable us to determine only ratios of inductivities. The inductivity of air, both electric and mag- netic, differs very little from that of a vacuum, so that for practical purposes we may consider the size of the units determined by experiments in air. We must notice that even if /z- is put equal to unity its dimensions remain in the equation and the dimensions of fju we have no means of knowing. As the matter of dimensions is always more or less arbitrary, we may make any supposition that we please, until we are led to contradictory results. Two different suppositions are convenient. We may, when dealing with electrical quantities, assume that the dimensions of the electrical inductivity are zero. This gives the electrostatic system of units. We may on the other hand, when dealing with magnetic quantities, assume that the dimensions of the magnetic inductivity are zero. This gives the magnetic system. Both these systems are due to Gauss, and when we use both systems for their respective kinds of quantities, we shall say that the quantities are measured in Gaussian units. This has been the case in the preceding chapters. When we come to deal with both electrical and magnetic quantities at the same time, we must choose one or the other of these assumptions, as we shall find in the next chapter- that both together are incompatible. 191. Susceptibility. The equation giving the apparent polarization of a medium of inductivity //«> surrounded by a medium of inductivity ^ is, (§ 186 (30)) " N F, 368 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. so that the polarization is proportional to the total force of the field, that is the sum of the external field and the field of force due to the polarization. The coefficient . 21 — is called the magnetic susceptibility, in the magnetic case. In the electric case the quantity K has never come into practical use. The equation I=KF, was the basis of Poisson's theory of magnetic induction, K being supposed a quantity inherent in the body, and equal to zero for air. We see however that K depends on the medium by which the body is surrounded, as well as on the body itself: /e21 may therefore be called the relative susceptibility of the body of inductivity /JLZ in a medium of susceptibility fa. If rc21 is positive, the polariza- tion is in the direction of the polarizing force, and the body is said to be paramagnetic, or simply magnetic. If /c21 is negative, the polarization is in the opposite direction to the force, and the body is said to be diamagnetic. Accordingly any body immersed in a medium of greater inductivity than its own will appear diamag- netic. If we consider always the polarization of a body with respect to a vacuum, so that /^ = 1, we may put fJi = 1 -f 4!7TK. Bodies are accordingly magnetic or diamagnetic as //, is greater or less than unity. It is evident that the assumption that K is zero for a vacuum is arbitrary, in the same degree as, but inde- pendently of the assumption that the inductivity of vacuum is unity, for we might assume all apparent polarizations to be the differences of the polarizations of bodies from the polarizations of vacuum. 192. Uniform Polarization by Induction. When a body of different inductivity from the rest of the medium is inserted into a field of force, the configuration of the field is disturbed owing to induction, the polarization due to which produces new forces FI which must be added to the forces of the undisturbed 191, 192] DIELECTRICS AND MAGNETIZABLE BODIES. 369 field FQ . We shall now examine in what cases the introduction of a polarizable body into a uniform field will produce such a resultant field that the polarization of the body will be uniform. Let the potential of the undisturbed field be F0 and the potential of the forces due to the induced polarization be F^, so that the total potential of the field is (i) F=F,+ Fi. If XQ, F0, ZQ denote the constant components of the force of the undisturbed field, we have (2) V«=C Let a, 13, 7, be the constant direction cosines of the uniform polarization, so that (3) ^ = /«> B = I/3, C=Iy. Then since / = icF we must have for the total potential (4) V=C'-Xx-Yy-Zz = C'--K( b and 6 < 7r/2 the couple is positive, or from X to F, that is, the ellipsoid tends to turn its longer axis parallel to the field, whether K is positive or negative. This is in contradiction to a statement frequently made, that diamagnetic bodies tend to set their longest dimension across the field. They do not do so if the field is uniform. If an ellipsoid be suspended by a torsion fibre in a magnetic field, the field will cause it to vibrate more rapidly when its long axis is parallel to the field, and more slowly when it is across the field, than it would do in the absence of the field. It is however extremely difficutoi|tf .not impossible, to obtain a magnetic field nearly enough uniform to show these phenomena in diamagnetic bodies, on accounl^F the extreme smallness of /c2. 194. Polarization of Sphere. In the case of a sphere for inside points we have by § 80, #2 - 3 so that (2) L=M = N=^, which is the self-demagnetizing factor. Accordingly the force is in the direction of the original field, rr_ ff> _ -ff> 3^ „ " *' 24—2 372 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. and the polarization is T= F- FQ F° = 30.-/O F 1 47T ' 47T/J! 47T 47TO * 3 /a-/4j 3 The self-demagnetizing force is If /A be infinite this becomes equal to — F0, so that the total force inside the sphere is zero. This is the case for a conducting sphere in an electric field, and is nearly the case for soft iron in a magnetic field. Outside the sphere we have a different form for H, so that ^_ 8ft 4-rrffi/ cos ' 8/T ~3~ ~7 The field due to the polarization is accordingly, by § 123 (7), the same as the field of a doublet of moment 4>7rRsII^=—R3Fi, and the total field outside the sphere is obtained by superposing this upon the uniform field F0. If p = oo the moment of the sphere is «. The lines of force due to a uniform field disturbed by a doublet FIG. 74. 194, 195] DIELECTRICS AND MAGNETIZABLE BODIES. 373 pointing in its own direction or the opposite are shown in Figs. 74 and 75 respectively. FIG. 75. The field of the sphere in a uniform field is shown for p/fa = 3 and p/fa = oo respectively in Figs. 76 and 77. These figures were originally given in Lord Kelvin's Reprint of Papers on Electro- statics and Magnetism, (p. 492), where the equations of the curves are discussed. The figures have been re-drawn for this book, the lines being drawn for equal increments of the flux-function M*, or p, § 103 (10). FIG. 76. 195. Infinite Elliptic Cylinder. If one of the axes of the ellipsoid is infinite, we have the case of an infinite elliptic cylinder. If c = oo , JV" is zero, and L, M, reduce to trigonometric forms. The force parallel to the £-axis is the same as that of the undisturbed 374 ELECTKOSTATICS AND MAGNETISM. [PT. II. CH. IX. field. This is a consequence of the distributions on the ends being infinitely distant. We may then measure K as the ratio of FIG. 77. the longitudinal magnetization to force, a method often used in practice, and accurate only when the cylinder is extremely long. 196. Ellipsoid of Revolution. In the case of an ellipsoid of revolution the form of the integrals simplifies, and inserting in the formulae of § 116 the eccentricity e=*/a? — fr/a we obtain for a prolate ellipsoid T"-^1"^!1! l+e fl I - e2 (2) M=N=*TT i— _ 'log " i — e and for an oblate ellipsoid (3) L = 4-7T j-2 g — sin"1 e [• , For e — 0 all these expressions become indeterminate, but on evaluating the indeterminate form they take the common value already found for the sphere. For e — 1 the expressions for the prolate ellipsoid become indeterminate, and on evaluation we find 195 — 197] DIELECTEICS AND MAGNETIZABLE BODIES. 375 This gives the case of the infinitely long circular cylinder, for which, as we have previously found, the longitudinal demagnetiz- ing factor vanishes, while for transverse magnetization it is equal to 2?r. When e — 1 the expressions for the oblate ellipsoid give L = 4?r, M = N = 0. This gives us the case of a disk magnetized normally, for which the demagnetizing factor is the largest possible, namely 4?r, or parallel to the faces, when the demagnetizing factor is zero. For a long prolate ellipsoid, for which e is nearly unity, we may conveniently use an approximate formula. Putting m = a/b for the ratio of the length to the diameter, since a m we have approximately (6) M=N= (log 2m - 1). fti/ A table of values of the demagnetizing factor is given by Ewing*, and a larger one by du Boisf. 197. Magnetization of Hollow Cylinder. We shall now consider a few cases of induction in which the induced magnetiza- tion is not uniform. In the first case let us consider the uniplanar problem of the transverse magnetization of an infinite homo- geneous circular cylinder, placed in a field such that the lines of force are the intersections of cylindrical surfaces with planes perpendicular to the generators of the cylinder. If the cylinder is circular the method of development in series of circular harmonics, § 94, gives the general solution of the problem. Let the cylinder be hollow, the inner radius being b and the outer a, the inductivity of the cylinder being /i2, and of the space within and without /4lt Let the undisturbed field, as before, be represented by F0 with potential F0, while the field due to the induced polarization is Fi with the potential F^. We shall suppose that the bodies producing the field lie outside the * Ewing, "Magnetic Induction in Iron and other Metals," p. 32. t du Bois, "Magnetische Kreise, deren Theorie und Anwendung," p. 45. 376 ELECTROSTATICS AND MAGNETISM. [FT. II. CH. IX. cylinder, so that the potential F0 and its derivatives are finite and continuous at the surfaces of the cylinder. Let it be developed at the outer surface in the infinite series of harmonics Then at points for which p < a it is given by the series, (2) Va=Ta + ^T1 + PlT1 + ...... Ct (I- The potential F* is represented by three different develop- ments in the three different regions, (1) p>a, (2) a> p>b, and (3) p < b. We will distinguish these by an affix. Since F$ vanishes at infinity, we have outside the cylinder (3) VM = ZAnp-"Ta. 0 In the substance of the cylinder we must take (4) V^= o while in the cavity, since Vi is finite at the center, (5) V^ o Since Vi is continuous, at the surface p = awe have*Fi(1) = F^(2), and as this must be identically true for all values of we must have for every term the coefficients of Tn equal. (6) Anarn = Bnan + Cna~n. In like manner, at the surface p — b, we have for every term, (7) Dnbn = Bnbn + Cnb~n. Beside the conditions of continuity, we have at each surface of the cylinder for the whole potential F= F0+ Vi. The potential of the ex- ternal field being continuous, with its derivatives as well, we have (9) 0 + = 0> 197] DIELECTRICS AND MAGNETIZABLE BODIES. 377 which being multiplied by /*2 and subtracted from (8) gives 9F0 At the surface p = a this gives, differentiating (2), (3), (4), by p, (ii) ^(-Ann 0 - /£, 2 (^a71"1 - CUa-) Tn = (/*2 - /^) no,-1 Tn, o o and consequently for every n (12) - frAnar* - ^ (Bnan - Cna~n) = (^ - ^), and at p = b, (13) - _ Tn, and consequently (14) - ^ Dnbn The four linear equations (6), (7), (12), and (14) determine the four constants An> Bn, Cn, Dn. Solving, we obtain for their values, putting If, f/7)\27l -^ 0 - 1 (15) Since the absolute value of M is greater than 1, and since b/a r>a, p-(n+v)Yn, a>r>b, n, r 80' dry we have for the equations of equilibrium, resolving along the three axes, Hdr + XxdSx + XydSy + XzdSz - XndSn = 0, (i) Rdr + Yxd8x + YydSy + YzdSz - YndSn = 0, Zdr + ZxdSx + ZydSy + ZzdSz - ZndSn = 0. Now the faces dSx, dSy, dSz are the projections of the face dSn on the coordinate planes, and accordingly dSx = dSn cos (nx), dSy — dSn cos (ny), dSz = dSn cos (nz). w. E. 25 386 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. If we now let the dimensions of the tetrahedron diminish in- definitely, the volume dr is of a higher order than the surface of any face and can accordingly be neglected, accordingly the equa- tions of equilibrium become, dSn dividing out, Xn = Xx cos (nx) + Xy cos (ny) + Xz cos (nz), (2) Yn—Yx cos (nx) + Yy cos (ny) + Yz cos (nz), Zn = Zx cos (nx) 4- Zy cos (ny) + Zz cos (nz), which proves the statement that the stress at any point, involving the action on a plane element in any direction at the point, may be expressed in terms of the nine components at the point, Xx, Yx, Zx, Xy) Yy, Zy, Xz, i z, Zz. Let us now consider the condition of any finite portion of matter r. Let the body-forces H, H, Z, per unit of volume be applied to each element. If now the forces Xn, Yn, Zn applied to each unit of surface are to produce the same effect as the given system of body forces, then the system of body forces with their signs reversed, together with the surface forces, would produce equilibrium. For equilibrium we must have, resolving in the ^"-direction, (3) '-.-. Let us now express Xn in terms of the nine components by the equations (2), (4) ( {Xx cos (nex) + Xy cos (ney) + Xz cos (nez)} dS Transforming the surface integral into a volume integral we obtain and if every portion of the body is to remain in equilibrium under the stresses, in order that the integral shall vanish for every field of integration we must have everywhere ax, sx, 200] DIELECTEICS AND MAGNETIZABLE BODIES. 387 In like manner we find (6') as the equations of equilibrium. In order to explain electrical and magnetic forces by means of stresses we must therefore be able to transform the expressions already found for 3, H, Z, into forms involving partial derivatives as above. Introducing into the expression for B the value of p from i (8 / 8F\ d f 8F\ a / dv and transforming the derivatives we obtain , 7r^^ W/ =sUHSH8 8« 80 The expression now has the required form of a sum of three deri- vatives. If we perform similar transformations on H and Z we shall find that the equations of equilibrium are satisfied by putting 3F3F 1 dy 4?r ~ 4 25—2 388 ELECTROSTATICS AND MAGNETISM. [FT. II. CH. IX. Since Yz = Zy, etc., it is easy to see that the couple tending to turn any element of volume about either of the axes vanishes, as is the case with ordinary elastic stresses. If the body is not isotropic this condition does not hold. We shall now apply the expressions found to determine the nature of the stress in two particular cases. First, let the element dS be perpendicular to a line of force. Then we have X Y Z and using these values in the equation (2) ^ l F r"~ F f 87TlAy* >F These components of Fn are equal to %F/S7r multiplied by the direction cosines of F, which is in the direction of the normal n. That is the force Fn is perpendicular to its plane. A plane possessing this property is called a principal plane of the stress. The stress being positive represents a tension. Accordingly the medium is in a state of tension along the lines of force, of an amount per unit of surface equal to $F/87r, which, it may be noticed, is the amount of energy of the medium per unit volume. Consider secondly an element tangent to a line of force. Then we have X Y Z •* -p cos (nx) + -p cos (ny) + j cos (nz) = 0. Multiplying this equation by F£/4>7r and subtracting it from the expression for Xn gives Xn = g {2£X - g^J cos (nx) + $ F cos (ny) + - $Z cos (nz) (10) 1 ^F - — {3L3T cos (nx) + £F cos (ny) + $Z cos (nz)} = - ^ cos (n#). 200, 201] DIELECTRICS AND MAGNETIZABLE BODIES. 389 In like manner CK Tjl ^V Fn = - Q— cos (ny), £w = - — cos (nz). O7T O7T Here again the components of Fn are equal to — gF/8-Tr multiplied by the direction cosines of n, or the force is normal to its plane. Consequently any plane tangent to a line of force is a principal plane of the stress, and the stress is symmetrical about the line of force. The negative sign shows that the stress is a pressure. The state of stress consisting of tension along the lines of force combined with an equal pressure at right angles to them was described by Faraday*, who expressed the matter in words that state in effect that the lines of force tend to contract and to repel each other. This may be illustrated by supposing the medium to be divided into filaments along the lines of force, and these again to be subdivided into short filaments. Then each short filament is a I4 'II* "*ll* '|E^3 polarized body which acts like a doublet, and h — 1 — 1. — 1r^^ I* *ll* 'II* 'II* 'I since unlike poles of successive elements are in juxtaposition, the filaments all attract each other endwise. For filaments lying side by side, however, since like poles are together, there is a sidewise repulsion. 201. Permanent Magnets and Electrets. Intrinsic Polarization. The fundamental laws of magnetic and electric induction may be summed up in the statement that in soft iron and in similarly acting bodies the force is lamellar, and //, times the force is solenoidal. Or in brief (i) curl.F=0, (i') div(/^) = 0. Iron for which this statement is true is said to be perfectly soft. When the external field affecting such iron is removed, the polarization disappears. As a matter of fact, this is an ideal condition not exactly realized by any sort of real iron, for when the external field is removed, a part of the polarization persists. This is called residual magnetization. The harder the iron or steel, the greater is the fraction of the induced polarization which * Faraday, Exp. Res. (1297). 390 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. persists. A substance in which, when the external field is removed, the whole induced polarization remains, is called perfectly hard, and a body consisting of such substance is called a permanent magnet. The inductivity of such a body is to be considered the same as of air. Such bodies do not exist any more than perfectly soft ones. We may however treat actual bodies as if they were formed by the superposition of perfectly hard and perfectly soft matter. The portion of the polarization which permanently remains is called the intrinsic polarization*. In order to carry out the analogy, Heaviside has proposed to call a dielectric per- manently polarized body an electret, and its polarization electriza- tion. Certain natural crystals when heated assume this condition. The permanent or intrinsic polarization now forms a real magnetic or electric charge, and if the intrinsic polarization be denoted by /0 with components AQ, B0, C0> we have for the real density dx dy dz with a similar expression for a. Comparing this with the expression for p in § 182 (15), we find ,, 47r \dx ty dz) ( dx dy dz or the divergence of the induction is equal to 4?r times the convergence of the intrinsic polarization. Comparing the expressions for the apparent density, that is the sum of the real and induced, in terms of the force F, § 186 (33), and in terms of the total polarization /, § 120 (6), we find ,_3Z dY dZ p - Accordingly (5) l(Z4 or more briefly, (5) div(F+47r/)=0. The solenoidal vector-sum, F+^TT!, has been called in § 121, the induction. We shall call it the Maxwellian induction, and denote it by $M, since it corresponds to the definition of the * Thomson. Reprint of Papers on Electrostatics and Magnetism, p. 578. 201, 202] DIELECTRICS AND MAGNETIZABLE BODIES. 391 induction given by Maxwell. It is solenoidal in intrinsically magnetized bodies as well as elsewhere. The induction, g, which is divergent in intrinsically magnetized bodies, and which is defined as pF, we shall call the Hertzian induction, and denote by %H. In magnetically soft bodies these two inductions are identical, but in intrinsically polarized bodies they differ. If we write equation (3) as (6) and from it subtract (5) we have (7) div ( g H - F) = 4-7T div (7 - J0). Now if we call /$ the induced polarization, we have as always (8) Ii = KFJj^lFi /=/„+/,. Inserting these in (7) div (gH - F) = 4vr div I, = div {(ft, - 1) F}, and transposing div F, (9) divg*=divO*F), agreeing with the definition of gH. 202. Heaviside's treatment of Intrinsic Polarization. The treatment given by Heaviside differs in several respects from that just given. According to that author the induction is always solenoidal, so that true magnetic charges do not exist. The only reason given for this assumption seems to the present writer insufficient, being, as stated by Heaviside, "to exclude unipolar magnets." It appears that the exclusion of unipolar magnets merely means that for any magnet the integral charge is zero, which simply means that the distribution is what we have called polarization, and lays no restriction on the divergence of the polarization or induction. It might be supposed that Heaviside's induction was what is here called the Maxwellian induction, were it not for the fact that he says that " we use always " g = fiF, In 392 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. order to make these two statements, which we hold to be mutually exclusive where there is intrinsic polarization, appear consistent, Heaviside proceeds in the following, as it appears to us, artificial manner. In our notation, Heaviside* considers the field F as made up of a part h = 47r/0/yu,, defined as the intrinsic force, together with a lamellar force which we shall denote by Fh (F in Heaviside's notation), In order to make the induction solenoidal, he then puts Ii = *Fht instead of Ii = /cF. Then the induction is defined, not as but as $ = Fh + 47T/. Inserting for / the sum of the intrinsic and induced polariza tions, this becomes This gives, in conjunction with the equation supposed to be fundamental, namely divg = 0, the equation — div (//J\) = — — div /j,h = — div /0, •rTT 47T which we may compare with our equation (6). Accordingly Heaviside's //J\ has the property of our Hertzian induction. The difference in Heaviside's treatment may be summed up as : 1. A different definition of the total field. 2. Induced polarization produced by only a part of the field. 3. The Hertzian induction considered solenoidal, even in case there is intrinsic polarization. We have stated the difficulties of Heaviside's treatment as they appear to us, without wishing to dispute the dicta of so weighty an authority. The theory as we have given it seems to be that of Helmholtz and Hertz, both of whom explicitly state that real magnetism exists in permanent magnets. Neither they, however, nor any other author, so far as known to the present writer, have * Papers, Vol. i., pp. 453—4. 202, 203] DIELECTRICS AND MAGNETIZABLE BODIES. 393 worked the matter out in detail as has Heaviside, nor have any problems been solved in which a difference becomes of importance. In either treatment, the flux of induction issuing from a magnet is the same, which is the quantity with which we are concerned in practice, the ambiguity existing only in the substance of the intrinsic magnet. The difference between intrinsic and other magnets is that in the former two independent vectors are necessary to characterize the state of the body, while in the latter one suffices. These may be taken as %H and gj,, as g^, /, or as $M, I. 203. Variability of /A. Hysteresis. Throughout this chapter it has been assumed that the value of //, at any point was constant for that point. This assumption is not borne out by the facts, but was necessary in order to make the subject amenable to mathematical treatment. It is found that p is a function of the strength of the field, and that for magnetic bodies, in which this phenomenon has been most carefully investigated, as the force increases, //, diminishes, finally tending towards the limit unity, so that the ratio of the induction to the force approaches unity. At the same time the difference between the induction and the force tends towards a constant maximum value, which is equal to 4?r times the greatest intensity of magnetization that the substance can assume. This is known as the intensity of saturation. For wrought-iron this intensity of saturation has been found to be about 1 700 c.G.s. units. The variability of p does not affect the validity of Ohm's Law, which determines the distribution of the tubes of induction, although it seriously complicates the mathematical theory. In fact no cases of magnetization have been worked out taking account of the dependence of /JL upon F. But this is not the only defect of our theory. It has been found that for a given value of F there is not a single determinate value of //,, but that the value depends not only on the actual value of F, but upon the values which have acted at the point in question at previous times. If we plot a curve having as abscissas the values of F at a given point at various times and as ordinates the values of g at the corresponding times, we may express this phenomenon by say- ing that the value of //, at any point of the diagram depends on the path by which the substance has been brought to the point, that is, on the whole history of the field at the point. This 394 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. phenomenon, discovered by Warburg*, and thoroughly investigated by Ewingf, was named by the latter Hysteresis, to denote the after- effects of the fields to which the substance has been submitted. Warburg and Ewing found that if the field was increased to a certain value, then decreased, and then varied successively between the same limiting values, the path of the representative point on the F-% diagram was a closed curve, which was re-traversed after the first periodic cycle. This is called the hysteresis-loop, and its area has an important physical significance. Such a loop is shown in Fig. 82. If instead of continuing to repeat the same cycle we vary F between different limits the point may take any position between the two limiting curves of the loop, as shown in Fig. 83, both these figures being copied from Ewing. FIG. 82. FIG. 83. If the cycle be so chosen that at some point, F, while decreasing, passes through the value zero, the value of / calculated as the corresponding value of 8/4?r is the residual magnetization. If the force Fis still further decreased, its value when 7=0, $ = F, is called, after Hopkinson, the coercive force, since it measures the negative force necessary to destroy the residual magnetization. Besides these phenomena of hysteresis, there is another more complicated effect, which causes the magnetization to arrive at its final value only gradually, taking a certain time to reach its permanent value. This is denoted by the name of viscous hysteresis, magnetic lag, or after-effect (Nachwirkung), to dis- tinguish it from the proper or static hysteresis just described. * Warburg, Wied. Ann. 13, p. 141, 1881. + Ewing, Phil. Trans. CLXXVI., p. 523, 1885. 203, 204] DIELECTRICS AND MAGNETIZABLE BODIES. 395 204. Dissipation of energy in Static Hysteresis. Since we have seen that p is not uniquely determined by the value of F, so it must be for the energy of the field. Accordingly the forces acting on polarized bodies cannot be derived from a single-valued potential, but must be non-conservative. In taking a body through a cycle of magnetization, accordingly, a certain portion of the work done upon it fails to be stored up as energy, and is therefore dissipated into heat. We may easily find an expression for the value of this dissipated energy. The potential energy of a polarized body in a field whose potential is V is, by § 126 (2), equal to or in terms of the field W=- \l\(AX + BY+CZ)dT. If we consider an element of volume dr, and suppose it moved to a point where the field is X + dX, Y+dY, Z+dZ, the work dW done upon the particle during the motion is accord- ingly equal to the increase in the value of the energy, (i) dW = - dr(AdX + BdY + CdZ). In the second position the values of A, B, C have changed to the values A+dA, B + dB, C + dC, but the change made by using these values in the expression for the work would be of the second order and may be neglected.* If instead of moving the particle we change the strength of the field the work done will be the same. Inserting the values of A, B, G in terms of the induction and force we obtain j (2) If now we vary X, Y, Z through a cycle of values, coming back to the value from which we started, the integral (3) 396 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. vanishes, since the value of X* at both limits is the same. The integral may be integrated by parts, giving Of this the integrated part vanishes, since, as found by Warburg and Ewing, after the cycle has been once traversed 3£ returns to the same value on traversing the complete cycle. We thus find that in taking the particle through the whole cycle of magnetic operations, and leaving it in its original state, we have done a quantity of work, which is equal, not to zero, but to the integral being taken around a closed loop. Each term of the integral must of course be obtained from a separate loop. The whole energy dissipated in the body is Yd® + Zd3\ dr. Of course the general theory is so complicated that it is not even to be assumed that when we have carried the magnetization through a closed cycle in one point of the body we have done so at all points. In practice we can calculate the dissipation only in the case of a uniformly polarized body, where A, B, C are the same at all points of the body and in the direction of the force. The cycle is then the same for all points, and the energy dissipated is equal to vol. of body x -:— I j 477-J The integral is evidently the area of the hysteresis-loop. This area is inde- pendent of the time of description of the cycle. In the case of viscous hysteresis there is an additional dissipation which depends in a complicated manner on the rate of description of the cycle. 204, 205] DIELECTRICS AND MAGNETIZABLE BODIES. 397 205. Hysteresis couple. In the examples of §§ 192—198, it is evident that a sphere or cylinder turned about an axis of symmetry in the field would experience no resisting couple, for no work would be done against the forces of the field. In like manner an ellipsoid would require on the whole no work to rotate it about an axis, for the forces hindering the motion in one part of the revolution would have corresponding forces helping the motion in another part of the revolution. If hysteresis exists, however, the case is quite different. Then the ellipsoid in a position in which its long axis makes a diminishing acute angle with the direction of the field experiences a mechanical couple tending to accelerate its motion. The magnetic force parallel to the long axis is then increasing, so that when the force has reached the same value in the symmetrical position in which the axis makes the same angle with the direction of the field, but on the other side, F being then on the decreasing branch of the hysteresis-loop, the value of the magnetization is greater, so that the mechanical force, which now retards the motion, is greater. Accordingly the motion is on the whole retarded*, and it is easy to see that- the mean retarding couple is proportional to the mean difference of the ordinates on the upper and lower branches of the loop, that is to the area of the loop. Upon this principle is based Ewing's Hysteresis indicator^, in which a long sample of iron is rapidly revolved between the poles of a magnet, and the mean couple between them measured by the pull on the magnet. The couple is, as seen above, independent of the time of revolution. * An effect of this sort was observed in diamagnetic and very slightly magnetic bodies by Mr. A. P. Wills, in the physical laboratory of Clark University, in the fall of 1895, and was discovered independently by Mr. Wm. Duane, in the physical laboratory of the University of Berlin. Wied. Ann. Bd. 58, p. 517, 1896. t Ewing, Journ. Inst. Elec. Eng. 24, p. 398, 1895. OF THF UNIVERSITY CHAPTER X. CONDUCTION IN DIELECTRICS. 206. Variable Flow. Relaxation-Time. We have hitherto supposed dielectrics to be perfect insulators. This can hardly be said to be the case, even for the best insulators. On the other hand, although, as we have seen, the greater the inductivity of a dielectric, the more nearly does it act, as far as concerns electro- static distributions, like a conductor, it is by no means likely that the inductivity of conductors is infinite. Still less is it likely that it is zero. We shall now consider the consequences of considering a dielectric to possess, in addition to its electrical iiiductivity u, an electric conductivity \. We shall now deal with currents which are not in the steady state, and shall require to assume that at any instant Ohm's Law determines the distribution of the currents, namely This assumption is justified by experiment. Instead of the sole- noidal condition for the current, however, we must obtain a new equation. This is obtained by the consideration that, if we consider a portion of substance r bounded by a closed surface S, the total charge within that surface increases in any interval of time by the amount of total current flowing into T through the surface, that is, if n is the internal normal ( I ) — 1 1 Ipdr = 1 1 [u cos (nx) + v cos (ny) -h w cos (nz)} dS du dv dw) , ^- + 5- + 3- 1 dr. dx oy oz) Since this equation must hold for any portion of space, we must have everywhere dp _ (du dv dw\ di = ~\fa+dy+W° 206] CONDUCTION IN DIELECTRICS. 399 But in a dielectric, (3) Differentiating (3) by t, and eliminating -£ from (2), we obtain for a conducting dielectric 3 f 1 d£\ d f 1 dg)) 9 f 1 d3\ (4) ^-V-+T -^rr + o- \V + T- -jfr + ^ -^ + -7- -^ = o. dx ( 4?r dt) dy ( 4>7r dt) dz ( 4?r dt ) If we put u, v, w, $, g), 3 m terms of the field, assuming that the substance is homogeneous as regards both p and X, this becomes a f i a 4?r r dt d* 4?r r dt or in terms of the density Integrating this differential equation, we have -^t (7) P = P«e » - Accordingly whatever charge the body has originally decreases in geometrical ratio as the time increases in arithmetical progression. The constant T= //,/47rX, which is the time it takes for the density at any point to fall to l/e of its original value, has been called by Cohn* the relaxation-time, a term used by Maxwell in connection with the Kinetic Theory of Gases. For ordinary metallic con- ductors this time is so short as to have hitherto defied observation. The importance of its discovery was recognized by the committee setting subjects for an international prize competition in 1893, who proposed this as one of the questions for investigation!. It appeared that no experimenter ventured to attack the problem, it being evidently considered too difficult. The finite relaxation-time was determined for so good a conductor as water in some remarkable experiments by Cohn and AronsJ, who are entitled to the credit of discovering the finiteness of T for conductors. * Cohn, Wied. Ann. 40, p. 625, 1890. t Elihu Thomson Prize, Electrician, 1892. J Cohn u. Arons. "Leitungsvermb'gen und Dielektricitatsconstante." Wied. Ann. 28, p. 454, 1886. 400 ELECTKOKINETICS. [PT. II. CH. X. 207. Method of Cohn and Arons. Consider a condenser A, which may or may not be connected in parallel with the con- denser B and the resistance wire R. Let the capacity of A be K, the inductivity of the dielectric p. Let the conductivity of the dielectric in A be X and in B zero. Then the charge of one of the plates 1 of A is in terms of the induction, § 182 (16), (8) ej = 7- 1 1 (£ cos (nx) + g) cos (ny) + 3 cos (nz)) dS. 8t On the other hand the quantity flowing through the dielectric in the condenser in unit time is (9) . —-1 = M [u cos (nx) + v cos (ny) + w cos (nz)} dS, Si so that, assuming X and p constant, (10) If we assume that an electromotive force is applied to the plates in order to establish a steady difference of potential F0 until a steady state of flow is attained, we have everywhere in the dielectric p = 0. If the electromotive force is suddenly re- moved, we have from that time on and accordingly the difference of potential of the condenser plates is di) F=FO*~'. If the difference of potential F can be measured by an electro- meter at any time t, we have ii t. (12) T If in the second place the condenser A is connected in parallel with the condenser B and wire of resistance R, we have for the charge e,f of the plate 1 of B, e^ = K'V where K' is the capacity of B. 207] CONDUCTION IN DIELECTRICS. 401 If after the steady state is established, we remove the electro- motive force and leave the system to itself, we have flowing through the wire R per unit of time the quantity F R' Accordingly we have for the decrease of the charges which when combined with the equation (8) ei = gives the differential equation «*(«, + «.') 47T\ F -3T ~f *--R- Substituting for the charges el} e^, their values in terms of the difference of potential V, we have which being integrated gives Putting R= oo , K' — 0 we obtain the solution (i i) just found. Considering the condenser B alone discharging through the wire, we obtain, putting K=Q, (17) V=V0e~^. A conducting condenser accordingly behaves, when left to itself, exactly like a perfectly insulating condenser discharging through a wire. The relaxation-time of such a condenser is KR, but for a conducting condenser, although we may use the same formula, the relaxation time is independent of the form or dimensions of the condenser, since, as we have seen in § 184, if K0 be the capacity of the condenser with air as a dielectric, we have The relaxation-time is accordingly a characteristic constant of the medium, and may be determined independently of other w. E. 26 402 ELECTROKINETICS. [FT. II. CH. X. media, whereas we may determine only the ratio of the inductivity of a medium to that of a standard medium. If we make two experiments with the above combination of condensers, one with A alone, which gives T, and a second with A and B which gives if we know K' and R we may from these two results determine K, and if the condenser is made in any shape suitable for calculating KQ from geometrical data, we can then determine p. In this manner Cohn found for water JJL — 73*6, the largest value of the electric inductivity yet found for any substance. In the case of metals, all that we know is that T is extremely small. This is of course due to the large value of X, so that whether ^ is large or small we have as yet no means of knowing. 208. Condenser with two Dielectrics. Absorption. In the preceding section we have seen that a charge residing in any part of a conducting dielectric will gradually disappear, and that no electricity will accumulate at any part of such a dielectric. We have considered only the discharge or leakage of a condenser, starting from a state of steady flow. We shall now consider the state which precedes the attainment of the steady state when an electromotive force is suddenly applied to produce a difference of potential between the -plates of the condenser. We shall also suppose that the condenser contains two dielectrics of different properties, and for simplicity we shall consider only a plane con- denser. Let the potentials of the two plates be Fi and F2, and let that of the plane separating the two dielectrics be F3. Let the thickness, inductivity and conductivity of the upper dielectric be d1} fa, \lt and of the lower dz, /z2, V The force in the upper dielectric will be the same at all points, Flt which however depends on the time. In the lower dielectric let the force be F^, also a function of the time. Let the currents in the two dielectrics be ql and qz respec- tively, and let Flt F2, q1} V> f°r the corresponding electric quantities. For the electric inductivity we shall use the letter e, leaving yu, for the magnetic inductivity. These distinctions have not before been necessary, since we have not at the same time considered both electrical and magnetic quantities, as we must do from now on. If we form the line integral of magnetic force from a point A to a point B, we have (I) fB I Ldx+Mdy + Nd**=£lA- which must be independent of the path AB, for otherwise, by changing the path infinitely little, we should, starting with the given value £1A, cause H^ to change by an infinitely small amount, and could thus cause £1B to take at the same point a series of con- tinuously varying values. The integral is accordingly the same for all paths that can be changed into one another by continuous deformation. If, however, the current separates two paths ACB, ADB, the integral is not the same for both. In other words, while the integral around any closed path not linked with the circuit is zero, the integral around a path linked with the circuit is not. But the integral around any two closed paths each linked once with the circuit is the same, for they may be continuously deformed into each other. Or in other words, we may connect two such paths 1 and 2, Fig. 84, by a path PQ. The integral around the circuit ABPQDCQPA, which is not linked with the current, is zero, but this is equal to the sum of the integrals PABP around 1 in the positive direction, together with the in- tegral QDCQ around 2 in the negative direction, while the integrals over the coincident paths FIG. 84. PQ, QP in opposite directions destroy each other. Accordingly We shall say that two geometrical circuits are linked positively, ELECTROMAGNETISM. 411 -f- Rig hi 210] when, given a direction of circulation about each circuit, the direction of circulation in one circuit agrees with the forward motion of a right-handed screw, whose rotation corresponds to the direction of circulation in the other circuit. Fig. 85 repre- sents two circuits linked positively above and negatively below. By an extension of the above reasoning we see that the integral around any circuit linked n times in the positive manner with the current is nJ, where J is the integral around any circuit linked once. Accordingly the potential at any point is an infinitely valued function, whose values differ from each other by integral multiples of J. We may however make the potential a uniform function, if we prevent passage from one point to another by paths not continuously deformable into each other, that is, if we reduce the doubly connected space about the current to a singly-connected one by means of a diaphragm covering the current circuit. Then no two paths can be separated by the current. If we consider the potential Fm- 86- at two points infinitely near each other but lying on opposite sides of the diaphragm, Fig. 86, to get from one to the other we must perform a closed circuit about the current, so that their potential differs by the amount J, accordingly in crossing the diaphragm, the potential is discontinuous, the amount of the discontinuity being where A is on the positive side of the diaphragm. There is, how- ever, no discontinuity nor lack of uniformity in the derivatives of 0. If we now consider all space, except a small sphere of radius R with center at the point P, and apply to it Green's theorem where for V we put the magnetic potential fl, and for U the function 1/r, where r is the distance from P, the volume integrals vanish, and the surface integrals are to be taken over the infinite 412 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. sphere, where they vanish, over the small sphere about P, where we have as in § 83, on making R decrease indefinitely, 1 r n and over the two sides of the diaphragm, where we have Since, however, dfl/dn is continuous, the first terms in the two integrals cancel each other, the normals n^ and n2 being in opposite directions, and since dni dn.2 this becomes and finally (3) Since f^ — H2 = J we have so that the action of the current is the same as that of a magnetic double-layer or shell of strength (5) d> = ^ = the solid angle subtended by the current circuit at the point P. The positive side of the shell and the one toward which the normal is to be drawn is the side toward which a right-handed screw advancing in the direction of the normal to the diaphragm, would move when rotating with the current. The line of force is positively linked with the current. Since the potential every- where, except in the substance of the conductor, satisfies Laplace's equation, the force is everywhere solenoidal, and the tubes of force are endless, and are all linked once with the current. 211. Electromagnetic Units. The determination of the factor A, which is a natural constant, is a matter of experiment. It is extremely small, that is, an enormous number of electro- static units of electricity must pass in unit time in order that the current may produce magnetic forces of appreciable amount. If, however, we choose a new unit for /, defined by the assumption A = 1, so that we get a new system of measuring currents known as the electro- magnetic system. The unit magnetic potential is defined as the potential at unit distance from the unit magnetic pole in vacuo, accordingly the electromagnetic unit of current is referred at once to a magnetic pole, instead of to an electrified point. From this definition of the new unit of current we may at once obtain a whole system of electrical units. We define the new unit of quantity of electricity as the quantity passing in unit of time when a steady current of one electromagnetic unit flows. From this definition of unit charge we obtain, as before, new units of field, of electric potential, of resistance, capacity, and the rest. Conversely if, measuring the current in electrostatic measure, we put A = 1 we shall get a new unit of magnetic potential, from which we may obtain a complete set of units for magnetic quan- tities, all referred to the unit of electric charge, instead of to the unit magnetic pole. We may thus measure electric quantities in the electromagnetic system, or magnetic quantities in the electro- static system, or as before, each kind of quantity in its own appropriate system, thus obtaining the Gaussian system. 414 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 212. Dimensions of the Units. If we denote the numeric of a quantity when measured in the electrostatic system by the suffix e and when measured in the electromagnetic or magnetic system by the suffix m, we have for the magnetic potential (1) nm = ^/eft> = /mft>, (2) £le = Ieco=-£lma>. Consequently the number A denotes the ratio of the numeric of a certain current when measured electromagnetically, to the numeric of the same quantity measured electrostatically, or I/ A is the number of electrostatic units of current in one electromagnetic unit. If m denote a magnetic charge, we have the dimensional equation, by § 190, the quantities being measured in either system. Also since the dimensions of solid angle are zero, the dimensions of ft are the same as of /, and rT-t r^-, f m (4) [f] '• Since the unit of electric charge in either system is obtained from the unit of current multiplied by the unit of time, and we accordingly have for the ratio of the two units of electricity or of current, inserting the suffix m in (5)* Now the fundamental assumption in defining the magnetic system was that the dimensions of p were zero. Also the assumption defining the electrostatic system was that the dimensions of e were zero. Accordingly the dimensions of ee, and of mm, both belonging to the Gaussian system, and defined by precisely the same considerations, namely (7) * Evidently any dimensional equation holds when either suffix e or TO is inserted on both sides. 212] ELECTROMAGNETISM. 415 are the same. Hence the dimensions of the quantity I/ A are accord- ingly the same as those of a velocity. All that has been said of course applies to any absolute system of units, and has no re- striction to the C.G.S. system. If the units of length, mass, and time are given, we can by definition immediately obtain the unit of electricity in either the electrostatic or electromagnetic system, and by experiment determine the number of electrostatic units contained in one electromagnetic. If the unit of mass is now changed, and we define our electrical units as before, the size of both units of electricity has changed, but in the same ratio, so that the number of one kind contained in one of the other is the same as before. If, on the other hand, we change the unit either of length or time, the two electrical units change, but in different ratios, so that the numeric expressing the number of one kind in one of the other is changed from its former value. It has, however, changed in precisely the same way that the numeric expressing any given velocity has changed, so that we may say that the number I/ A represents a certain definite velocity, which is totally independent of the units chosen. When the units of mass, length, and time have been settled upon, the numeric of this velocity may be given. This velocity will be denoted by v. It is to be noticed that the determination of the quantity v depends upon the determination of a certain numeric, the units being settled upon, and that there is nothing of the nature of an actual velocity involved. We shall, therefore, not as yet be under- stood to speak of v as a velocity, but merely as a quantity whose numerical expression changes like that of a velocity, with any change of units. The quantity v is thfc most important electrical natural constant. Numerous determinations of its value have been made, the first by Wilhelm Weber* and Rudolf Kohlrausch, in 1856. The number now generally accepted is v=3 x 1010cm./sec. Electrical and magnetic potential are defined in terms of work, so that (8) [eV] = [mCl] = [ML*T-zl which agrees with the other possible definition Weber, Elektrodynamische Maassbestimmungen iv. 1856 ; Werke, Bd. in. p. 609. 41 6 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. From (8) and (6) we obtain (9) [eeVe] = (emVm], and there are v electromagnetic units of potential in one electro- static unit. Capacity is defined as ratio of charge to potential, so that (10) from which (10') or there are v2 electrostatic units of capacity in one electromagnetic unit. Resistance is defined as ratio of potential to current, so that or there are v2 electromagnetic units in one electrostatic unit. 213. Practical System. The absolute system of units was due to Gauss, and was introduced to practice by Weber. The system was first made practicable for general use by the exertions of the British Association, which issued copies of the unit of resistance, and decided on various multiples of the C.G.S. electro- magnetic units for practical units. Its action has been seconded by international congresses, at Paris in 1881, 1884 and 1889, and at Chicago in 1893, which determined on the following multiples of the electromagnetic units : 1 Volt = 108 C.G.S. electromagnetic units of Potential. 1 Ohm = 109 „ „ „ Resistance. 1 Ampere = 10"1 „ „ „ Current. 1 Coulomb = 10-1 „ „ „ Electric Charge. 1 Farad = 10~9 „ „ „ Capacity. 1 Joule = 107 „ „ „ Work. 1 Watt = 107 „ „ „ Activity. The prefixes mega and micro are used before the preceding names of the units to denote respectively multiplication and division by 212 — 214] ELECTROMAGNETISM. 417 a million. These units form a consistent system, so that electrical relations involving quantities measured in these units require no numerical factors. For instance, a current of one ampere is produced when an electromotive force of one volt is impressed in a circuit whose resistance is one ohm, and the activity of one watt thereby exerted dissipates energy at the rate of one joule per second. 214. Electrostatic compared with Practical Units. From the above definitions with the value of v given and equation (9') we find 1 C.G.S. electrostatic unit of Potential = 300 Volts. From (10') 1 Farad contains 9'10U C.G.S. electrostatic units of Capacity. The electrostatic unit of capacity is the unit of length, accordingly 1 Microfarad = 900,000 cm. of Capacity. A sphere of nine kilometers radius in free space would have a capacity of one microfarad. From (6) 1 Coulomb = 3'109 C.G.S. electrostatic units of Electric Charge. From (ii7) 900,000 Megohms = 1 C.G.S. electrostatic unit of Resistance. From equation (7) we may find the dimensions of e and m, when those of e and /* are settled upon. Any convention that may be made gives us a possible system of units. It must be noticed, however, that there is always a relation between the dimensions of e and p, From equations (4) and (5) Squaring this and dividing by we obtain w. E. 27 418 THE ELECTROMAGNETIC FIELD. [PT. III. CH, XI. Consequently the dimensions of the product of the electric and magnetic inductivities must in any system be those of the square of the reciprocal of a velocity. The absolute dimensions of either factor are arbitrary. Attempts have been made to settle the absolute dimensions of e or //,, but they are evidently based upon misconceptions of the theory of dimensions. The two common assumptions are, that This gives the electrostatic system. Secondly we may assume = 1 € = - P' ~ V2 ' This gives the electromagnetic system. We shall, when dealing principally with the magnetic properties of currents, use the electromagnetic system, but when dealing equally with elec- trical and magnetic phenomena, to avoid ambiguity, we shall, following Helmholtz and Hertz, use the Gaussian system, measuring all electrical quantities in the electrostatic system, all magnetic quantities in the magnetic system, and introducing the factor A, with the numerical value 1/v. A complete table of dimensions of the various units is given at the end of Chapter XIII. 215. Potential due to Circular Current. The potential at P due to a current being H = /&>, where a> is the solid angle subtended at P by the current circuit, if P is situated at a dis- tance so from the center of a circular current of radius R, on the line through its center 0 perpendicular to its plane, we have for the area of the segment of the sphere of unit radius about P cut off by the right cone whose vertex is P, and base the current, (l) ft> = 2?r I sin 6dO = 27r(l — cos a) Jo ' 27T (l - This may also be obtained, according to § 123, by differen- tiating the expression for the potential of a disc at a point on the axis. 214, 215] ELECTROMAGNETISM. 419 The force in the direction of the axis is L^ 90^ 27nR2/ At the center of the circle 27T/ From this expression comes the definition often given of the unit of current as that current which, flowing in a circle of unit radius, produces the field 2?r at its center, or less correctly, the current, which, flowing in an arc equal to the radius in a unit circle, produces unit field at the center. The expression for the force is an example of the proposition that similar geometrical circuits traversed by equal currents, produce at corresponding points forces inversely proportional to their linear dimensions. For at corresponding points the solid angle, and therefore the potential is the same. In the circuit of n times the dimensions, the potential changes by equal amounts for displacements of n times the length, hence for equal displace- ments the change is l/n as great, and the force is n times smaller. When the point is not on the axis of the circle, the cone, having an oblique section circular, is elliptic, and we must cal- culate the area of the spherical ellipse cut out by it from the unit sphere. This involves an elliptic integral. We may however develop the result in an infinite series of zonal spherical harmonics, as in the case of the potential of a disc, in § 102. Developing the above expression for o> at points on the axis by the binomial theorem, we have 27—2 420 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL Accordingly for points not on the axis, at a distance r from the center of the circle, T r» / ~D\ a. ~t O * i ~D\ R ^ r>R, - 274 3 2.4.6 In order to find the direction of the force we must differentiate this in the directions parallel and perpendicular to the axis, and take the resultant. A figure of the lines of force is given by Maxwell, Plate 18. 216. Infinite Straight Current. Law of Biot and Savart. If we have a current flowing through a straight linear conductor of infinite length, we may consider the circuit com- pleted by conductors lying at an infinite distance all in the same plane. The solid angle subtended by the circuit at a point P will be that sector of the unit sphere with center at P included between the plane through the straight conductor and P, and a plane through P parallel to a given plane, which is assumed to be the plane of the circuit. This angle being <£, we have the ratio of the solid angle &> to the surface of the unit sphere equal to the ratio of the plane angle to the circumference of the unit circle, But $ is equal to the angle made by a plane through P and the conductor with a fixed plane through the conductor. Conse- quently the equipotential surfaces are planes through the con- ductor, and the lines of force are circles whose planes are per- pendicular to the conductor. The line integral of force about a circle of radius r is the value of the force H, which is tangential to the circle, times the length of the circumference, and this must be equal to 4t7rl, 47T/ = 2-irrH. Accordingly the value of the force is H&- r This is the law of Biot and Savart*. * Biot et Savart, Ann. Chim. Phys. 15, p. 222, 1820. 215 — 217] ELECTROMAGNETISM. 421 217. Force due to any Linear Current. If the potential at a point P is fl and at a neighboring point Q is fl + SH, where the distance PQ = Sh, and if H is the magnetic force at P, we have (2) S£l = This change in the potential is the same as the change that would be made in the potential at P by moving the whole circuit parallel to itself the same distance Sh in the opposite direction. The change Sfl is proportional to the change Sco made in the solid angle subtended at P due to the motion of the circuit, which is easily seen to be exactly the solid angle subtended at P by the narrow ribbon of cylindrical surface whose edges are the initial and final positions of the circuit, and whose generating lines are equal and parallel to Sh. But any arc ds has described in the motion an area dS of a parallelogram equal to (3) dS = dsSh sin (ds, Sh), and if n be the normal to this element of area, we have for the element dSco of the solid angle subtended by it at P, ,. dScos(nr) , -, . ^. (4) doco = -- — --- - = dsoh sin (ds, oh) cos (nr), where r is the distance of the element from P. Consequently integrating around the ribbon (6) sn If we consider that each element of the current of length ds contributes to the field the potential d£l and the force dH, we have, by (2), / \ JTTZI. fj-cr *i.\ j*r\ TrdsShsin(ds, Sh)cos(nr) (7) - dHSh cos (dH, Sh) = dS£l = / — ^—J - - - ^- - . The numerator is the volume of the parallelepiped whose sides are r, ds, Sh. It therefore vanishes if the direction of Sh coincides with that of r. There is accordingly no component of the force in the direction of r, or the force is perpendicular to r. In like manner if Sh has 422 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. the direction of ds, the force vanishes, so that the force is per- pendicular to ds. If Bh is perpendicular to the plane of r and ds. we have cos (dH, Bh) = 1, sin (ds, Bh) = I, . . JTT rdscos(nr) rdssm(r,ds) dH = -7*- —J§- Accordingly if we call da- the component of ds perpendicular to r, the magnetic force due to the whole conducting circuit will be obtained if we suppose each element ds to contribute to the field the amount (9) ** = £, which has the direction perpendicular to the element ds and the radius r. The total field is the vector sum of all these infinitesimal parts. The proper sign to be chosen may be found by considering the way in which the lines of force are linked with the current, and we find that the direction of the force is given by the rotation of a right-handed screw advancing with the current in the direction of da. The complete specification may be most con- cisely stated by saying that the force due to the element ds is 1/r3 times the vector product of Ids and r, the vector r being drawn from the element ds to the point P, (10) dH = ^-VJs.r. The resolution of the field into elementary fields is artificial, for the field is of course due to the whole closed circuit. Moreover the resolution may be performed in an infinite number of ways, for it is the integral of the above differential taken around the whole circuit which gives the field. We may consequently add to the differential above the differ- ential of any function of the coordinates of the element ds, for in integration around the circuit this function returns to its original value so that the integral vanishes. If the coordinates of a point in the current circuit are xlt yl} zl} these of P, x, y, z, since the direction cosines of r and dsl are 217] ELECTROMAGNETISM. 423 respectively x-xl y-yi z-zl r > r ' r ' dxl dy± dz^ dsl ' dsl ' dsl ' we have for the components of the vector-product representing the field due to an element ds1} dL = -8 (dvi (z - *0 - d*i (y - 2/i)}» \ (u) dM = — {dzi(x-ocd-dxi(z-zd\, dN= -3 [dad (y - yd - dy^ (x - a^)}. We may obtain the same result by the use of Stokes's Theorem, § 31. Since the component of the field in the direction h is Let the constant direction cosines of h be a, fi, 7, and those of n be X, /-t, v, variable over the surface of the diaphragm. Then a a a a 8^=a (13) o — ^ ^ ~r M ?; — ~r dn da? 8 Now since r' = (a? - we have _ d%! ' "dy dyl ' dz 424 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. SO that (I4) _( l a aw a a avi dhdn V 9a?i 8^ 8^/ V 8^ ^8^ 'faJ\r a.(i) 3^(1) a- (1) ^aS + ^ + ^>8 Now since 1/r satisfies the equation I ; . - we may write with similar substitutions for 82 (l/r)^2 and 82(l/r)/8^2. Making these substitutions, and arranging the terms differently, we obtain *£ dhdn (15) '1\ „/! a(- ,r/ V^ pVH Consequently if we put 217] ELECTEOMAGNETISM. 425 (16) Vh w" making Qh the resultant of Uh, F^, Wh equal to / times the vector product of the unit vector h and the vector parameter of 1/r, the force in the direction h is (17) = - J|(curl QA) cos (curl &, n) ^. But by Stokes's theorem this is equal to the line integral (18) - (Uifa + Vhdyi + WhdzJ^-QhCos (Qhds) dSl, around the current circuit. Accordingly attributing to each element ds the amount of field (19) dH cos (dH, h) = adL + 0dM + jdN and since this must hold for every value of a, /3, 7, equating their coefficients we obtain T dL = dM = a - fyi which give the values obtained in (11). 426 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 218. Forces on Conductor carrying Current. The magnetic energy of a pole m at P in the field due to a current is (i) where flm is the potential due to the pole. For any number of of poles, in like manner Hm being the potential due to them all, (2) W = which is the flux of force through the current circuit in the negative direction, due to all magnets. The potential energy tends to decrease, consequently a current in a magnetic field tends to move so as to make the surface integral a maximum, that is, to embrace the largest possible number of tubes of force linked with it in the positive direction. This statement of the mechanical action of magnetic forces on a current is due to Faraday. 219. Mechanical Force acting on Element of Circuit. We may consider the forces acting on the whole circuit as the resultant of the forces acting on each element dslt with the same degree of arbitrariness as in the case of the field due to the current. By the principle of reaction the force on ds1 due to the presence of a unit pole P must be equal and opposite to the force dL, dM, dN on the unit pole, due to the current element ds^ Consequently if dB,, dH, dZ, are the components of the mechanical force acting on ds, I da = -^ \dy-L (*| — z) — dzl (y-^ — y)}, (3) dH =- {dzl (X -x)- dx^ (z, - z)}, dZ = — [dssl (2/1 — y) — dy-L (x± — x)}. T^ But 1 #! — x T 1 yi — y ,., 1 zl — z _ -\r w> are the components of the fiel,d at ds^ due to the unit pole at P. 218, 219] ELECTROMAGNETISM. 427 Consequently dx = I(Nmdyi-Mmdz,\ (4) dU = I (Lmdz, - Nmdx,), and the whole force due to the presence of any number of magnetic bodies producing a field L, M, N is the resultant of all the individual actions (5) That is : the mechanical force on the element is the vector product of the current element Ids1 and of the magnetic field where it is situated*. Suppose that the magnetic field is due to a second element ds2 of strength 72 at a distance r from ds^. Then since by (i i) § 217, putting ds2 for ds1} xl,yl) ^ for a?, y, -z, dL = -2 [dyz (zl - z2) dN = ^ {dxz (y± - 7/2) - cfa/2 (^ - a?a)}, we have for the mechanical force acting on dsl, by (5), (6) &Z=1^[dy1{dx2(yl-y2)-dy2(xl-x2)} — dz-L [dz2 (^ — a?a) — Adding and subtracting the term dxldx^(xl — x^/r* this maybe written (7) + dx, dx, + - dy, + dz, * = - 2 x - (cos (rx) cos (ds-ids*) — cos (rfs^) cos (rc?^)}, r being drawn from ds^ to c?52- * It would be hard to devise a simpler rule for remembering the direction of the force than the one given on p.' 12. 428 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. In like manner 1 2 x 2 a * 2 * 2 cos (ry) cos (ds^ds^) - cos (ds.2y) cos cos rs cos ss^) — cos cg-z cos The resultant (ZlR has a component /j/aCfeiCfoa COS in the direction of r, and one of magnitude cos (r, in the direction ds2. This resolution into infinitesimal forces is unfortunate on account of the lack of symmetry with regard to the two elements. 220. Mutual Energy of two Currents. The whole force acting on the circuit 1 is found by integrating the expressions (7) already found for c£2E, c£2H, d2Z, around both circuits 1 and 2, .. (8) %= IJZ \ -r^- [dxl dx2 + dyl dyz + dz J iJ 2 3®\ with similar expressions for H and Z. If we now suppose the circuit 1 displaced or deformed in any manner, so that a point #1, 2/i > z\> ig displaced by the amount &EJ, Sylf 8&i the circuit 2 being fixed, the forces H, H, Z do the work (9) dy±dy© , . - | I JiJa 219, 220] ELECTROMAGNETISM. 429 The second factor in the second integral may be written and we may then perform the integration around the circuit 1, integrating by parts, obtaining r I ^ cj = — (c?#2 &PI + dy2 oyi •+• T ' . -L- The integrated part vanishes, for the factors 8^, 8y1? 8^ are the same for the beginning and end of the circuit. Accordingly the expression for the work becomes denoting the change made by changing x^y-^z^ keeping x^y^z^ constant. We have accordingly obtained the work as the change due to the motion in the value of a line integral around both circuits. Consequently the mechanical forces are derivable from a force-function, and the integral represents the negative mutual potential energy due to the magnetic forces acting between the two currents. ( 1 1 ) - W = /!/2 I I - (dxidx* + cfa/icfa/2 + dzl r T [ f J J cos This form of the integral was given by Franz Emil Neumann* in 1845 and is generally known in Germany by the name of the Electrodynamic Potential of the currents 1 and 2. * Neumann, "Allgemeine Gesetze der inducirten Strome." Abh. Berl. ATcad. 1845. 430 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 221. Various Resolutions into Elementary Forces. The value of the integral will not be changed if we add to the integrand any expression, where F is any function of r, for it will disappear when integrated around either circuit. If we put F = r, dr _ dr dx^ dr dy± dr dzl dsi dx-L dsi di/i dsl d^l dsi If, Fig. 87, we drop a perpendicular from ds2 on the tangent at dsl} and call the length of the tangent thus cut off p, we see by infinitesimal ^ geometry that - dsl cos (r, ds^) = dr, ds2 cos (dslt ds2) = dp. dst Fm. 87. Accordingly (12) But Consequently (13) cos(ds and ), (14) 9V dp -£- = COS - = co< p = r cos (r, t _9g_ d_ f dr\ dr dr 1 (dr dr = IT~ ^ — H cos (^u r (9^! 9s2 cos (r, ^^ cos (r, ds2} — cos (c?^ , ds2) d*r and multiplying this by an arbitrary constant (1 — &)/2, and adding to the integrand in ( 1 1 )* (15) - W = IJ> ( f i JlJ2r (l-k) cos (r, The value & = 1 gives Neumann's form of the integral, from which may be obtained the resolution into elementary forces already * Helmholtz, Wiss. Abh. Bd. i. p. 567. 221] ELECTROMAGNETISM. 431 given. For k = - 1 we get a resolution into forces proposed by Weber and C. Neumann, and for k = 0 one implicitly suggested by Maxwell. Let us examine the case k = — 1. (.6) - W-IjJ [ JlJ ™ . COB 2 r 1 dr dr , , i2 - 5- o~ ds^Sz. JiJirfad* From this we obtain *w r r f f f ! 8r 3r s ! 8r 88r X 9r 8 (i 7) - $W = ^/s M- 5- r- Sr - - 5- = --- 5- o~ J i J 2 IT 3*i 9s2 r ds2 dSi r d*2 9«2 Integrating by parts, the second term around the circuit 1, and the third around the circuit 2, the integrated parts vanishing in both cases, T t f f1 8r 8r 8 A 1/2 -^-^— ^- + ^ - J i J 2 1^2 3*i 3s2 3«! V^ (18) T f [ ( I dr dr 2 92r = Ij/2 -^ -- - r- ^— + - J iJz ( rz ds!ds2 r ds Since the integrand contains the factor e>r, work is done only when the distances apart of some of the pairs of elements are changed, and we may resolve the action into attractions between dsi arid ds2 of the magnitude drdr 92r x = /i/2 — (2 cos (c?5j, ds.2) — 3 cos (r, ds^ cos (r, c?52)}. This form for the elementary forces was given by Ampere*. Accord- ing to this form, we see that parallel elements perpendicular to the line joining them attract each other with a force Parallel elements having the direction of the line joining them repel each other with a force * Ampere. "M6moire sur la thSorie math6matique des phenomenes Slectro- dynamiques." Mem de VAcad. T. vi., 1823. 432 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. while mutually perpendicular elements exert no action on each other if either is perpendicular to the line joining them. 222. Currents distributed in three Dimensions. We have seen in § 210 that a current I is equivalent to a magnetic shell of strength * = ^ 47T' where J is the line integral of the magnetic force around a circuit positively linked once with the circuit, and in the electromagnetic system / = . Accordingly J=47T/. If now we consider steady currents distributed in any manner in a conducting body with current density q, the integral of magnetic force around any closed curve depends only on the tubes of flow with which it is linked, being equal to 4?r times the total current through the curve. Consequently (i) ^Ldx + Mdy + Ndz = 4?r M {u cos (nx) + v cos (ny) + w cos (nz)} dS, the surface integral being taken over any surface bounded by the curve. But by Stokes's theorem lldx + Mdy + Ndz dN\ - cos (nx} + - asv cos (ny} The surface integrals can be equal for all surfaces bounded by any curve whatsoever only if we have everywhere 8^ dM -= -- -5- , dy dz dL dN (2) 4>7TV =-= --- -5— , dz dx dM dL -5 -- -5- . ex cy 221 — 223] ELECTROMAGNETISM. 433 These are the fundamental equations of electromagnetism. In the Gaussian system, we must introduce the factor A on the left. By these equations the solenoidal vector q is expressed as the curl of the magnetic force H. The magnetic forces cannot be derived from a potential except where there is no current, but must be found by integration of the partial differential equations (2). In order to show how this may always be accomplished, we shall prove a general theorem. 223. Vector Potentials. Helmholtz's Theorem. Any uniform, continuous, vector point-function vanishing at infinity may be expressed as the sum of a lamellar and a solenoidal part, and the solenoidal part may be expressed as the curl of a vector point-function. A vector point-function is completely determined if its divergence and curl are everywhere given. Let R be the given vector, with components X, F, Z. Let us suppose it possible to express it as the sum of the vector parameter of a scalar function <£ and the curl of a vector-function Q, whose components are U, F, W. Then Y_84 , dW_dV — ^ • ^ "i > ox o oz 7_. ~dy+dz d dV dU ^= 3 +^ -- -0~' oz doc oy Finding first the divergence of JR, ,. -p dX 8F dZ div R = 3- + _ + _ = A<£, dx dy dz for the curl of any vector is solenoidal, § 35. But by § 85 (18) we know that if $ and its first derivatives are everywhere finite and continuous, we have Since R is continuous by hypothesis, div R is finite, so that 9F dZ dr w. E. 28 434 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. Consequently the lamellar part of R is determined by its divergence. Secondly finding the curl of R, say o>, with components f, 77, f, a^_8F=a_/a^ <¥ _ ^~9# cte "cty \9fc 30 Since Q is as yet undetermined except by the partial differential equations (i) we may impose on it the condition of being sole- noidal, dx dy dz Hence r^glf 9y 3^ and in like manner 3-ST 8^ "==aJ-te ?-9F ^- ATT ?~te~^- But since -B is continuous, curl R must be finite, and therefore as before (?) The vector Q, whose components are l/4?r times the potentials of the scalar functions £, 77, £, the components of G>, is derived from o)/47T by the operation Pot, considering o> as a vector, so that we may write (8) Q = ~ and call ^irQ the Fecfor Potential of o). Since the solenoidal part of R is the curl of Q, we shall also say that Q is the vector potential belonging to R. We accordingly see that the solenoidal part of R is determined by curl R, and accordingly the vector 223, 224] ELECTROMAGNETISM. 435 is uniquely determined by its divergence and curl. This theorem was given by Helmholtz in his celebrated paper on Vortex Motion*. 224. Symbolic Formulae. These relations may be con- cisely expressed by means of Hamilton's and Gibbs's symbols V and Pot (§ 78). In words we may say that any solenoidal vector is the curl of the vector potential belonging to it, which is the vector potential of l/4nr times its curl. By virtue of the definition of Hamilton's operator we have the vector equation (9) B = V so that we may call the sum of the scalar and the vector Q the quaternion potential belonging to R, from which R is derived by the single vector operation V. Inserting the values of <£ andQ, (10) R = V - (_ Pot divJ£ + Pot curl R) so that the operator (Pot curl - Pot div)/4?r is the inverse of V, when applied to a vector-function. For a lamellar vector we have and for a solenoidal vector (12) divE = 0, E = ~V Pot curl R = ^- curl Pot curl R. 4nr 4?r Taking the curl of o>, we find in like manner curl o> = curl2 R = — &R, (R being solenoidal) so that (13) R = ^- Pot curl2 R. 4-7T In fact since the operations of definite integration and partial differentiation are commutative, the operations Pot and curl must be. * Helmholtz. "Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen," Crelle's Journal, Bd. 55, 1858, p. 25. Wiss. Abh. Bd. i. p. 101. 28—2 436 THE ELECTROMAGNETIC FIELD. [FT. III. CH. XI. 225. Magnetic Force from Current. Applying Helm- holtz's theorem to the solenoidal vector H, the magnetic force, and calling the components of the vector potential F, G, H* and using the fundamental equations (7), together with § 222 (2), we obtain _dH_dG ~ > (IS) "' G =///>' oo *-///> or the vector potential belonging to the magnetic force is the vector potential of the current density. 226. Energy of Magnetic Field of Currents. The mag- netic energy of the field is by § 118 (10), (i) Wm = and introducing the vector potential this becomes - ~- dy dzj \bz foj \dx 00 Integrating by parts for any volume r bounded by a closed surface 8, * It is to be noticed that the letter H is here unfortunately used for both the resultant magnetic force and one component of the vector-potential. This is because we have followed Maxwell in using the letters F, G, H. The ambiguity need cause no confusion. 225, 226] ELECTROMAGNETISM. 437 = [ I {(ME - NG) cos (not) + (NF - LH) cos (ny) + (LG-MF)cos(nz)}d8 This important theorem in integration may be abbreviated as (4) jjj(H curl Q-Q curl # } dr =JJV.HQ cos (w, V. The integral representing the energy is extended over infinite space, and the surface integral vanishes at infinity. Inserting the value of curl H in terms of the current density, § 222 (2), we obtain (5) Wm = and since no portion of space contributes to the integral unless it is traversed by currents, we may take the integral simply through conductors carrying currents. The components of the vector potential are however themselves triple integrals over the same portions of space, so that if we distinguish a second point of integration by an accent, we have the double volume integral r* = (x - xj + (y- yj + (z- /)2, where each point of integration traverses the whole volume occupied by currents. This form of the energy corresponds to the form in terms of density given in § 117 (5), the integrals being there taken through all distributions of matter. If we perform the volume integration by dividing the space up into current tubes, of infinitesimal cross-section S, ds being the length of the generating curve, and I=qS the total current in the tube, we have for the element of volume dr = Sds, so that the integral becomes 1 ffffrfU'coB (dads') (7) 2JJJJJJ- -IT J. 438 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL both ds and ds traversing all current tubes. The sextuple in- tegral is here interpreted as a line integral around every current tube and then an integration for the double infinity of tubes for each variable s and s. If the currents consist of two linear circuits, or closed tubes of infinitesimal cross-section and strengths /! and 72, the sextuple integral reduces to a double-line-integral, and since both variables s and s' are to traverse both circuits, we may divide the integral up into four parts according as s or s coincide with sl or s2, (8) The second and third integrals are equal, for it is evidently a matter of indifference which point of integration is associated with either circuit, so that we may write for the sum of these two terms TT f [ cos(dslds.2) , , JlJ 2 f where each point of integration goes once around one of the circuits. This term is equal to the negative of the mutual potential energy of the electromagnetic forces acting between the two currents, as found in § 220 (i i). In like manner the first and last terms, where each point of integration goes once around the same circuit, are the negatives respectively of the potential energy of either current in its own field, from which the electromagnetic forces acting between its different parts may be calculated. If we call the integrals T _ [ f cos(cfoefe') , , , r _ [ f cos(dsds') l~iiJi r S S> 2~J2J2 r we may say then that the magnetic energy of the field due to both currents (9) Wm = \LJf 226] ELECTROMAGNETISM. 439 is the negative of the total potential energy. But the potential energy tends to decrease, and if the current strengths are constant, while the circuits are moved or deformed, their position and form being specified by a certain number of geometrical parameters qs, the forces Ps according to these parameters are given by (10) 2,P,fy. = - B W = 8 Wm = (11) ^-W&^ oq8 s s The magnetic energy of the field then tends to increase, and we find the system behaving in the same manner as a cyclic system during an isocyclic motion, § 70. The energy which must be furnished to the system during a motion caused by the electro- magnetic forces must be double the amount of work done by the electromagnetic forces, which is equal to the loss of potential energy, and must be furnished by the impressed electromotive forces that maintain the currents. We have already seen that in the case of concealed motions we cannot always tell whether energy is potential or kinetic, and that in cyclic systems the kinetic energy has the properties of a force function for either isocyclic or adiabatic motions. We are therefore led naturally to consider a system of currents as a cyclic system, and, instead of considering W as potential energy, to consider Wm = — W as kinetic energy. We shall henceforth call it the electrokinetic energy, and denote it by T. These considerations, assimilating an electrical system to a mechanical system, are due principally to Maxwell, and by means of them we shall in the next chapter be able to deduce the laws of induction of currents. If in the integral (5) we integrate over current-tubes in the manner just explained, for udr we must put q cos (qx) Sds = Idx etc., so that we obtain for each current (12) T= (Fdx + Gdy + Hdz\ where the integral is around its own circuit, but Ft G, H are the definite integrals over all currents, as previously used. Apply- ing Stokes's theorem to the above line-integral, we obtain 440 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. F ffl\ dy over any surface bounded by the current. But by the equations § 225 (14), (14) T = 1 [{{L cos (nx) + M cos (ny) + N cos (nz)} dS, or the electrokinetic energy of a system of currents is equal to one- half the sum of the strengths of each current multiplied by the total flux of magnetic force through its own circuit in the positive direc- tion. The part of the flux due to the current itself constitutes the term ^LI2, while for any two currents 1 and 2, the portions con- sisting of one -half the strength of either times the flux through its circuit due to the other current, being eo.ua! to the two middle terms of (8), are equal. We may consequently express the mutual kinetic energy of two currents as the strength of either multiplied by the flux through its circuit of the magnetic force due to the other. 227. Mechanical Forces. We may deduce the mechanical forces acting on conductors carrying currents from the expressions found in § 219 (7). Calling the forces per unit of volume H, H, Z, and writing for Ida; the value in terms of the current density udr, we have (I) jjJB*r = The first terms in the first and second integrals destroy each other. The second terms may be written respectively, since the accented quantities are independent of the unaccented, and 226 — 228] ELECTROMAGNETISM. 441 Consequently we get fff~,j ffff (W dF\ fdF (2) &d,T= l/MvU -- -5- - w U --- = fff(vN-wM)dr, and we obtain for the mechanical forces on the conductor per unit volume, H = vN—wM, (3) H=wL-uN, Z=uM- vL. The mechanical force per unit volume is the vector product of the current density and the magnetic field. 228. Effect of Heterogeneous Medium. Let us consider what changes are necessitated in our equations by the presence of magnetizable bodies, so that the magnetic inductivity //, is not con- stant throughout space. In the reasoning of § 210 we supposed the magnetic force to be both lamellar and solenoidal in all space not traversed by currents. As soon as we have variations in the inductivity, the force is in general no longer solenoidal, but the in- duction is. We cannot, however, apply the reasoning unchanged to the induction, for this, in general, is not lamellar. The reasoning connecting the current strength with the work of carrying a pole around a closed circuit is however unchanged, and if the circuit lie in any other medium than air, the work is the same as if the circuit lay in air, namely zero if the circuit is not linked with the current, kirnl if linked n times positively. For consider a circuit composed of two infinitely near circuits each embracing the current once, corresponding points of the two lying infinitely near each other on opposite sides of a surface separating air from another medium. Then if we carry a pole around the circuit in air in one direction, and back around the circuit in the other medium in the opposite direction, since the double circuit is not linked with the current no work has been done. For otherwise, in going around the double circuit in one direction or the other, we might store up energy, as much as we pleased, by repeating the operation. But this would be in opposition to the principle of conservation of energy, which says that the energy is definitely determined when the positions and strengths of poles and currents are given. Consequently our 442 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. electromagnetic equations § 222 (2) remain unaltered. When we consider the energy and the mechanical forces, however, we have changes. The potential due to a current is no longer proportional to the solid angle subtended by it, and accordingly we can no longer deduce the forces as simple line-integrals. We must now write for the energy of the field, by § 180, (i) T = so that if a given current is placed in an infinite homogeneous medium, since the distribution of the force is independent of the medium, as long as it is homogeneous, the induction, and therefore the energy, are directly proportional to the inductivity. Contrast this behaviour of a current with that of a permanent magnet, which in different homogeneous media always emits the same total flux of induction, while the force and therefore the energy are inversely proportional to the inductivity. The flux of force emitted by the conductor carrying current is constant. Since the magnetic force is no longer solenoidal, it can no longer be represented as the curl of a vector potential. The in- duction, on the contrary, can be so represented, and the vector potential belongs to the magnetic induction. aff_8^ dy dz ' _ J\- ~~ ,-v °°~ ^ • ox dy On account of this change it is no longer possible to integrate the equations § 222 (2) in the same simple manner as in § 225, for while the current is the curl of the magnetic force, it is the induction that is the curl of the vector potential. Taking the curl of the induction, mm (Mr_Mf\ afc.^ dy dz r\8y dz J dy dz or using § 222 (2), -.Jrfe-Jf|6- dy dz 228] ELECTROMAGNETISM. 443 so that the vector potential is not related simply to the current, but the magnetic forces still occur in the differential equation. We have the same difficulty as occurs when we undertake to find the potential of the field when the inductivity varies. As we there made use of an apparent density, so here we may define an apparent current as dy dz so that the vector potentials are '-//£*• «-//£*• *- If each magnetizable body is homogeneous, fix dy dz and the apparent currents are the true currents multiplied by the inductivity, except at the surface separating two media, where the derivatives of fi, and consequently the values of &F, A6r, A77, are infinite. We have the above form for F, G, H only when AF, etc. are finite, and when they are infinite at a surface we must proceed as in the case of a surface distribution of matter, that is we must consider an apparent current-sheet between the two media. Con- sidering two surfaces infinitely near each other and situated on opposite sides of a surface of discontinuity of ^ at a distance dn from each other, and integrating the equation (3) over the volume of the thin sheet between them we obtain* — -^— cos (nz) I dndS dn 'J - 91) cos (ny) - (3R' - 3») cos (n*)} d£ * The second integrand in (4) is equal to the first since n is the direction of most rapid (infinitely rapid) change in the functions 9t, 2ft, in the infinitely thin sheet. 444 THE ELECTROMAGNETIC FIELD. [FT. III. CH. XI. £', S3ft', 91', being the components of the induction on the side toward which n is drawn, S, 93t, $1, on the side from which it is drawn. We must now, as shown in § 85 (18), add to the volume integral already found for F the surface integral J^f/W dF\dS + ~' which is the effect of an apparent current whose ^-component per unit of surface is l/4?r times (W - 91) cos (ny) - (W - W) cos (nz) = 33' {cos (33V) cos (ny) — cos (33'y) cos (nz)} — 33 {cos (33^) cos (wy) — cos (33?/) cos (ws)}. Now the normal component of the induction is continuous, its tangential component being Discontinuous, while the tangential component of the force is continuous. The normal plane tangent to the line of force is the same in both media, and the amount of the discontinuity in the tangential component of the induction is 33' sin (S3 n) - 33 sin (33n). Referring now to the definition of a vector product, we see that the first parenthesis above is the ^-component of the vector product of the induction and a unit vector in the direction of the normal, which vector product has the magnitude 33' sin (33'ft). The apparent current is accordingly in the surface, perpendicular to the normal plane containing the line of force where it crosses the surface, and its magnitude per unit of surface is l/4?r times the discontinuity in the tangential induction. If the lines of force are normal to the surface, the apparent surface current vanishes*. If, however, there is a surface carrying a true current-sheet, by the same reasoning, applied to equations § 222 (2), we find a discontinuity in the component of the force tangent to the surface and perpen- dicular to the current of amount 4?r times the current density. 229. Mutual Energy of Magnets and Currents. If we have permanent magnets and currents situated in a homogeneous medium of unit inductivity, we may represent their mutual energy in two ways. We may in the first place consider the magnets to * This apparent current-sheet was overlooked by Maxwell, and it was not until the appearance of the Third Edition of his Treatise that the correction was made by J. J. Thomson. 228, 229] ELECTROMAGNETISM. 445 be traversed by apparent currents and current-sheets, as in the preceding section in the case of temporary magnets. We there introduced the discontinuity in the induction, but we might have introduced the intensity of magnetization. In the case of the permanent magnet this will be more convenient — in either case the form of the vector potential will be the same. We have for the potential due to a magnet in a homogeneous medium of unit inductivity, § 122 (3), <„ o where A', B', Cf are the values at a, 6, c, dr' = dadbdc, r* = (x - of + (y- by + (z- c)2. The field at any point x, y, z has the component <„ L. d® Now the derivatives of 1/r with respect to a, b, c are the negatives of its derivatives with respect to x, y, z, so that we may write L ^ -• ^ o Bo2 dxdy But since 1/r is harmonic we may put I =_rw I 9y2 and thus the integral becomes '-) *(- r) ,, \r vr da) dy / dz\ dz dx (4) '9® ^!§1, Lf/rfr.'© ,-a©L -sr- * — rT-^JJJ\c is— ^ irrT- 446 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. But since //, = 1 we have and accordingly the components of the vector potential may be taken as (5) Every element of volume produces at x, y, z a portion of vector potential equal to 1/r2 times the vector product of its magnetiza- tion by its vector distance from the point #, y, z. The mutual energy of currents and magnets is then obtained by the equation § 226 (5), omitting the factor ^. This method of treatment is that of Maxwell*. From the above form for the vector potentials we may easily express the solenoidal vector F, G, H as itself the curl of another vector potential. For again replacing derivatives of 1/r by a, 6, c by derivatives by x, y, z, (6) .J-— so that if we introduce the vector potential of magnetization, with components (7) * Treatise, Vol. n., Art. 405. 229] ELECTROMAGNETISM. 447 the vector potential belonging to the magnetic force is its curl. as_8Q ~ ' SQ SP •" ==^ -- ^; ^r~ • ex oy This leads us to a second manner of obtaining the mutual energy, due to Helmholtz. The expression for the energy of a permanent magnet in a magnetic field L, M, N is, § 126 (2), (9) W = - ^l + BM + CN) dr, where the volume of integration is that occupied by magnets, or it may be extended to infinity, since elsewhere 4 =£ = (7=0. We may transform the integral into one taken throughout the space occupied by currents. If we introduce the vector potential of intensity of magnetization, we have from (7), if the magnetiza- tion is everywhere finite, A = - - AP, 4-7T (10) -- AE. 4>7T Introducing these values of A, B, C into the integral (9), (ii) w = (L^P + M AQ + NbR) dr, and transforming each term by Green's theorem in its second form, the surface integrals vanishing at infinity, (12) W = 1 6) + (uCf - wA') which differs from the result of substituting (5) in § 226 (5) in the same way as (15), above. We have thus seen how we may replace every magnet by an apparent current * Helmholtz, Ges. Abh. Bd. i. p. 619. 229, 230] ELECTROMAGNETISM. 449 4?r 4?r \dy 0*J' ^-^AG=-^A'---i' which would produce the same magnetic effect. It was for this reason that Ampere was led to the hypothesis that all magnetism was due to currents of electricity circulating about the molecules of matter. The above formulae all refer to currents and magnets placed in a homogeneous medium, and as has been already seen, lose all their simplicity when the inductivity varies. For although we may still calculate the vector potentials due to the induced magnetization, the process will be complicated, and in general impracticable. For this reason, and because both scalar and vector potentials are quantities whose physical significance is much less apparent than that of the strength of the field, Heaviside and Hertz have been led to avoid the employment of potentials, and to deal directly with the electrical and magnetic fields. We have however introduced the vector-potentials here on account of their important mathematical relations, and the fact that they have been so much used by the highest authorities. 230. Magnetic Field due to Current- Sheet. We have found that in a current-sheet the amount of electricity that flows in unit time across a curve connecting any two points in the sheet is equal to the difference of the current-function SP at those two points. This quantity is the same whatever the curve connecting them, unless there is an electrode lying between. We shall sup- pose that a sheet has no electrodes, so that the current flows in closed circuits in the sheet. We may find the magnetic field of such a sheet, at points not lying in the sheet, by the consideration that the strip of the sheet bounded by the curves NP" = const, and ^ -I- dW = const., dW being a constant difference in the values of the current-function for the two curves, is equivalent to a linear current of strength dW. Such a current, by § 210, is equivalent to a magnetic shell of strength dW. The whole current sheet may therefore be replaced by an infinite series of magnetic shells, whose edges only are given, the form of the shells being in- w. E. 29 450 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. different, so long as the attracted point lies outside them. These shells may be considered to form a continuous body, which is, being divided into shells, lamellarly magnetized, the potential of magnetization being equal to the current-function ^ (§ 124). The magnetic potential Q is accordingly at outside points, by §124(n), «HH6 But since the form of the magnetic shells is indifferent, as long as their edges are of the given shape, we may consider them all deformed so as to coincide with the current-sheet, as is illustrated in Fig. 88. The shells overlap each other continuously, so that there are more shells laid on the sheet the greater the values of "VF. Fm. 88. As we cross the sheet, the poten- tial fl is discontinuous, as in the case of a single magnetic shell. As in that case also, the normal component of the magnetic force, being continuous for all the shells, is continuous on crossing the sheet. The tangential component in the direction of the lines of flow is also continuous, but, as we found at the end of § 228, the component perpendicular to them experiences a discontinuity equal to 4?r times the current-density, that is 4>7rdW fin*. This may also be very simply obtained by taking the line-integral of magnetic force around any circuit composed of two infinitely near portions lying on opposite sides of the current-sheet and coinciding with an electrical equipotential line, the integral being equal to 4?r times the difference in the values of the current-function at the two points where the circuit cuts the sheet. 231. Examples. Coefficients of Induction. Toroidal and straight coils. We shall now calculate the energy due to currents in a few simple cases. The coefficients of the half-squares and products of the current-strengths in the expression for the electrokinetic energy, are called, for reasons to be explained in the next chapter, coefficients of induction, or more briefly, inductances, distinguishing coefficients of half-squares by the name self-in- ductaiice, coefficients of products by the name mutual inductance. 230, 231] ELECTROMAGNETISM. 451 Any self or mutual inductance is the magnetic flux through a circuit due to unit current in its own or another circuit respec- tively. We shall first consider a solid of revolution bounded by a surface generated by revolving any closed plane curve about an axis in its plane not cutting it. Such a solid may be called a tore. If the tore be uniformly wound with wire carrying a current, so that every winding lies very nearly in a plane passing through the axis of revolution, Fig. 89, we may very approximately consider the layer of wire as a current sheet, the difference of value of the current function between any two points being ml, where 7 is the current in the wire, and m is the number of turns of wire between the points. By reason of symmetry the lines of magnetic force must be circles whose planes are perpen- dicular to the axis of revolution, and whose centers lie on the axis. Consequently the strength of the field H is a function only of the distance p from the axis, and the line integral of the field- strength around any line of force is equal to the constant value of H on that line times the circumference of the circle. If n be the total number of turns of wire on the tore, any circle lying in the substance of the tore is linked with the current n times in the same direction, so that the value of the above line-integral is (i) 4w7rJ = 2>rrpH. This gives as the value of the force for internal points A circle lying outside the tore, however, is not linked at all with the current, so that the line integral is zero, and therefore the force H must be zero. Such a closed coil or toroidal current-sheet accordingly emits no tubes of force, but all its tubes lie within the doubly-connected space of the tore. The force accordingly has a discontinuity at the sheet equal to 4?r times dW/dn, which is the amount of current crossing unit of length of a circle co- inciding with a line of force, or nl/Zwp. If the tore be filled with a homogeneous medium of magnetic inductivity //, the induction at any point will be pH = ^n^I/p. This whole reasoning 29—2 452 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. has supposed that the external medium is homogeneous, but since there is no field there, the value of the inductivity is immaterial. If z, p be rectangular coordinates parallel and perpendicular to the axis in the plane of any orthogonal cross-section of the tore, the whole flux of induction through the section is (3) pl and p2 being the least and greatest values of p on the contour of the section for a given value of z, and being given a^ functions of z by the equation of the contour. If another circuit be wound in any manner about the tore, embracing it n' times, the flux through it is n' times that just found, and the mutual electrokinetic energy of a current /2 in it and a current /x in the former winding is, according to the last sentence of § 226, (4) T The mutual inductance of the circuits is accordingly (5) J If the second coil coincide with the first the flux through itself is so that the self-inductance of the toroidal coil is (6) . L = 2K The electrokinetic energy is 'dpdz P For a coil of square cross-section whose side is 2a and whose mean radius is R, R 4- n. (8) L 231] ELECTROMAGNETISM. For a circular cross-section of radius a ra ft i ^/aa _ 2z _______ (9) Z = 2n> log - - ===== cfc = 47m> (E - V^2 - a2). J —Ct Ji — V ft - ^ If in equation (2) we insert the number of turns of wire per unit of length of the line of force, m, since n = Zirpm, (10) . H=4f7rml, or the force depends only on the amount of current per unit of length. In case the radius of the tore is increased indefinitely, so that we get an infinitely long straight coil, m is the number of turns per unit of length of the coil, and we have within a uniform field of the magnitude 47rm/. If any coil of n' turns be wound on outside, the mutual inductance will be It is noticeable in all these cases that it is of no importance whether the outer coil is in contact with the inner or not, for in any case it is threaded by the whole flux of force. If there were any field external to the tore, the case would be different. It is however necessary that the tore be entirely filled by the medium of induc- tivity p. The formulae of this section are applicable to induction coils and transformers, providing the coils are endless. The line- integral of magnetic force 4>7rnl is called the magnetomotive force, and the problem of finding the magnetic induction in the tore is the same as that of finding the current in a tore of conductivity IJL in which there is an impressed electromotive force of the amount 4-Tm/, the lines of flow being circles. In case the cross section of the tore is small compared to its radius, we may neglect the curvature of the coil, and find the reluctance (§ 184), by § 174, so that we have / \ T j 4.- TT-I Magnetomotive force ^irnl ( 1 1 ) Induction Flux = — & _. . - = — =— . Reluctance I ?8 This formula is used in practice in finding the flux in the field magnet of a dynamo-electric machine, although it is accurate only in the case that we have treated, where all the tubes of force are encircled by all the current turns, so that the numerator is the same for every tube. Any tube being partly in iron and partly in air, the reluctance of any infinitesimal tube is found by the formula for the resistance of conductors in series, as 454 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL In order to find the influence of the ends of a uniform straight coil of any cross-section, we may consider that each current turn is. replaced by a plane shell, so that the whole current sheet is replaced by a uniformly magnetized cylindrical magnet with intensity of magnetization d for a2 and changing the sign, we have The portions for the pairs of sides BG, B'C', and BC, D'A', are ob- tained from these by changing ^ into 12. We have then considered j ust half of the two circuits, so that, adding these four parts and multiplying by two, we obtain the value of the inductance M = 8 a - The attraction of the two circuits for each other when traversed by unit current is obtained by differentiating this expression by a. 233. Pair of Parallel Circles. If the circuits are circles of radii Rlt R2) their planes being perpendicular to the line joining their centers, of length a, we may put IR L_ I so1 = Rlcosfa, #2 = ^2 cos <£2, a VL\ I T/J = R! sin fa , 2/2 = ^2 sin fa> z\ =0, 22 = a, cos fa - R2 cos 2)2 + (A sin 0X - R.2 sin cos (fa - fa), M =/T Jo Jo cos (fa - fa) R^dfadfa V a2 + ^2 + ^22 - ^^i^2 cos (0! - fa) ' The integration with respect to fa amounts merely to multipli cation by 2-7T. If we put fa-fa = 2^ — 7T, d (fa — fa) = 2ctyr, COS (fa -fa) = - COS 2l|r, the integral becomes 232—234] ELECTROMAGNETISM. 457 and writing KZ we have finally & -E(K) K where E and F are the elliptic integrals T2 = 1 V 1 — tf2 sin2 -v/rcfyr, -^W = Jo Jo Vl — /e2 sin2 i/r These definite integrals are functions only of the parameter K, and their values have been tabulated by Legendre for various values of K. If we put <# + (&• r2 and r2 are the maximum and minimum distances of points on the circumferences of the two circles from each other. The expression M/^TT *jRJRq being a function only of K and therefore of 7, has been tabulated by Maxwell as a function of 7. (Treatise, Vol. 2, Art. 701.) We may also find the value of M in a series of zonal spherical harmonics by means of the series of § 215 by differentiation with respect to r and integration over a spherical segment bounded by the second circle. For a full treatment of the properties of circular coils the reader is referred to Maxwell's Treatise, to Mascart and Joubert, Lessons on Electricity and Magnetism, and to Gray, Absolute Measurements in Electricity and Magnetism, where a great variety of formulae will be found. 234. Non-linear Currents in Parallel Cylinders. If the expressions in the two preceding sections be used to find the self-inductance of a linear circuit we find a difficulty, for on putting a = 0 in § 232, the expression becomes logarithmically infinite, while on putting a — 0, R^ = J?2 in § 233, K becomes unity, the elliptic integrals reduce to trigonometric, and F (K) becomes logarithmically infinite (log tan w/2). This is easily seen to be the case for any linear circuit, for if dsi and dsz traverse the same circuit there is an infinite element in the integrand, and, 458 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. considering the element in which it occurs as straight the integral fds J • becomes logarithmically infinite. We may see the reason for the self-inductance becoming infinite in another way by considering Biot and Savart's Law, for as we approach a linear conductor the force is inversely proportional to the distance from the conductor. The flux therefore increases like the logarithm of this distance, and is not finite when we approach the linear conductor indefinitely. We may avoid this difficulty by con- sidering conductors of finite cross-section, for in that case the corresponding element of the integral, in which the integrand becomes infinite, I — is not infinite, as was proved for an ordinary potential, § 76. We shall now consider currents flowing in three-dimensional conductors in the form of cylinders of infinite length whose generators are all parallel. We might treat the problem by the application of the law of Biot and Savart to each infinitesimal tube of flow, but we shall prefer to make use of the general equations § 222 (2), and § 228 (2). It is evident that the lines of force are in planes perpendicular to the conducting cylinders, which we shall take for the XF-plane, so that N—0 and the field is independent of the coordinate z. The problem is accordingly a two-dimensional problem, and all the quantities concerned are independent of z. Since u = v = 0 we have F= G = 0 so that our equations are dM dL -= -- ~— , das dy from which results, if ^ is constant, &H (4) -47^ = ^ But this is Poisson's equation for the logarithmic potential, §91 (10), 234] . ELECTROMAGNETISM. 459 so that its integral is (5) H=C-2 n^wlogrdadb, where r2 = (x - a)2 + (y - &)2, and C is a constant, which, though infinite, does not affect the value of the force. If the conductors are concentric circular cylindrical tubes and the current-density is uniform, we may find the magnetic force without finding the vector-potential, in the same way as in § 231, for it is evident that the lines of force are all circles in planes perpendicular to the conductors. At points outside the outer tube, at a distance p from the axis, the line integral of magnetic force (which we will denote by P instead of H, to prevent con- fusion with the vector-potential) around a circle is 4-7T/, *-" where (7) is the total current through all the conductors. Accordingly at external points the field is the same as if the current were con- centrated in the axis of the conductor. If different tubes are made part of the same circuit, so that all the current flowing in one direction is returned in the other direction by concentric conductors, the total current is equal to zero, and the force is zero at all external points. Such a double tubular conductor accordingly, like a toroidal coil, emits no tubes of magnetic induction. For this reason, when it is wished to protect delicate magnetic instru- ments from the action of strong currents, the circuit should be formed of concentric conductors. The mutual inductance of any external circuit with such a concentric conductor is accordingly zero, so that, as we shall see in the next chapter, no currents would be induced in the concentric conductors by external cur- rents. Such a conductor would thus be suitable for telephone circuits. In the space outside the conductors the magnetic potential is (8) n 460 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL which we know to be a harmonic function, as is conjugate to the function log p, both being derived from the function log (x + iy). In the substance of the conductors, there is no magnetic potential. We may find the force by evaluating the expression for the vector- potential, or as above, except that now the line of force does not surround the whole current, but only a portion of it. If the conductor is a solid cylinder of radius R, %7TpP = 4-7T . TTp^W = 4-7T/ -^ , (9) P-%p. . The integral (5) represents H only when //, has the same constant value everywhere, for if it has discontinuities we must add a part corresponding to the apparent current as shown in § 228. In the case just treated, however, the apparent current vanishes, for the induction is tangent to the surfaces of the conductors. In the general case, if p is constant in the space outside of the conductors there is a magnetic potential, and the equations (2) and (3) become dy 'dx showing tfiat the function //.fl is conjugate to the vector-potential H, which is accordingly the flux-function for the magnetic induc- tion. The method of functions of a complex variable is accordingly applicable to problems connected with the field of cylindrical conductors. For instance, Fig. 65 represents for external points the lines of force and equipotential lines of the field due to two circular cylinders carrying equal currents in opposite directions. No one of the circles in the figure however represents either of the conductors, whose centers are at the points + a. The surface of a cylindrical conductor is tangent to lines of force only when it is alone in the field, or accompanied by concentric conductors. Within conductors, although there is no magnetic potential, equations (2) and (3) show that R is still the flux-function for the induction. If 8 is the area of the cross-section of any conductor, the vector-potential at any point, whether external or internal, is by 234] ELECTROMAGNETISM. 461 (5) equal to C - 2/*wS log r = C - 2fjLl log r, where r is defined by the equation (11) S logr = 1 1 logrdadb. But from the interpretation of a definite integral as a mean, § 23, we see that logr is the arithmetical mean of the logarithms of the distances of all the points of the cross-section from the fixed point x, y. Now defining the geometric mean of n quantities as the nth root of their product, we see that r is the geometric mean of the distances of the points of the area from the point #, y, for its logarithm is the arithmetical mean of their logarithms. If TI and r.2 be the geometric mean distances of a point from two areas Si and SZ) r3 the geometric mean distance of the point from both areas taken together, we have by the definition, (n), ( 1 2) (S1 + &) log r, = S! log n + S9 log r2. By means of this principle we may find the geometric mean distance from a complex figure if we know it for the various parts of the figure. This method is due to Maxwell*. We shall first find the geometrical mean distance from a circular ring of infinitesimal width. Let p be the radius, e the width of the ring, and h the distance of the given point from its center. Inserting polar coordinates in the equation ( 1 1 ), ffcr (13) 27r/>e log r = \ log (h2 + p2 — 2hp cos <£) ped. Jo This integral assumes different forms according as h is greater or less than p. Taking out from the parenthesis the square of the greater of these, and integrating, we get (14) \ogf = logh+-^j, h>p, or (15) lo£ * = loS ? + 4^ ^ Q ' P>h> where J is the definite integral T27T J (a) = I log (1 + a2 - 2a cos <£) d, * Trans. Roy. Soc. Edinburgh, 1871—2. 462 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. which is a function only of the parameter a, and we are to put a = p/h when h > p and a = h/p when h < p. We can easily show that J(a) = 0 if a < 1. For / (a) = Jj (a) + J"2 (a), where r» Jj(a)= log(l -f a2 - 2a cos Jo /2 (a) = f ^log J 7T (1 + a2 - 2a cos Substituting <£ = TT + (/>' gives TTT J"2 (a) = log (1 -f a2 + 2a cos (f)') dfi, Jo and since the variable of integration is indifferent, we may drop the accent. The integral being now between the same limits as in Jj we may add the integrands, giving J(a) = ["log {1 + a4 - 2a2 (2 cos2 - 1)) d. Jo Now substituting 2<£ = $ we obtain J(a) = I r\og (1 + a4 - 2a2 cos ') d$ = * Jo Repeating the process we get and letting ?i increase indefinitely we obtain, if a < 1, /(«) = 0 if /(O) is finite. But «/(0) = 0. We accordingly obtain from (14) and (15) the result that the geometric mean distance from a circular line is, for an outside point, its distance from the center, and for an inside point, the radius of the circle. By means of this result we may find the mean distance from the area of a ring of finite width, of internal radius R^ and external Rz. For a point outside the ring the mean distance is its distance from the center. For a point in the space within the ring, by ( 1 1 ) or (12), /&2 logp.pdp -Ri = i TT [R? (log Rf - 1) - R? (log Rf - 1)}, log f = Ti f^2 i<>g R* - w log A - * (R? - m- 234] ELECTROMAGNETISM. 463 For a point in the area of the ring itself, we must divide the ring into two, one within and one without the given point, so that TT (£22 - RS) log r = TT (hz - R?) log h + TT {R? log R2 - A2 log h - \ (R? - h% (17) logr = i{ The vector-potential is always, for uniform flow, and since this is the flux-function for the induction, by reason of the equation -¥^- — ^ -^, we obtain the induction perpendicular to the radius, by differen- tiating according to — h, so that which agrees with the result (9), in which JRj is equal to zero, The electrokinetic energy of the system of currents is, by § 226 (5), and inserting the value of H from (5), (19) T=^jjjCwdadbdz-jjjjlnww' logrdadbda'db'dz, If we integrate with respect to z from — oo to oo , we obtain an infinite result for the energy, but for a finite length I the energy is proportional to I, so that the energy per unit of length of the conductors T/l is given by the above expressions omitting the integration with respect to z. Each point of integration = Aext, IJb = BeV, where At B and X are constants to be determined. Inserting these values in the differential equations (5), the factor eP appear- ing in every term may be omitted, giving us the simultaneous equations (Ll\ + Rl)A+M\B = 0, M\A + L\ + RB = 0. 474 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. These equations can be satisfied for values of A and B differing from zero only if the determinant of the coefficients, M\ , £2X + R vanishes. But this being expanded gives us the equation (7) (L,L2 - M*) X2 + (R,L, + R.L,) X + R,R2 = 0, a quadratic to determine X. If we call its roots \ and X2, we have , ,, 2 ,2 ,, - AT2) x = Both roots are real, for we can write the quantity under the radical sign both terms of which are positive. Both roots are also negative, for since the electrokinetic energy is intrinsically positive, we must have L,L, - M 2 > 0. Having found the value of X either of the equations (6) will give us the ratio of the constants A, B. If we choose the value Xx the first equation gives (a\ ?!. - -^Ai + -Ri Ai M\, If we choose the value X2 we obtain a different ratio (10) B?___Ll\2 + Rl A,' M\2 The theory of linear differential equations shows that the sum of particular solutions is a solution, and that the general solution is given by where the constants Alt Blt Aa, Bz are connected by the equations (9) and (10). We may now determine the absolute values of these 237] INDUCTION OF CURRENTS. 475 constants by means of the initial values of the currents Jx and 72. These being 1^ and /2(0) we have for the induced currents when t = 0, !1<«>=I1®-I1v=A1 + A.i, //o)=/2(o)_/2(i)=J5i+jg2. These equations with (9) and (10) determine the four constants, so that the solution is complete. The most important case is that in which there is no electromotive force in one circuit, while the other circuit originally open, and containing an electromotive force E, is suddenly closed. The latter circuit is called the primary, and will be taken as that denoted by the suffix 1, the former the secondary, with the suffix 2. We accordingly have /,«» = J2(o) = /2d) = 0, /! w = E/R, and E \ If RA-&L, \ R! ( 2 W( J?2A - R.L.Y + 4RAM* If _ R&-RA _ \ 2W(£2A-^i4)2 + WUf2 / 7" — — £/M ^f _ . ^ Since \ and X2 are negative, the induced currents die away as the time goes on. The function vanishes when has a maximum or minimum when t = — loer and the curve representing it has a point of inflexion for These three points are equidistant, and, since Xj and X2 have the same sign, are real if Ci and (72 have opposite signs. This is the case for the secondary current J2, but the primary current has both coefficients negative, and consequently has no maximum nor 476 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. inflexion, but rises continuously, the appearance of its represen- tative curve to the eye being the same as in the case of a single circuit. The growth of the currents is represented in Fig. 92. FIG. 92. The total quantity flowing in the secondary is (i3) I""/,*- Jo EM EM This result may also be obtained by direct integration of the second of equations (4), with E2 = 0, M(I^-I^ (14) To find the effect of breaking the primary current, we have /!(0) = E/Rj., /x(1) = 0, so that the whole quantity passing in the secondary on breaking is the same as on making. This is one of Faraday's fundamental results. The manner of variation of the secondary is, on the contrary, very different from that on making. After the break we have to consider the primary circuit as sup- pressed, so that the secondary is to be considered by itself, and varies according to equation (3) above, where Ilf the final value, is zero, and 70, the initial value, is to be found from the above value of the time-integral of the secondary, . _R2t 6 2 dt - J 0 EM 237, 238] INDUCTION OF CURRENTS. 477 The fact that the secondary jumps abruptly from zero to its maximum value 70 at starting may be reached from considering the preceding case with ^=00. The time between the secon- dary's taking the value zero and attaining its maximum and the time from then to the inflexion is (log Xa/^O/^i - ^-2), which is less the greater R1} vanishing for ^=00. The effects here described may be illustrated by means of any of the mechanical models described in § 71. For instance suppose that the mass raj, Fig. 30, is revolving with a uniform angular velocity, the centrifugal force, which represents the electromagnetic force, being just balanced by an applied force so that the distance of raa from the axis remains constant. If ra2 is at rest and we suddenly apply a force to the upper bar so as to increase its angular velocity, the lower bar will begin to turn in the reverse direction, the velocity representing the secondary induced current. If on the other hand the upper bar is suddenly retarded, the lower begins to move forward in the direct sense. Similar effects may be produced by suddenly changing the distance of either m1 or ma from the axis, corresponding to a relative motion of the two circuits, producing a change in the mutual inductance. We have not in this section explicitly considered this case, but since if the change is made suddenly, and the circuits then remain at rest, the differential equations are the same as those we have used, and the solution is obtained from those here given. 238. Periodically-varying Electromotive-force. (1) SINGLE CIRCUIT. Suppose that in the circuit is included a variable electromotive-force varying proportionately to the cosine of a linear function of the time, as would be the case if a coil of wire should rotate in a uniform magnetic field about an axis in the plane of the coil, and perpendicular to the direction of the field. Then the equation for the current is (i) L -j- + RI = EQ cos cot. A convenient way of treating such an equation is by replacing the trigonometric term cos wt by the exponential ei. The time T is called the period, and its reciprocal, the number of periods in unit time, n = co/Sir, is called the frequency. In like manner the quantity Aei = 0), it becomes the resistance. It has been proposed by Hospitalier* to call the coefficient of i in the ratio E0/A, the reactance. The mean value of a quantity varying harmonically taken over any exact number of periods is zero, while in virtue of the formulae I rT 1 rT 1 1 fr . ~ I cos2 wtdt = m I sin2 cotdt = ~ , ^ sin cot cos cotdt — 0, -L Jo J- Jo 2 JL JQ * Hospitalier, L'Industrie Electrique, May 10, 1893. See also, Steinmetz and Bedell, Trans. Am. Inst. EL Eng. 1894, p. 640. 480 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. the mean value of the square of such a quantity is one-half the square of its amplitude, and the mean of the product of two such quantities of the same period and a difference of phase equal to a right angle is zero. The quadratic mean, or square root of the mean square of a variable current or electromotive-force is called the effective or virtual current or electromotive-force. Its value in the case of a harmonically -vary ing quantity is accordingly the amplitude divided by V2- The activity, or power absorbed by the circuit, is E 2 El— -~ cos cot cos (cot — a), J and its mean value, by the above formulae, To the solution (5) is to be added, in order to obtain the general solution of ( I ), the solution of the equation with the right hand member equal to zero, obtained in the preceding section, but as the current thereby represented rapidly dies away, the resulting state of the alternating current is that which we have found. A number of circuits in parallel, to which a single harmonic electromotive-force is applied, receive virtual currents inversely proportional to their respective impedances — if the frequency is great enough the distribution is almost independent of the resistances of the branches, the impedance being sensibly equal to the reactance. (2) Two CIRCUITS. Suppose fchat we have two circuits, one of which, the primary, contains a harmonic electromotive-force, while the secondary contains no impressed electromotive-force, except that due to induction. The equations then are 238] INDUCTION OF CURRENTS. 481 or making use of complex variables as before, (g) Af + Jf' A particular solution is given as before by putting I giving (Ljco + RJA ' + (L&» + ft) 5 = 0. Eliminating 5 from these equations we get do) Comparing this with equation (3) above we find that the current in the primary is the same as if the secondary were absent, and the resistance and self- inductance of the primary were Rf and L', where (II) These results were first given by Maxwell in 1864 in his celebrated paper "A Dynamical Theory of the Electromagnetic Field*." They constitute the basis of the theory of the alternating current transformer. We see from equations (n) that the effect of the presence of the secondary circuit is to cause an apparent increase of resistance and decrease of self-inductance in the primary. Both of these effects cause a decrease in the angle of lag of the primary current behind the electromotive-force, and accordingly, by (6), an increase of power. Inserting the values ( 1 1 ) in (6) we obtain for the power * Phil Trans. Vol. CLV. W. E. 31 482 'THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. As we increase the ratio L2co/R2 the values of the apparent resist- ance and self-induction approach the limiting values ' M 2 These values are nearly approached in actual transformers, par- ticularly when fully loaded with a number of lamps in parallel in the secondary. Now although we have in general L^2 > M*, still when the primary and secondary are toroidal coils wound on the same core (§ 231), so that very nearly the whole induction-flux due to either is linked with the other, we have very nearly The transformer is then said to have no magnetic leakage. In this case the apparent inductance Ln' is reduced to zero, the current does not lag, and takes on the largest value that it can have, namely The expression (12) for the power becomes, if we neglect the square of R2/L22 (A - M*/ltf\ ' as we see on dividing numerator and denominator by Z22a>2 and then adding and subtracting the term R/M^/L^ in the deno- minator. If there is no magnetic leakage, this increases as R2 decreases, until it reaches the limiting value \EI 2 JV while if there is magnetic leakage, the power absorbed is a maximum when _ „ M2 *+&z? becoming equal to and thence decreasing as R2 decreases. 238] INDUCTION OF CURRENTS. 483 The power when R2 is zero is only which, for high frequencies, may be much less than the maximum*, being, when co is great enough, sensibly equal to the maximum value multiplied by 2^ The second of equations (9) gives B - Mia The modulus of the quotient, being the quotient of the moduli, \B Mto (16) shows that the amplitude J2(0) of the secondary current is equal to the amplitude of the primary I-f* multiplied by Mco divided by the impedance of the secondary. Inserting the values of R', L ', from (17) gives for /2(o), (18) \B\=If\ In the case of no magnetic leakage this becomes and if we may neglect R^L^ or R2/L2a) in comparison with unity we have the simple form (20) This is the practical equation of the transformer. By § 231, (5) and (6), we have * J. J. Thomson, Elements of the Mathematical Theory of Electricity and Magnetism, p. 409. 31—2 484 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. where n± and n2 are the numbers of turns in the primary and secondary coils. If w2/Wi is so small that its square may be neglected, we have E ^ (at) lfc The ratio w2/7ii is called the ratio of transformation. The argument of the ratio B/A, being the difference of the arguments of the numerator and denominator, shows that the secondary lags behind the primary current by the phase-angle which approaches two right angles as the ratio L^co/R^ increases. The efficiency of the transformation, or the ratio of the activity in the secondary ^RzI2(0)a, to that in the primary, is, by (18) and (12), which, neglecting R^/M^co2, becomes (22) that is, in practical cases, nearly unity. 239. Circuit containing a Condenser. In the cases heretofore considered the only energy of the system has been electrokinetic. If the circuits are connected with conductors upon which charges of electricity can accumulate, we shall in addition have electrostatic, or potential energy. As the simplest case let us consider a single circuit whose ends are connected to a condenser of capacity K. If the charge of one plate of the condenser at any instant is q, then the current flowing into that plate is defined as The electrostatic energy of the system is (§ 143) (2) F 238, 239] INDUCTION OF CURRENTS. 485 which gives rise to the difference of potential, or electrostatic electromotive-force impressed in the circuit in the direction of the current, *--£--*• Accordingly the differential equation for the current is from which, substituting from (i), we obtain the equation for the charge, Again, assuming q = ext we obtain the quadratic for X (6) whose roots are R ~ Z2 KL' We have now to consider two cases. CASE I. R* > 4tL/K. Both roots real We then have (8) q = Ae^J 4- Be*zt, and as \ and X2 are both negative, the charge, and likewise the current (9) / = \iAe*it + Afclfe***, die gradually away. If there is a permanent impressed electro- motive-force E0 in the circuit, we must add the quantity E0K to the charge, which, however, does not affect the current. Determining the constants A and B by the conditions that there is initially neither current nor impressed electromotive- force, and that the initial charge is qQ, we have Oo) 1 = ^ while if there is no initial charge, but an impressed electromotive- force E0j we obtain • 486 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. In either case, the curve representing the charge as a function of the time has a point of inflexion distant from the origin by the amount while the curve of current has one at an equal distance farther on. The curves of charge and of current are represented in Fig. 93. Fm. 93. CASE II. R* < 4L/K. Both roots complex. If we write 4Z2' = _RL v = /JL we have for the roots (13) 1~^ W> 7^ = yit - lVt and we may write the solution (14) q = &* (A cos vt + B sin vt). In this case the charge not only dies away, but periodically changes sign, performing a damped harmonic oscillation of the period v) cos vt + (Bfi - Av) sin vt}. We have for the current 06) ,.|-,- Determining the constants so that the initial current is zero, and the charge qQ, we have (17) q = q0eflt (cos vt — — sin vt] . 239] INDUCTION OF CURRENTS. 487 This case is represented in Fig. 94. The charge is zero at times such that FIG. 94. which is later than the time of the vanishing of the current by the phase difference 0, which approaches Tr/2 the smaller /z. We may specify the damping, or decrease of the charge or current, by the relaxation-time of the damping factor &*, namely r = 2L/R, or by the logarithmic decrement, that is the logarithm of the ratio of a maximum value to the absolute value of the next following minimum. Since the maximum and minimum values of the parenthesis in (17) are equal and opposite, and separated by intervals of time T/2 = TT/V, the ratio of the absolute values of q is e " , and the logarithmic decrement X, nrrr nr (18) X = \/zs KR* If R = 0, there is no damping, X = 0 and the period is Introducing these values of TQ and X we may write (15), (20) r=: so that if the damping is small it affects the period only by small quantities of the second order. 488 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. We have in this case a type of the very important class of phenomena known as electrical oscillations, of which we shall presently give the general theory. The theory here given was published by Lord Kelvin* in 1855 and by Kirchhofff in 1864. The theory was confirmed experimentally in a qualitative manner by FeddersenJ in 1857, by observations on the electric spark arising when a Ley den jar is discharged, and by Helm hoi tz§ in 1869, and Schiller || in 1874, under conditions admitting of quan- titative results. More exact determinations in absolute measure have been made by Lodge and Glazebrook by a method involving the spark, and by the author, by a method similar to that of Helmholtz. We have seen that the occurrence of oscillations is due to the presence of both kinetic and potential energy. If there is no kinetic energy, L — 0, and we reach the case treated in § 207, while if there is no potential energy, we have the case of § 237, to which we may pass by putting K = oo . A mechanical model of an oscillation may be obtained from any mechanical system possessing both potential and kinetic energy, such as a pendulum or a heavy body moved by a spriog. The stronger the spring the quicker is the oscillation, so that we may assimilate the reciprocal of the capacity of the condenser to the elasticity of the spring. The self-inductance of the system, on the other hand, is the analogue of the mass, or inertia of the mechanical system. The analogy of the resistance may be obtained by making the system move in a viscous medium, so that the motion is retarded by a force proportional to the velocity. 240. Periodic Electromotive force. Resonance. If into a circuit joined to the plates of a condenser is introduced a harmonically-varying electromotive force, we have for the cur- rent, instead of (4) of the preceding section the equation cos cot. (i) L -J- + RI + -JJ: lldt = * Thomson, " On Transient Electric Currents," Phil Mag. June 1853 ; Math. and Physical Papers, Vol. i. p. 540. t Kirchhoff, " Zur Theorie der Entladung einer Leydener Flasche," Pogg. Ann. Bd. 121, 1864 ; Ges. Abh. p. 168. I Feddersen, "Beitrage zur Kentniss des elektrischen Funkens," Dissertation, Kiel, 1857; Pogg. Ann. 103, p. 69. § Helmholtz, "Ueber elektrische Oscillationen," Wissemch. Abh. Bd. i. p. 531. || Schiller, Pogg. Ann. 152, p. 535. 239, 240] INDUCTION OF CURRENTS. 489 Proceeding as in § 238, we write (2) L + RI + and assume for the particular solution I — Aei(at, which inserted in (2) gives (3) From this we get, by comparison with § 238, for the impedance, and for the lag of the current behind the electromotive force, (5) so that the solution of (i) is j. E0 cos (a>t - a) In order to obtain the general solution we must add to this result the solution of the equation with E0 = 0 from the previous section. An oscillation whose period is that of the force, as in our present case, is called a forced oscillation or vibration, in contradistinction to the case of the previous section, where, no force being applied, the period is governed by the constants of the system, and the oscillation is called a free oscillation. If there is damping, the free oscillation soon dies away, leaving only the forced oscilla- tion. We see by (6) that if there is no condenser, K — oo , we obtain the case of § 238, and the current lags, while if on the other hand L = 0, the lag is negative, or the current advances by the phase-angle 1 a = tan"1 Ka>R' The reason of this is of course that in the differential equation the inductance is multiplied by the derivative, and the capacity- reciprocal by the integral of the current, which, when the electromotive force is an exponential with imaginary exponent, introduce the factor iw into the numerator or denominator respectively, producing opposite effects on the argument of A. 490 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. Thus the tendency of the inductance and capacity is to neutralise each other's effects, Exact neutralization is produced, so that there is neither lag nor advance, when 1 1 27T In this case the impedance is the smallest possible, and the magnitude of the current is a maximum, being the same as would be given by Ohm's Law for steady currents with a closed circuit. The period of the electromotive force which gives this result is exactly that of the free vibration which would be natural to the system if there were no damping. Under these circumstances the system is said to be in resonance with the force. The magnitude of the current is inversely proportional to the resist- ance, and if there were no damping would be infinite. For this reason resonant oscillations, either mechanical or electrical, may be very intense. By connecting two similar circuits with two similar Ley den jars, Lodge has caused the oscillatory discharge of one jar to produce such violent resonant oscillations in the other circuit that a considerable spark-discharge is produced. The phenomena of resonance have been demonstrated in a number of interesting papers by Pupin*. In order to show how the resonance depends on the agreement of the frequency of the impressed force with that of the free vibration, we give in Fig. 95 a graphical representation of the current as a function of the frequency. If we call a)m the value of ft) which gives the maximum current, fc: «-'=31 the amplitude of / is / A/ ft) ft)TOV KRZ \com w / In Fig. 95 are plotted the values of the factor of E0/R as ordinates, the abscissas being those of o)/a>m. The different curves are, beginning at the outermost, for integral values of the ratio . /•=• / R from 1 to 10. The resonance is sharper the larger this ratio. * Pupin, "Electrical Oscillations of Low Frequency and their Kesonance." Am. Journ. Science, April, May, 1893. 240, 241] INDUCTION ,OF CURRENTS. 491 241. General Theory of Electrical Oscillations. We shall now consider the question of electrical oscillations in the ,-, FIG. 95. most general case of a network of linear conductors, conducted with any number of conductors K which may carry electrostatic charges. These may be grouped in pairs to form condensers, as in the last section, or they may be entirely independent of one another. Of the linear conductors, any one may form a closed circuit unconnected with the others, and affected only by current induction, or may end at points of embranchment with other conductors, or upon any of the conductors K. For brevity we shall call the linear conductors wires, and the conductors K accumulators. We shall suppose that the net contains p points of embranchment, k of which are connected with accumulators, for all wires which end on the same accumulator are to be considered as meeting in an embranchment. Let the number of wires be I. Then if all the wires form a part of the same net, the number of independent meshes is I — p + 1, for we see at once that the smallest number of lines that can join p points to form a closed net is p, giving one mesh, and that after the first mesh every additional line adds a mesh*. For every wire r between points a and 6 we have an equation where Eab is the impressed electromotive-force from a to b and Va and Vb are the potentials of the points a and 6. There are I equations of this sort. * By independent meshes we mean such that circulation about any one is not the resultant of circulation about any number of others. For instance the outer boundary of a plane net is not independent of its meshes. 492 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. For every point of embranchment a we have an equation (2) / ...... a ~, at the currents being now marked with double suffixes to denote the points between which they run, as in § 171, and ea denoting the charge of the accumulator connected with the point, or zero if there is no accumulator. These p equations are not all indepen- dent, for adding them all together, every current appears in both directions, so that the left-hand side in the sum is identically zero, giving which is merely the statement that the total charge of the system is unaffected by the flow of currents. There are accordingly p — I independent equations (2). For every accumulator Ka we have an equation, § 138 (10), (4) V* =Pia,ei + pMei + ...... + pkaek = -5— . v@a From the equations (i) the F's may be eliminated by Kirch- hoff's principle, § 179. If, traversing any closed circuit, we add the equations (1) for each wire, every V appears with both signs, so that on the right we obtain the sum of the E's around the circuit. We shall thus obtain as many equations as there are independent meshes in the net, I— p + 1. Other equations may be obtained in the same manner by traversing any unclosed circuit ending on two accumulators. All the potentials at embranch- ments passed over are eliminated except those of the two ends. The number of equations to be obtained in this manner is one less than the number of accumulators, or k — 1. We thus obtain in all I — p + k = n equations, and there are the same number of inde- pendent variables. We may take as parameters to characterize the system a set of currents, one circulating in each mesh, so that the actual current in any wire is the sum or difference of the currents in the two meshes to which that wire is common. The time-integral of any mesh-current shall be taken for one of the parameters q. Besides the l—p + \ q's thus defined, we will choose k — 1 others, denoting the integral currents along any series of wires joining the accumulators two and two, the whole series forming a chain with two ends. The charge of any accumulator is 241] INDUCTION OF CURRENTS. 493 thus the difference of the two •] is the electromotive-force of induction around dt \dqsj the circuit s, for and every dlr/dq8' is zero except in the case of the currents which bound the circuit, for any of which dlr/dqs' is either plus or minus unity. The dissipation function, § 64, (7) becomes also a homogenous quadratic function of the q"s in which the product terms will in general appear. The dissipative force will also be represented by — ^— ? , for which is again the sum of the products El around the circuit. The terms dt\ *M_ are accordingly what we get by adding the equations (i) for all the wires bounding the mesh s. Since any charge is equal to plus or minus one of the q's of the second sort, or to the difference between two, W, the electro- static energy, becomes a homogeneous quadratic function of these q's. Again — = — is the electrostatic electromotive-force belong- ing to qs, for dq8 ~ *r der dq8 ' ) while ~ is zer oqs lators at the beginning and end of qs, where the derivative has the Now by (4), —— — Vr) while ~ is zero except for the accumu- ocr oqs 494 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. values minus one and plus one respectively. We shall write our three functions (5) where the M'B are linear combinations of the inductances of the wires, the It's linear combinations of their resistances, and the ps linear combinations of the coefficients of electrostatic potential of the accumulators. The values of the coefficients of the three functions are such that each of the functions is positive for all possible choices of its variables. We may now apply Lagrange's equations for any parameter qs. d fdT\ dF dW (6) -j-A*-. ' ) + 5-' + -^r- = ]®s> dt \dqs J dqs dqs where Es is the total external electromotive-force around the circuit. Performing the differentiations this becomes a linear differential equation of the second order with constant coefficients. We have one such equation for each parameter qs. We shall first find the free oscillations, that is the solutions with every Es = 0. As in the case of the simple examples of § 237, a particular solution may be obtained by assuming for every qs, (8) «. where X is the same for all the qs. Inserting these values in (7) we obtain +pu) ox + ...... + (Mln\* + Rm^+Pm) an = 0, + ...... + (M2n\* + Rm\ + p™) an = 0, (9) ........................................................................ (Mm\* + Rnl\ +pnl) a,+ ...... 4- (Mnn\* + Rnn\ +pnn) an = 0, a set of linear equations to determine the ratios of the a's when X is known. If these are to be satisfied by other than zero values of 241] INDUCTION OF CUKRENTS. 495 the as, however, the determinant of the coefficients must vanish, namely (10) lfnX2 0. nl\* + Rnl\ + pn, ...... Mnn\2 + Rnn\ + pn This is an equation of order 2n in X, from which the odd powers are absent if F= 0. We shall denote its roots by Xi, X-2, ...... ^m- If we multiply the rth equation (9) by ar, and take the sum for all r's, we obtain ( 1 1 ) \^r^sMrsaras + \^r^sRrsaras + %r%sprsaras = 0. The double sum by which X2 is multiplied is the value of the function 2T when for every qs' is substituted as. We shall denote this by 2T (a). Similarly the coefficient of X is 2F (a) and the term independent of X is 2TT(a). But by the fundamental property of the three functions, each must be positive. The equation (i i), KT(a) + \F(a) +W(a) = 0, shows us at once that X can not be real and positive, for that would involve the sum of three positive terms being equal to zero. Secondly, if F = 0, that is, if the resistance of every wire is zero, and X is a pure imaginary. In this case &* and e~M are trigono- metric functions, representing an undamped oscillation of the same period for all the parameters q. Thirdly, if F is large enough, X can be real and negative. In this case each parameter q gradually dies away to zero, the relax- ation time being the same for all. This corresponds to Case I of §239. Fourthly, if either W or T is zero, instead of a pair of roots we have a single one, which is real and negative, the cases correspond- ing respectively to § 237 or § 207. Fifthly, in other cases, that is when neither T, F, nor W vanish, and F is not too large, X is complex. We shall prove that then the real part of X is negative. 498 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. damping of any particular simple oscillation are the same for all the coordinates, and the n factors of the amplitudes and the n phases are to be determined from the initial values of the q's and of their first time derivatives. We will now consider the case of forced vibrations. On account of the linearity of the equations, if we find a solu- tion qr(l) for a particular set of values Eg(l] of the right-hand members of our equations (7), and a second solution q8(^ for a second set E^\ then the sum qr(l) + qr(2} will be the solution when the right-hand members are Es(l) + E®. We shall, therefore, consider the effect of each impressed force by itself. Suppose first then that in each circuit there is impressed a harmonic electromotive force, Es cos cot, all of the same period. Then we have the equations of which the 5th is (20) Assuming qr = agei(at these reduce to (- Mua>* + Rnico + pu) Oj + ... + (- Mlnco2 + Rlnico + pln) an = E n a set of linear equations to determine the a's. If we call the determinant of equation (10), D (X), and Drs (X) the minor of the element of the rth column and 5th row, we have as the solution of (21), <-> --^^- Since D (X) = 0 is the determinantal equation for the free vibration, whose roots are Xj, X2 ... \Zn, we have (23) D (X) = C (\ - \) (\ - X2) ... (X - X2w) = CUg (X - Xg). Accordingly the denominator D (ico) is (24) D (ia>) = CUS (ico - Xs) = CUS {- The minors Drs (iw) are rational integral functions of ia), and the numerators are therefore complex quantities, which reduce to real 241, 242] INDUCTION OF CURRENTS. 499 ones if the R's are zero. Calling the modulus of a numerator Br) and its argument 0r, (25) 2sDrs(ia>)Es = £reier, 6r is a small angle if the resistances are small. We thus have (26) «" = oir]: where (27) A.= {/vM Retaining now only the real parts, we have for the solution (28) qr> Thus if the resistances are small, all the oscillations are in nearly the same phase. If the frequency of the impressed force coincides with that of any one of the free oscillations, o> — vs — 0, and one factor of the denominator reduces to /^, so that if the damping of that oscillation is small, the amplitude is very large, or infinite if there is no damping. This is the case of resonance. (Resonance may also be defined in a slightly different manner as occurring when ico is one of the roots of the equation D (\) = 0 in which all the R's have been put equal to zero. This corresponds with our example in § 240. In practical cases the difference is very small.) If now we have a system acted on by electromotive forces each one of which is the sum of any number of harmonic components of different periods, any component may cause resonance with any free oscillation of the system, so that resonance may occur in a large number of ways. 242. Examples. Two Circuits. We shall illustrate the principles of the preceding section, aside from the examples that have already been given in §§ 239, 240, involving one degree of freedom, by an example of two circuits. Consider an induction coil in which both the primary and secondary contain a condenser in series. This is the case of the so-called Tesla high-frequency coil, in which a Leyden jar produces an oscillatory discharge through the primary, while the ends of the secondary are usually connected with a small capacity, say a pair of knobs. We shall 32—2 500 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. take for ^ and qz the charges of the two condensers, so that the currents are (1) ^ = ^f' /2 = W' We accordingly have T: (2) F r~i£+i£ -*K^* KZ' and the differential equations for the free oscillations are (3) The equation for the frequencies is (4) M\* = 0, or (5) + + + + As this equation is of the fourth degree, we shall treat only the case of no damping, which, as we have seen, will not cause a large error in the determination of the frequencies. Putting then JJj = E2 = 0 the equation becomes ^ "^ _ \ — v> or, as we may otherwise write, (7) -\ 4 ' V~.i — A i — m-~"Tt/ -\2 If the two roots of this quadric are A !_ V' V' = 0. 242] INDUCTION OF CURRENTS. 501 we have for the periods, /7T _27™ /77/_2<7™ = ^T' * ~~^' so that r = If we introduce the periods of the two circuits alone, T, = 27T N/2iA, T2 = 27T and a quantity 6 which is nearly a mean proportional between them, these periods become T = (9) Sp- in case Ti = T T2 and Tf - T 2&>, we have, developing the square roots by the binomial theorem, the approximation, <94 a'-av-fais (ii) In this case the longer period is nearly that of the longer individual period, being somewhat longer, while the shorter period is some- what shorter than the shorter individual period. This is probably (13) 502 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. the usual case of the Tesla coil, where only the longer oscillation plays much part. For a further treatment of this example, the reader is referred to articles by Oberbeck* and Bliimckef. We shall now consider the forced oscillation. Let there be an impressed force JE0 cos wt in the primary circuit, there being none in the secondary. Then we have for the secondary = 1 + i^ The amplitude of the secondary current /2(0) = co . \ a2 \ is We get resonance when co2 is one of the roots of the quadratic In case there is no condenser in the secondary, we have and there is then but one frequency for resonance, <'5> "^.(ZA-jfy This is the practical case of a transformer or induction coil, and is treated by J. J. Thomson in his Recent Researches in Electricity and Magnetism, Chapter VI., to which the student is referred for further examples of this subject. For a treatment at length of the subject of oscillations, the student may consult Rayleigh, Theory of Sound, Chapters IV, V. and X.B, and Routh, Advanced Rigid Dynamics, Chapter II. * Oberbeck. "Ueber den Verlauf der electrischen Schwingungen bei den Tesla'- schen Versucken." Wied. Ann. 55, p. 623, 1895. t Bliimcke. "Bemerkung zu der Abkandhmg des Hrn. A. Oberbeck." Wied. Ann. 58, p. 405, 1896. CHAPTER XIII. EQUATIONS OF ELECTROMAGNETIC FIELD. ELECTROMAGNETIC WAVES. 243. Localized Electric Force of Induction. In the preceding chapter we have developed the theory of current induc- tion in linear circuits, on the basis of the treatment of a set of currents as a mechanical cyclic system, and we have thus arrived at equations which are justified by experiment. We have found for the electromotive force of induction in any circuit, d fiT\ dp where p, the electro-kinetic momentum corresponding to the circuit, is by the results of § 226 defined as the total flux of magnetic induction through the circuit, that is the surface integral (2) p = {£ cos (nx) + S3JI cos (ny) + SJJ cos (nz)} dS over any cap bounded by the circuit. If we consider the electromotive force around the circuit as made up of electric forces acting at each point of the circuit, just as in § 166 we considered the electromotive force due to electro- static action as the line-integral of the electrostatic field-intensity, we may here consider the electromotive force as a line-integral around the circuit, (3) E{ = l(Xdx + Ydy + Zdz} = JF cos (Fds) ds. The vector F whose components are X, Y, Z is a quantity of the same nature as the electric field-intensity, and we shall not in future distinguish whether it is of electrostatic or electrodynamic origin. If we apply Stokes's theorem to the line-integral in (3) we 504 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. convert it into a surface-integral which, in virtue of (i), must be equal to the negative time-derivative of the surface-integral in (2). / \ ((\ftz aF\ / \ ftx ^Z\ / x fiY dX\\ . (4) [His -- -5- cos (nan + -= -- ~- cos (nv) + [-= --- ^- }\ dS JJ(\dy d*/ \dz d®J \dx dyj) S ^^ d8' As we assume that the circuit does not change geometrically with the time the differentiation with respect to t may be passed under the sign of integration, and operates only on the quantities 8, SR, 91. Since the two surface-integrals may be taken over the same surface, and the equality holds for any portion of surface whatever (as we may choose any cap over any circuit), the integrands are necessarily equal at all points of space, necessi- tating the equations _as _a^_a_F dt ~8y a*' _89tt_az_az " dt ~ 8* das' _ =_ dt ~~ dx dy ' These equations, which are more compactly expressed by (6) _ are the general equations of induction, and are justified because of their leading, by the reverse process, to the equation (i), which is directly verified by experiment. A direct experimental verification of equations (5) has been given but recently. If we wish to introduce the vector-potential belonging to the magnetic induction, by § 226 we have the alternative expression forp* (7) p = l(Fdx + Gdy + Hdz). Comparing this now with the line-integral in (3) gives us (8) l(Xdx + Ydy + Zdz) = - ^ f(Fdx + Gdy + Hdz). * See the definition of p following equation (3), p. 469. 243] EQUATIONS OF ELECTROMAGNETIC FIELD. 505 From the equality of the line-integrals we must not conclude the equality of the integrands, for the line-integral of any lamellar vector point-function around a closed path vanishes. We accord- ingly obtain F=-a!+r> where (Z', F', Z') is a lamellar vector. If X, F, Z denote the whole electric force, when the state of the magnetic field is not changing it becomes the electrostatic force, so that the components X', Y', Z' must be the negative derivatives of the electric potential. Accordingly the equations are _ dt a* • These are the equations as given by Maxwell*. We shall however prefer the form (5), not containing either potential, as introduced by Heavisidef and Hertz J. Since the electrostatic field has no curl, it need not be considered separately in equa- tions (5). If however there are impressed electromotive forces X, Y', Z' not of electrostatic origin, such as those due to chemical or thermal effects, and X, F, Z still denote the total field, we must replace X, Y, Z in equations (5) by X - X' ', F- F', Z-Z'. (Heaviside, Vol. i. p. 449.) In a closed conductor undergoing electromagnetic induction there are not necessarily differences of electric potential, for * Treatise, Art. 598, equations (B). t "Electromagnetic Induction and its Propagation." Electrician, Feb. 1885, Papers, Vol. I., p. 447, eq. (20). J " Die Krafte elektrischer Schwingungen behandelt nach der Maxwell'schen Theorie." Wied. Ann., 36, p. 1, 1889. Jones's trans., p. 138. 504 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. convert it into a surface-integral which, in virtue of (i), must be equal to the negative time-derivative of the surface-integral in (2). dY\ , . ftX dZ\ , , ftT dX\\ . cos <"*> + - cos (ny} + - ds m/aZ d JJfe - -s = " it cos + cos + cos As we assume that the circuit does not change geometrically with the time the differentiation with respect to t may be passed under the sign of integration, and operates only on the quantities S, 90R, $1. Since the two surface-integrals may be taken over the same surface, and the equality holds for any portion of surface whatever (as we may choose any cap over any circuit), the integrands are necessarily equal at all points of space, necessi- tating the equations _a§ =a£_aF dt ~dy dz ' dt _=_ dt da) dy ' These equations, which are more compactly expressed by 39* (6) -g-curil1, are the general equations of induction, and are justified because of their leading, by the reverse process, to the equation ( I ), which is directly verified by experiment. A direct experimental verification of equations (5) has been given but recently. If we wish to introduce the vector-potential belonging to the magnetic induction, by § 226 we have the alternative expression for?* (7) p = f(Fdx + Qdy + Hdz). Comparing this now with the line-integral in (3) gives us (8) j(Zdx + Ydy + Zdz) = - 1 f(Fda> + Ody + Hdz). * See the definition of p following equation (3), p. 469. 243] EQUATIONS OF ELECTROMAGNETIC FIELD. ,505 From the equality of the line-integrals we must not conclude the equality of the integrands, for the line-integral of any lamellar vector point-function around a closed path vanishes. We accord- ingly obtain where (X't Y', Z') is a lamellar vector. If X, Y, Z denote the whole electric force, when the state of the magnetic field is not changing it becomes the electrostatic force, so that the components X', Y', Z' must be the negative derivatives of the electric potential. Accordingly the equations are y__aF_8j dt das' dO dV „_ dH dV "dt~fo- These are the equations as given by Maxwell*. We shall however prefer the form (5), not containing either potential, as introduced by Heaviside-f and Hertz J. Since the electrostatic field has no curl, it need not be considered separately in equa- tions (5). If however there are impressed electromotive forces X, Y', Z' not of electrostatic origin, such as those due to chemical or thermal effects, and X, Y, Z still denote the total field, we must replace X, Y, Z in equations (5) by X - X' t Y-Y', Z-Z'. (Heaviside, Vol. I. p. 449.) In a closed conductor undergoing electromagnetic induction there are not necessarily differences of electric potential, for * Treatise, Art. 598, equations (B). t "Electromagnetic Induction and its Propagation." Electrician, Feb. 1885, Papers, Vol. i., p. 447, eq. (20). J "Die Krafte elektrischer Schwingungen behandelt nach der Maxwell'schen Theorie." Wied. Ann., 36, p. 1, 1889. Jones's trans., p. 138. 506 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. example in the case of a circular ring placed perpendicularly to the force of a varying uniform magnetic field the electric potential is constant. If however the circuit is broken, current flows for a very short time until the electric force vanishes; there is then produced a disturbance of charges producing differences of potential to be calculated from the equations _a_F=ap _aF_a£ _^_T_^ dx~~dt' dy ~ dt ' dz~~dt' Conductors connected to the broken ends of the circuit, for instance the plates of an electrometer, will then show a difference of potential. 244. Displacement Currents. If we compare the equa- tions (5) with the equations § 222 (2), = -- -5— , oy oz dL dN dM dL 4-7TW = - -- — , 06 oy we notice that they are analogous in having the right-hand sides equal to the curl of the electric and magnetic field respectively. We make the analogy still more complete by introducing the conception introduced into the theory by Maxwell of the electrical displacement current *. Suppose that we have a condenser charged with electricity. There is then a field of electric force, the lines of force running from the positively charged plate to the negative. The electric induction is, by § 182 (i 6), g = 47TCT. If now the plates be connected by a conducting wire, the positive charge passes from the positive plate along the wire, until a state of equilibrium is reached. During this period the electric induction between the condenser plates is diminishing and finally reaches zero. The hypothesis of Maxwell is that the change of the * "A Dynamical Theory of the Electromagnetic Field (11)," Phil Trans. Vol. CLV. 1864. 243, 244] EQUATIONS OF ELECTROMAGNETIC FIELD. 507 induction produces the same magnetic effect as would be produced by a current of current-density at every point of the field, which together with the current in the wire would form a closed circuit. As the equations § 222 (2) were deduced from the magnetic effect of closed currents, some hy- pothesis is necessary if we are to deal with unclosed currents, and Maxwell's hypothesis is justified by its remarkable consequences. Since Maxwell calls the vector g/4?r the electrical displacement, he 1 ?f% terms the vector — — the displacement current. The consequence of Maxwell's hypothesis is that in the dielec- tric we must introduce the components of the displacement 1 83P 1 8g) 1 33 . , m the tions ( 1 1 ), giving d$==dN_dM >B ^e*AAlH dt 'by dz ' dVjL_dN ?«*.« dt dz dx ' dj$ = dM_dL dt dx dy ' These equations are now completely analogous to the equations (5) except for the difference of sign on the left, the two sets being represented by (13) If the dielectric is conducting, we must introduce both the conduction and the displacement current, so that the equations are 8£ 8^ dM ^r + 4nrw = ^- — -^— , dt dy dz 8g) , dL dX (14) -^ -- — -, dt dz dx 83^ dM dL — + 4<7rw=- -- — . dt dx dy 508 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. Differentiating these equations respectively by x, y, z and adding we obtain so that the total current is solenoidal, like the flow of an incom- pressible fluid. Integrating (15) through a portion of space r bounded by a closed surface S, = [u cos (nx) + v cos (ny) + w cos (nz)} dS, which by § 182 (17) becomes (17) gj J J J pdr = g- = II (M cos (nx) + v cos (ny) + w cos (ras)} d£ That is, the increase of charge of any portion of space is equal to the electricity brought in by conduction. This agrees perfectly with our previous conceptions. Our statement made in § 129 that electricity is not incompressible is also reconcilable with Maxwell's statement that the total current, the resultant of the conduction and displacement currents, is like the flow of an incompressible fluid. By the analogy between the equations (5) and (12) we 1 r)9^ might call the vector j— -^r *ne magnetic displacement current. Magnetic conduction-currents do not exist, although they have been introduced into the equations by Heaviside* for the sake of symmetry. 245. Complete System of Equations for Media at rest. We may now collect all the fundamental equations of the theory as it has been developed. Before doing this it will be convenient to make a slight change in our units. It will be recalled that in the whole of Part III since the introduction of the electro- magnetic system of units we have considered all quantities, whether electrical or magnetic, to be measured in that system. Up to the present this has been most convenient, and in prac- tical cases dealing with electro-magnetism and electro-magnetic * " Electromagnetic Induction and its Propagation." Electrical Papers, Vol. i. p. 441. 244, 245] EQUATIONS OF ELECTROMAGNETIC FIELD. 509 induction this will generally be true. We are now, however, about to consider a new class of phenomena, and it will be convenient to use the Gaussian system, that is, to measure all electrical quantities in electrostatic units, and all magnetic ones in magnetic units. We shall therefore be obliged to reintroduce the factor A, § 210, which will multiply the electric currents, and divide the electrical forces, according to § 212, equations (6) and (9). Equations (14) and (5) thus become* . a* dN dM . as dz 97 A ^T + QtTrAu = -= -- ir- , — A^- = ^ -- -r— , dt dy dz dt dy dz .eg) dL dN . m dx dz (A) A ^ + 4i7rAv = -^ ---- — , (B) -.4— = - -- ^-, dt dz dx dt dz dx , 93 dM dL . 99t dY dX A ~ ~ -^ -- 5- , ^r- - - . dt dx dy dt dx dy These are the equations of cross-connection between the elec- tric and magnetic fields and thus show that in non-conductors the curl of the force of either field determines, or is determined by, the time-variation of the induction of the other. If we know the state of the field at any instant we may accordingly find it at any subsequent instant. For we have the three sets of equations expressing the Fourier-Ohm laws, £ = eX, 8 = /juL, u = \X, (C) g) = eF, (D) W = pM, (E) v=\Y, The letter e denotes the electric inductivity, which in Chapter IX, where we did not distinguish electric and magnetic quantities, was denoted by ^. It will be noticed that equations (A) and (B), which are the fundamental equations of the theory, contain no quantities that are intrinsic to the media, but only those quantities which completely specify the electric and magnetic state of the fields. The equations (C), (D), and (E), on the contrary, contain the quantities e, //, and X, which denote properties of the media. These latter equations are not fundamental to the theory, as they may under certain circumstances be replaced by others. In addition * In Hertz's papers the right-hand members appear with the opposite sign, since Hertz employs the left-handed arrangement of axes. (Cf. Fig. 1.) 510 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. we have for the electric and magnetic energies and the dissipativity, or heat generated per unit of time, (F> T = (m + Mm + H = II '(Xu + Yv + Zw] dr. 246. Eolotropic Media. The equations (C), (D) and (E) have been established on the supposition that the medium is isotropic, that is that it has the same properties in all directions at any point. In some bodies, such as certain natural crystals, this is not true. The assumption next in order of simplicity to that made in Chapter IX is to assume that the energy per unit volume is a homogeneous quadratic function of the components of the field ^ [eaXf + e^F2 + 633^2 + enYZ + e3lZX + elaZ7}, the six coefficients being properties of the medium. If we then apply the reasoning of § 180, we find that our results are the same as before, providing that we define the inductions by the equations £ = enX -f e12F + euZ9 8 = /AU 3 = €S1X + e32F+ e33^, 91 where €rs=esr: prs = psr. The inductions thus defined have all the properties that we have hitherto predicated with regard to them. It has been pointed out by Pupin* that these are not the only possible generalizations of the equations (C) and (D). Media which are not isotropic are called eolotropic. A body may also be eolotropic with respect to conduction, in which case * Pupin. "Studies in the Electro-magnetic Theory." American Journal of Science, Vol. L., p. 326, 1895. 245 _ 247] EQUATIONS OF ELECTROMAGNETIC FIELD. 511 We shall in the future, as we have done in the past, consider only isotropic bodies. 247. Consequences of the Equations of the Field. Propagation. If we differentiate the equations (A) respec- tively by x, y, z and add, we obtain the consequences of which we have discussed in Chapter X. If the medium is an insulator, the relaxation- time is infinite, and 8?+§9+83 dx dy dz is independent of the time. Applying the same process to the equations (B), we obtain dx dy dz independent of the time, and the value of this divergence is zero, except in intrinsic magnets (§ 201). We shall now deduce the more important consequences of the equations, proceeding from the simpler to the more complicated cases. We shall first, therefore, consider the phenomena in insu- lators, in which the equations (A) and (B) are exactly symmetrical. On account of the dual nature of the relations of the two fields it follows at once that every effect of electrodynamic induction in producing electromotive forces has an analogous effect in the pro- duction of magnetomotive forces by electric displacement currents. For instance a closed iron ring placed in an electrostatic field varying with the time would become magnetized. Effects of this sort have not yet been observed, on account of the extreme small- ness of the factor A, by which the displacement current is multi- plied. For the same reason, electrostatic forces produced in insulators by the variation of magnetic fields have not been successfully observed, although the attempt has been made by Lodge*. The justification of the equations (A) has been given by other results. * Lodge. " On an Electrostatic Field produced by varying Magnetic Induction. " Phil. Mag. (5) 27, p. 469, 1889. 512 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. If we perform upon the equations (13) the operation of curl, which is typified by the result of differentiating the third of equations (B) by y and subtracting it from the second differentiated by z we obtain, after adding and subtracting u . s (m m\ .„ d ftx ar A3i(^-^) = *X-S-X(te + Sy Now supposing the medium to be homogeneous, that is e and /-t constant, making use of the equations (C) and (D), and supposing there is originally no electrification, we have «+?!+§*.« da dy dz and making use of the first of equations (A) we transform ( I ) into Proceeding in like manner we obtain for the other components, (2) We thus find that in insulators each component of the two fields satisfies a differential equation of the form (3) ?*-*A*, , . where a — I/ A *J pe. Since this is an equation of great importance in mathematical physics, we shall investigate its general solution. Let us multiply both sides of the equation by the element of volume dr and integrate throughout the volume bounded by a closed surface $, applying the divergence theorem to the right-hand member, 247] ELECTROMAGNETIC WAVES. 513 If the surface S is a sphere of radius r with its center at the point P. we have where by r we denote the values of <£ at points on the surface of the sphere of radius r, with center P. Introducing polar coordinates into the left-hand side of equation (4) also we may write it Now differentiating this and the transformed right-hand member (5) by the upper limit r changes our equation (3) into The surface integral which appears on both sides is 4?r times the mean value of the function <£ on the surface of the sphere of radius r. Calling this mean value 0r we have the equation which, on performing the differentiations and dividing by r, may be written _., If we now introduce two new independent variables u = at -f r, v = at — r, we have, putting rr — ty, dt ~~ dudt 3vdt~ du dv ' _ _ _ __ dr ~~ du dr dv dr ~~ du dv ' dtz =_ , r W dudv w. E. 33 514 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. so that our equation (9) becomes do) I*, a dudv Of this equation the general solution is (11) ^ = 9i(u)+g9(v)t where g^ and g2 are perfectly arbitrary functions of their arguments. We consequently have for the solution of (9), ( 1 2) rr = g1 (at + r)+ g2 (at - r). When r = 0 we have Q = g1(at)+g2(at), and this being true for all values of t the functions gl} g2 are not independent, but one is the negative of the other, whatever the value of the argument. Putting then gi = g, 92=-g> we have (13) r$r = g(at + r)-g (at - r). Differentiating by r, (14) ^r + rd^ = g'(at + and again putting r = 0, we obtain _ But r is the mean value of over the surface of a sphere of radius r with center at P, and the mean value over a sphere of radius zero is the value at P itself. Accordingly (16) r, Suppose that for a certain initial instant, for which we shall take t = 0, the values of the function <£ and of its time derivative ~ are given as functions of a point in space, (JU (20) Inserting in the equation (19) it becomes but when r — at the value of the left-hand side is by (16) equal to at all times may be calculated for every point P if we know the mean value of d(f>/dt at a time earlier by the interval at for all points on the surface of a sphere of radius at about P. as well as the rate of variation of the mean value of (f> as the radius of the sphere is altered. Suppose that initially <£ and ^ are both zero except for ot a certain region whose nearest point lies at a distance r: from P and whose farthest at a distance r2. Then as long as t rja as well. Accordingly there is no disturbance except between the times rja and rjat or the quantity (f> is propagated in all directions with the velocity a. It may be easily shown that <£P is finite if F and / are finite everywhere. We might have obtained the same result in a more simple manner by transforming A<£ to polar coordinates in equation (3), making <£ independent of the angular coordinates, when the equa- tion becomes * Nouveaux Memoires de VAcademie des Sciences, t. in. 33—2 516 THE ELECTROMAGNETIC FIELD. [FT. m. CH. XIII. Of this a solution is (24) as has been shown (12). Accordingly for all points and times for which at — r has the same particular value we have the same value of r^, or a particular value of travels outwards with the velocity a. The value of is inversely proportional to the distance r traversed. The solution makes the value of infinite at the point from which it is propagated. This is only an apparent difficulty, for just as the potential due to a single mass-point is infinite at the point, but is never infinite when the mass is continuously distributed with finite density, BO here if we consider a finite region in which is not zero, the infinite value will not occur, as is shown by our general solution (22). We see, accordingly, that a state of electric or magnetic field existing in any region of space has its action propagated with the finite velocity a = 1[A V/i€ in all directions, and inasmuch as by the equations (A), (B), the time-variation of one field is propor- tional to the curl of the other, the second term of (22) shows that a curl of one field in any part of space causes a propagation of a field of the other kind. The conclusion that electrical and magnetic actions are pro- pagated with a finite velocity is the great and remarkable consequence of Maxwell's theory, and was enunciated by him in 1864 in his celebrated paper on the Dynamical Theory t>f the Electromagnetic Field. From this, and the other consequences of the equations, he was led to enunciate the theory that light was an electromagnetic phenomenon. In feet, the equation (3) is, as we have shown, the equation of wave motion, and is the basis of any undulatory theory, whether of light or of sound. The manner in which the equations give us a theory suitable for light and not for sound will be discussed in § 249. Since the velocity of propagation is I/ A Vep, in air the velocity should be I/ A — v, or the velocity which corresponds to the ratio of the two electrical units of quantity. Determinations of this purely electrical quantity, as refinements in measurement in- creased, gave results showing a surprising agreement with the 247, 248] ELECTROMAGNETIC WAVES. 517 determinations of the velocity of light, so that many German authors are accustomed to speak of A as the reciprocal of the velocity of light. It seems preferable, however, to keep the definition of A and v purely electrical, as we have given it in § 212. A further confirmation of the electromagnetic theory of light was sought in the fact that the index of refraction, being inversely proportional to the velocity, should in non-magnetic bodies, for which ft = l, be proportional to the square root of the electric inductivity. This relation was experimentally verified for a sufficient number of transparent dielectrics to make it appear that the agreement was not accidental, although many exceptions were found. Nevertheless, although these considerations made the electro- magnetic nature of light very probable, the theory of propagation of actual electrical disturbances with finite velocity remained unverified by experiment until 1887, when Hertz began the publication of his remarkable researches*, which have since carried conviction of the truth of Maxwell's theory of electricity and magnetism to the most conservative parts of the scientific world. For an account of them the reader is referred to Hertz's collected papers on "Die Ausbreitung der elektrischen Kraft," or to the English translation by D. E. Jones. 248. Transfer of Energy. Poyn ting's Theorem. We shall now form the equation of activity for any portion T of the field. If E= W+ T be the total energy, H the dissipativity, we have (i) + H = ~t + HJ(Xu +Yv + Zw) dr. Since e and JJL do not vary with the time we have, by (C), (D), 9®* i Y^3P -*- +x * "Ueber die Ausbreitungsgeschwindigkeit der elektrodynamisohen Wirkun- gen." Wied. Ann. 34, p. 551, 1888, trans, p. 107. 518 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. Inserting these and the corresponding values in the integrand, and replacing the time derivatives by the curl-components from equations (A) and (B), we have dL dz ( Ar r -- o- 1 *f -w 1 3 -- o-* Now integrating by the theorem of § 226 (4), this becomes (3) d + H + (XM-YL)cos(nz)}dS. We have supposed that there are no intrinsic electromotive forces, but if there are, the components X, Y, Z in (i) must be replaced by X-X', Y-Y', Z-Z' (§243), except in the last integral representing the dissipativity, consequently that integral will not be entirely cancelled as above, but there will remain the term (4) representing the activity of the impressed forces, in addition to the surface-integral. If we extend the integral in (3) to a space to whose boundaries electric and magnetic actions do not extend, since the integrand in the surface-integral vanishes we have (s) £+*-« as the equation of activity (cf. § 64 (6)), whose H is the present E, while H of (5) is the 2F of § 64, (8).) If the fields are not zero at the surface S, the equation (3) shows that the energy in the volume will be accounted for by supposing that a quantity of energy per unit of surface S enters the volume r in unit time. We may therefore call the vector 248,249] :.; ELECTROMAGNETIC WAVES. 519 whose components are the energy-current-density. The equation (3) accordingly states that the quantity of energy R is transferred per unit of time across the unit of surface tangent to the direction of both the electric and the magnetic force. This is Poynting's* remarkable theorem. It has been remarked by J. J. Thomson, Heaviside and Hertz that this determination of the energy current is not the only possible one, since we may add to the above vector any solenoidal vector without changing the surface-integral in (3). Hertz has also pointed out that, as this makes energy flow at all points where fields of both kinds exist, it involves the continuous flow of energy (in closed tubes) when a magnetic pole and an electrified point exist in each other's presence. In many cases, however, the notion of the motion of the energy here given is a very fruitful one. It has been further developed in several papers by Wienf. The vector R is sometimes called the radiant vector. 249. Plane Waves. Let us again consider a perfect insu- lator. The equations § 247 (2) are all satisfied by any function of the argument s = Ix + my -\-nz- at, where a = l/A Veyu.. (i) = (f> (loc 4- my + nz — at). For we have • and therefore * Poynting, Phil. Trans. 2, p. 343, 1884. t Wien. "Ueber den Begriff der Localisirung der Energie." Wied. Ann. 45, p. 685, 1892. "Ueber die Bewegung der Kraftlinien im electromagnetischen Felde." Wied. Ann. 46, p. 352, 1892. 520 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. if we take (4) The quantities X, Y, Z, L, M, N accordingly have the same set of values for all points for which (5 ) Ix + my + nz — at = const. But this is the equation of a plane whose normal has the direction cosines I, m, n, and whose distance from the origin is at + const. The plane is accordingly travelling in the direction of its normal with the velocity a= 1/J. V//,e. The disturbance is accordingly a plane electromagnetic wave, whatever the nature of the function <£. The six functions are not independent. For let (6) X = fc, F=02, £=3, Z=fi, Jf=^, ^=^3. Then inserting these in the equations (A) and (B) we have - y ^ & =mi/r3' - n^r/, (7) -*'-»#/- \/ a system of linear equations to determine the ratios of ^>x's and i/r"s. Multiplying the equations (7) or (8) in order by I, m, n, and adding either set, we obtain (9) l# + mfe + nfc = 0, %' + m^r/ + n^ = 0. These are two differential equations with regard to the variable s, integrating which gives Ifa + m2 + n$3 = Clt Ifa + m^2 + nfa = C2) that is (10) lX + m7 + nZ=Clt IL + mM + nN = C2. This shows that the component of either field resolved parallel to the direction of propagation is constant as we travel in that direction as well as in the plane of the wave, and is therefore 249] ELECTROMAGNETIC WAVES. 521 constant throughout space. But such a constant field is not propagated at all, and we shall therefore disregard it, and put both constants equal to zero. Both fields are consequently perpen- dicular to the direction of propagation. It is for this reason that Maxwell's theory is appropriate for an explanation of light, which, as the phenomena of polarization show, is due to transverse undu- lations. Although the forces of the two fields lie in the wave-plane, and are constant over any particular wave-plane, it does not follow that their directions are the same in all wave-planes, that is for different values of s. We shall however assume that their directions are the same in all wave-planes, and we will call the direction cosines of F, alf /3it ylt and of H, a^, /32, ry2. Such a wave is said to be plane-polarized. Then we have (ii) Z = «!<£, F=&, £ = 7i& 1 = 0^, M and our equations (7) and (8) take the form (12) - Multiplying the equations of the first set respectively by «2j &, 7a and adding, we get (H) or the electric and magnetic forces are mutually perpendicular, as well as perpendicular to the direction of propagation. There are accordingly two directions, either of which might be chosen to define the plane of polarization, and it rests with experiment to decide between them. Squaring and adding either equations (12) or equations (13) gives e<£/2=/iip. Extracting the square root and integrating, 522 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. putting the integration constant equal to zero for the same reason as before. Accordingly the fields are , , V6 V€ V — , ,— , . — . Vjj, V > V/A The two fields are accordingly propagated together. Comparing the energies of unit volume we find 07T O7T O7T or the energy is equally divided between the two fields. The radiant vector VFH is of course in the direction of propagation. 250. Propagation in a Conductor. In § 247 in deducing the equations of propagation we have supposed the conductivity to be zero. If we do not make this assumption in substituting the value of the curl-components on the left of equation (i) we obtain and in like manner all the components of both fields satisfy the equation The general solution of this equation has been given by Boussinesq, but it is too complicated to be given here. A particularly interesting special case will be treated below. In the mean time we shall content ourselves with the consideration of disturbances which are harmonic functions of the time. For this purpose we shall assume (3) = **U(a,y,*), when our equation becomes (4) ^2 ( The equation (5) 249, 250] ELECTROMAGNETIC WAVES. 523 has been made the subject of a treatise by Pockels. We shall here consider only the case in which U depends on a single rectangular coordinate #, the circumstances being the same all over each plane perpendicular to the X-axis. In a conducting dielectric the value of k2 is complex. In metals we know nothing regarding the value of the electric inductivity e, for whereas electrostatic phenomena may be explained by supposing it to be infinite, in variable states this is far from being the case. In fact in all experiments that have been performed with electric waves thus far the value of co has not been great enough to make the influence of the term containing e appreciable in comparison with that containing X. (See § 206.) We shall therefore neglect the first term, so that our equation of propagation is (6) tatajfctff-A*' ot This is the equation for the conduction of heat, as given by Fourier. We shall consider it in some detail below (§ 254), but shall now return to the consideration of the equation in which k2 is the pure imaginary (8) te The solution of equation (7) is (9) U^C^ Since we have *Ji = (1 + i)/V2 the value of k is (10) k=±A V27rXyLto) . (1 + i). Accordingly the real and imaginary parts of i (/27rA/xw. furnish us with particular solutions of (6). We thus obtain sin (mt - A V2^^T. a?), cos ^t + A 27rXyLto) . x)t 524 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 0! Of these the first two, being of the form e~df(x—at), represent damped waves travelling in the direction of increasing x with the velocity a = -^-A . The periodic factor at any time repeats its values when x is increased by the distance 1 = — \/ ^ A V \JJLCO ' which is called the wave-length, the frequency being n = o)/2,7r. The damping factor e~An*t"*-x which causes the amplitude of the wave to decrease in geometrical progression as the distance travelled increases in arithmetical, has the relaxation-distance, or the distance in which the amplitude diminishes in the ratio I/e, . A v27T\fJL(0 The last two particular solutions represent waves travelling in the opposite direction with the same velocity and damping. Since the velocity depends on the frequency, there is no definite velocity of propagation in a conductor. On account of the damping, harmonic disturbances of high frequency rapidly die out, consequently alternating fields penetrate but a short distance into conductors. This was shown by Maxwell*, but its importance was brought out by the researches of Heavisidef, Lord RayleighJ and Hertz §. We shall now consider the relations between the two fields. If the components X, Y, Z, L, M, N are equal respectively to the complex constants Alt A2, A3, Bl} B2, B3) each multiplied by ei»t+k*t inserting in the equations (A) and (B) we obtain 4f7rA\Al = 0, — A^ia)Bl = 0, (12) ^TrAxA^ — — kB3, — AfjLia)B2 = — kA3) = kB2) — ApiwB3 — kA2. Eliminating A2/B3 or A3/B2 we obtain the value for k already found. We thus see that the directional relations of the fields * Treatise, Art. 689. t "The Induction of Currents in Cores." Electrician, 1884. Papers, Vol. i., p. 353. t "On the Self-induction and Eesistance of Straight Conductors." Phil. Mag. 21, p. 381, 1886. § " Ueber die Fortleitung elektrischer Wellen durch Drahte." Wied. Ann. 37, p. 395, 1889. Jones's trans, p. 160. 250, 251] ELECTROMAGNETIC WAVES. 525 and the direction of propagation are the same as in insulators, but the ratio of the two fields is H 4>7rA\ The magnetic field accordingly lags in phase by one-eighth of a period behind the electric, while in an insulator the fields have the same phase. 251. Reflection of Waves by a Conductor. We shall now consider the effect of a train of plane waves in an insulator striking the plane surface of a conductor which is parallel to their plane. We shall suppose the conductor to extend to infinity in one direction. Let us take the plane # = 0 as the face of the conductor. Let the waves be harmonic in the insulator, for which x < 0, and let the electric force be parallel to the F-axis, the magnetic to the Z'-axis. Then in the wave approaching the con- ductor we have VfJU In the conductor we have F= At the plane x — 0 the boundary conditions to be satisfied are that the tangential components of both forces and the normal components of both inductions are continuous. The latter com- ponents being zero we have only the first two conditions to satisfy. There are not enough constants to enable us to satisfy them both, it is accordingly necessary to add to the terms representing the disturbance in the insulator other terms representing a wave travelling in the opposite direction, or a reflected wave. We therefore take . V/4 V/JL 526 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. in the insulator. Our boundary conditions are then, dividing by ««, (4) a -A Accordingly we have (5) 1- 1 + 2-7TX//, 2-irXfjb Since these two ratios are complex, at the surface of the con- ductor there is a difference of phase between the direct, reflected and transmitted waves. As we increase the conductivity of the conductor the ratio ft'/ft approaches the value — 1, in which case the electric force vanishes at the boundary, which is a node for the electric field, while the magnetic field is a maximum. On the other hand, as we increase the frequency of the oscillation or the magnetic inductivity of the conductor the ratio ft'/ft approaches + 1, or the magnetic force vanishes at the boundary, while the electric is a maximum. If we put we find for the ratios of the amplitudes of the reflected and trans- mitted to that of the direct waves ft' ft ft Accordingly whether r be very great or very small the reflected waves tend to become as great as the direct, while the larger r the less is the magnitude of the transmitted waves. 251, 252] ELECTROMAGNETIC WAVES. 527 The experiments of Hertz* confirmed the above results, the boundary of the conductor being more nearly a node for the electric than for the magnetic force. If the amplitude of the reflected waves approaches that of the direct ones, the two systems will interfere and produce a set of standing waves, with nodes at regular distances from the conductor. This field was explored by Hertz by means of a resonator, composed of a circle of wire with its ends terminating in two small balls near together, constituting a con- denser. This system has a certain period of its own, and what has been said in § 240 applies to it. It was tuned to the period of the waves, and being placed anywhere in the field would have currents induced in it by the harmonic electromotive forces of the field. Thus where the force is a maximum sparks appear between the balls of the resonator, disappearing at the nodes. For the further theory of the resonator the reader is referred to Poincare, Les Oscillations Electriques, J. J. Thomson, Recent Researches in Elec- tricity and Magnetism, and Drude, Physik des Aethers, 252. Spherical Oscillator. We have hitherto considered waves in insulators, without considering how they are produced. In the experiments of Hertz the waves were produced by disturb- ing the charges in a peculiar dumb-bell-shaped conductor, and allowing oscillations to set in, which were propagated outward through the air. A satisfactory theory of the oscillations in Hertz's oscillator has not been given. We may state the general problem : given a charge disposed in any manner not in equilibrium upon a conductor of any form, find the nature of the oscillations that ensue while the conductor is settling down to its state of equi- librium, together with the fields to which they give rise. This problem is a very complicated one, and has been solved for very few cases. We shall consider the case of a conducting sphere, the charge upon which is that induced by placing the sphere in a steady uniform electric field. The field is supposed suddenly destroyed, and the charge then oscillates until equilibrium is attained. We shall suppose e = /A = 1. * "Ueber elektrodynamische Wellen im Luftraume und deren Reflexion." Wied. Ann. 34, p. 609, 1888. 528 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. Since both fields are propagated according to the equation and since any derivative of a solution is also a solution, we may take - dydt' which satisfy the solenoidal condition and the equations (A) and (B). If we assume to be a function only of r and £, we have for a diverging wave Differentiating <£ by the coordinates we obtain 9<£ _d(f>dr _d(f> z dz~~drdz~'drr ' _ a/i d$\d_r = a_/i a^\^ ~ * _ = _ dr (r dr ) fa dr \r dr) r (3) 82<^> = _a_ /i a^\ 3r = _a_ /i a^\ ^ " dr\r dr) 'dy 'dr \r dr) r = .« . 8^2 ~ Z dr \r dr) dz r dr dr \r dr) r r dr' so that the forces are v— ^ t^- ^$\ xz *£. — z~ I "^ ] j dr \r drj r u) T-L(l*£\yL ~dr(rdr) r ' dr \r drj r r dr The field is thus the resultant of two parts, the first, equal to drrdr' 252] ELECTROMAGNETIC WAVES. 529 parallel to the radius, and the second, equal to parallel to the ^-axis. At the surface of the sphere, r = R, if the conductivity is large,, the lines of force are normal to the surface, so that this second component vanishes, and we have -- o— -, r dr dr2 r dr that is When t — 0, the electric forces are derivable from a potential, which is, by (i), equal to — ~ (since A$ = 0). But by § 194 (7) the potential is, in the case supposed, proportional to z 9 r*=- Consequently initially Introducing the value of from (2) this gives Consequently the function / is constant for all values of its argu- ment greater than R. Hence the value of <£, remains equal to (7/r so long as r — at>R, and the field remains unchanged. When t = (r - ^)/a, the wave arrives at the distance r, and to determine the field at subsequent instants we must deter- mine the values off for values of its argument less than R. Let us make use of equation (5). Differentiating =f(r-at) f(r-at) dr r r* ?* =/> - at) _ 2/(r-aQ 2/(r-'tt)=V(xtt\ b_K'R a2" KR ' that is /y/2 /v,2 KfR^ = KRx-. t t Accordingly the time necessary to produce a given potential at a distance x from the origin is proportional to KR multiplied by the square of the distance. We quote Lord Kelvin's words : "We may be sure beforehand that the American telegraph will succeed, with a battery sufficient to give a sensible current at the remote end, when kept long enough in action; but the time required for each deflection will be sixteen times as long as would be with a wire a quarter of the length, such, for instance, * Proc. Boy. Soc., May 1855 ; Math, and Phys. Papers, Vol. n. p. 61. 536 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. as in the French submarine telegraph to Sardinia and Africa. One very important result is, that by increasing the diameter of the wire and of the gutta-percha covering in proportion to the whole length, the distinctness of utterance will be kept constant ; for R varies inversely as the square of the diameter, and K (the electrostatical capacity of the unit of length) is unchanged when the diameters of the wire and the covering are altered in the same proportion." (The so-called " KR-L&w " has been applied to the theory of telephony on long-distance land-lines, to which it is not at all applicable, as has been shown by Heaviside. The use made of this law by the chief electrician of the English telegraphs would have prevented long distance telephony in England, even had there been any long distances.) Guided by the conclusion announced above, we shall insert a new variable, u = x/*Jt, and attempt to satisfy the equation (6) by a function of u alone. We have dV = dVduss 19F^ dt du dt 2 du V#* ' aF=aFaM = aFj^ dx du fix du * ' d#2 ~ Vtf du* d® t so that our equation becomes KR dV d*V or KR = u. dV KR , dV n .™u* lo%-= •+C0nst' = The integral of this equation is dV KR , lo%du-= •^+C0n or, integrating from 0 to u = xj*Jt, r^t , .6—" du. . 254] ELECTROMAGNETIC WAVES. 537 If we now write u instead of u */KR/Z this is (13) F=C" Jo This definite integral is a function of its upper limit, and therefore of x and t, satisfying equation (6). For x > 0 and t — 0 the value of the integral is VTT/S*. As we may add any constant to V, we will put f 9 /t (14) F=F0(l--^ \ NirJQ Thus for x > 0 and t = 0, F= 0. For x = 0, t > 0, the value of the definite integral is zero, so that V— F0. Consequently the solution (14) represents the result of connecting one end of the cable with a constant battery, and leaving it permanently con- nected. The definite integral in (14) is the transcendent known as the probability-integral, for which numerical tables have been cal- culated. From these the values of V have been plotted, showing the potential at the different points on the cable, Fig. 96, the different curves being for times 1, 2, 3, 4, 5 times KR. It is to be noticed that however small the interval of time from the instant of connecting the battery, the disturbance is felt somewhat at all points, however remote. Thus the velocity of propagation would be infinite, if we could speak of a velocity. This shows that the This may be shown as follows. We have J: Consequently f 00 f 00 Ixdy. 0 J o Changing to polar coordinates, Too T = e-x*dx = Jo J = f00 t" e-W+ Jo Jo fir*-! Therefore 538 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. theory is only an approximation, for it is hardly imaginable that the velocity should be greater than in free space. FIG. 96. The current is obtained, according to (7), by differentiating with regard to as. We must therefore differentiate the integral by its upper limit, multiplying this by its derivative by x. Accordingly (15) /-FA •L — " n \ I ' ini ;= FIG. 97. 254] ELECTROMAGNETIC WAVES. 539 The values of the current for different points are shown in Fig. 97 for times £, J, £, 1, 4 times KR. When x— 0 we have IT i so that when t = 0 the value of the current is infinite, instan- taneously. The rise of the potential at any given point other than x — 0 is shown by the outside curve in Fig. 98, taken from Lord Fm. 98. Kelvin's paper, the abscissas representing the time, the ordinates the potential. The potential rises asymptotically to its value at the origin, but the current rises to a maximum, which occurs at the time t = KRxzj%, and then dies away to zero. The maximum values of / are inversely proportional to the distance from the origin, and to the resistance R. Since our differential equations are linear, disturbances due to different initial states are merely added. If the battery is con- nected, instead of permanently, for a definite time T, and the cable then put to earth, the effect is the same as if, in the preced- ing case, after a time r we permanently apply the potential — F0 at the origin. If the preceding solution be called V(t) the present will be V(i)— V(t — r), consequently we may obtain the graphical 540 THE ELECTROMAGNETIC FIELD. [FT. III. CH. XIII. representation by taking the difference of ordinates of the outside curve in Fig. 98 and the same curve pushed to the right the distance r. The other curves in Fig. 98 represent the potential at x when the battery is applied for times 1, 2, 3, 4, 5, 6, 7, times KRx*. Since any derivative of a solution of (6) is a solution, the derivative of (14) by x is also a solution, and F= represents the result of instantaneously connecting a battery and then insulating the end of the cable. The distribution at any time is of course shown in Fig. 97, and while V is initially infinite at the origin, the total charge .00 K \ Jo Vdx = q0 is finite, and remains constant throughout. 255. General case of Telegraphic Equation. The telegraphic equation (5) has been treated byHeaviside, Poincare*, Picard f, and Boussinesqj. We shall give the solution of Bous- sinesq, not only because he has given the general solution of the more general equation § 250 (2), but because his method obtains the solution by an ingenious artifice from Poisson's solution § 247 (22), and the knowledge of other methods required by the processes of Poincare and Picard is unnecessary. Let us put 1 R «7 so that our equation is * Poincare\ " Sur la propagation de I'electriciteY' Comptes Rendus, 117, p. 1027, 1893. t Picard. "Sur liquation aux derives partielles qui se rencontre dans la th^orie de la propagation de I'61ectricit6. " Comptes Rendus, 118, p. 16, 1894. $ Boussinesq. "Integration de liquation du son pour un fluide indefini." Comptes Rendus, 118, p. 162, 1894. 254, 255] ELECTROMAGNETIC WAVES. 541 and let us suppose that initially the state of the line is given, that is the potential and current are given at all points, by (2) Let us transform the equation by putting (3) F= e* u. Accordingly so that if we put p = — b the equation becomes The initial conditions now become The method of Boussinesq may be applied to the more general equation which we shall accordingly consider, putting whenever we choose u independent of y. The artifice employed is the introduction into the function u of one more degree of freedom, by making it depend upon another parameter z which is finally to be given any constant value we please. Let then u satisfy the auxiliary equa- tion (8) «2^ = 62«. so that the equation (7) becomes, taking this into account, A 542 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. We will suppose that the initial conditions are now, u = 3> (x, y, z), (10) du , Z- + b*& Then the solution of (9) is, by § 247 (22), inserting explicitly the rectangular coordinates and the direction cosines cos a, cos f3, cosy, of the radius r in the functions <&at, at, (i i) u(x,yyz) — j-g- I \t&(x+at cos a, y + a* cos fi,z + a£ cos 7) dco + 7— I lt (x + a£ cos a,y+at cos fi, z + at cosy)d 0, y, z) = F(x, y)cos~z + H (x, y) sin - s, (12) f / \ / \ ib 7 / ^ . ib (x, y,z) = g (x, y)cos- z + h (x, y) sin — ^, Ov CL and for ^ = 0 these reduce to <& = F, 0 =/, so that for t = 0 we have the proper values of ut ^- . We therefore insert the values (12) in the integral (n). Now as we integrate over the whole sphere, the sine terms, being odd functions of z, disappear, while the cosine terms, being even functions, give us double the value that we should get by integrating over the hemisphere for which z>0. Accordingly, giving z the constant value zero, (13) u — ~- - JTT 1 1 tF(x + at cos a, y + at cos ft) cos (ibt cos 7) dco ~*~ 9~ \\tg (x + at cos a, y + at cos /3) cos (ibt cos 7) c?a>. In the telegraphic equation F and g are independent of y, and are therefore constant on all small circles of the sphere normal to the X-axis. We will therefore employ polar coordinates, a the angle made by r with the X-axis, and % the angle that the plane of r and the X-axis makes with the XZ-plane. Then , v cos 7 = sin a cos ^, dco = sin a 255] ELECTROMAGNETIC WAVES. 543 and * 1 3 rir r2 » _ __ I I tF(x + at cos a) cos (ifa sin a cos %) sin adadx ZTT ot J Q J ^L 2 7T i r* r * ~ I tg(x + at cos a) cos (16^ sin a cos v) sin adadx- ?r J o J -^ e 2?r 2 Let us put t'&£ sin a = f . The definite integral IT 1 (16) /o(f) = - is one of the set (17) /,(£) = n^X cos(? cos which may be evaluated by developing the cosine into a series and integrating each term. We thus obtain (i8) /„(?) = - 2 wg.0./-| an infinite power-series in f, the coefficient of f2? being (2?)! where IT (19) /M-i 2 an integral which we can evaluate. Integrating by parts, writing the integrand (sin2^ ^ cos %) cos2^"1 %, 544 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. from which follows the reduction-formula 20-1 (21) ± = T Integrating (19) again by parts, writing the integrand sin IT IT cos2Jp (ix) of a pure imaginary argument is usually denoted by Ip(x) (not the Ip of the preceding). Putting then _ at cos a = X, — at sin a = dX, f = ib Vtf2 — X2/a2, our solution becomes (29) -at + J (* + V Jo (b v-XYa2) dX. Performing the differentiation by t (§ 26), since 70(0) = u = i {.P (x + aO + ^(a? - a^)} 22 (30) + "1 -JI (^ + x) /0 + - 9 ZdJ-at This solution was obtained by Heaviside*)- and by Poincare', by entirely different processes. We shall now suppose that initially there is no current in the line, and that the potential is zero except between two points a?i < a*. That is F (cc) = 0, except when a^ < x < xz> * See Gray and Mathews, Treatise on Bessel Functions. t " The General Solution of Maxwell's Electromagnetic Equations in a Homo- geneous Isotropic Medium." Phil. Mag. Jan. 1889, p. 30 ; Papers, Vol. n. p. 478, eq. (40). w. E. 35 546 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. Then for x — #2 > at or #! — x > at all the functions occurring have the value zero, therefore V is zero. This at once distinguishes the solution from that for the case of the cable, for the disturbance does not arrive at x until the time fp J_ f*\ t = — - on the right, Cv or t = ^-? on the left. a When x — x± > at > x — #2 we have This represents the disturbance while the wave is passing over, for a point on the right. In like manner for a point on the left, for #j — x < at < #2 — x, (33) V Finally, at later instants, x — xl < at or #2 — x < at, a—bt rx—xl / % (34) V= -YJ^ F (a - X) (b + | This represents the disturbance after the wave, travelling with velocity a, has passed on. Accordingly the solution, while repre- senting a wave travelling with the velocity a, as in free space, differs from that case in that there remains a residue, or tail to the wave, which does not fall to zero however great the time. The exponential factor shows that the disturbance, both in the wave proper, and in the tail, is continually becoming attenuated. Thus when successive impulses are transmitted, each leaves a tail, which interferes with all the succeeding waves, and the possibility of telephonic speaking depends not only on the attenua- tion and distortion with the distance, but on the magnitude of the tail of the wave. The tail also explains the discrepancies that existed between the results of the attempts made to determine 255] ELECTROMAGNETIC WAVES. 547 the velocity of electric waves by means of telegraph lines, what was generally observed being more probably the maximum dis- turbance than the front of the wave. In order to give a concrete idea of the nature of the propagation, and to afford a means of comparison with the electrostatic theory, we shall suppose that the function V is constant and equal to F0, from a^ to x.2. We shall also change our units of time and length, by taking the relaxation-time r = 1/6 for the unit of time, and the relaxation distance, d — ar = ajb, as the unit of length. Accordingly putting ., t , , , x — #2 b (x — #2) b\ «=- = &«, *2=^_ = _A__'( __^ we have (35) F for a point on the right while the wave is passing over. This equation was given by Heaviside in 1888, who carefully refrained from giving his method of deduction, remarking " since, although they were very laboriously worked out by myself, yet as mathe- matical solutions, are more likely to have been given before in some other physical problem than to be new*." Inasmuch as not only Heaviside's results but any others were overlooked by the three French mathematicians quoted, who published results six years later, we may conclude that in the English writer modesty and original productiveness were more strongly developed than historical research. (This modesty is not maintained on the same plane throughout.) Inserting in the value of V the series for /0 (dropping accents), _ O=oo//2 I/2V7 (36) and developing each term by the binomial theorem, we obtain * "Electromagnetic Waves." Phil. Mag. 1888; Papers, Vol. n. p. 373, eq. (52). 35—2 548 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. Changing the order of summation and putting q=q'+p> _ »=oo o' = oo / 1 \ p /2O' ,.2» (38) /,^-2 *.»*£ which by the definition of Ip* is equal to p— oo / _ T \p . .2jp (39) -- Differentiating by £ and adding to itself, which by the formula connecting the derivative of Ip with the contiguous function [Gray and Mathews (141)] may be written »=- -i^ / , Inserting in the integral (35), and integrating from x to t, (42) F- The terms free from a; may, by writing out the sums for Ip, and collecting terms, be shown to be equal to ff. We, therefore, finally obtain for V / x (43) From this the values of F have been calculated and plotted by Mr W. P. Boynton in Fig. 99, which shows the distribution of potential along the line to the right, for times t = 1, 2, 3, 4, 5 times * From (27) we have for the present Ip (f), 255] ELECTROMAGNETIC WAVES. 549 the relaxation-time. This may be compared with Fig. 96 showing the electrostatic theory. The rise of potential at particular points, FIG. 99. as a function of the time, is shown in Fig. 100, which is the analogue of the outer curve in Fig. 98. The different curves FIG. 100. are for the points at 1, 2, 3, 4 times the relaxation-distance from the start. The potential produced by connecting the battery for a definite time, and then removing it may, as before, be obtained by taking the differences of two curves relatively displaced. In this way the effect of the initial potential shown by the rectangle in a is shown in 6, c, d, e, Fig. 101 for the times *2, *4, *6, 1*6 times the relaxation- 550 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. time, the dotted lines showing the wave as it would be if there were no resistance. The last figure,/ Fig. 101, shows the effect of short- ening the duration of a signal, the tail left being noticeably smaller. a — X X X \ d - X r — T : i f FIG. 101. From these figures we may obtain an idea of the distance to which telephony is possible, if we know the relaxation-distance of the line. With ordinary land-lines the relaxation-distance is of the order of several hundred kilometers. This has made speaking possible between Boston and Chicago. Obviously it is of importance to make the relaxation- distance d = v . 2L/R as great as possible by making the distance between the wires great, and using large copper wires. 256. Terminal Conditions. In the preceding examples we have considered the line to be without end. In many practical 255, 256] ELECTROMAGNETIC WAVES. 551 cases we wish to know what goes on in a line of finite length, when the ends are connected to any electromagnetic systems whatever, both when the systems are left to themselves and when electro- motive-forces are applied. Space is lacking for more than the briefest possible treatment of this matter, which is very fully treated in Heaviside's papers on wires to which reference has already been made. The method of procedure is the same in every case. We shall make use of the equations <•> si ,,dv <2> -te = Kdi- Let us seek particular solutions of the form (3) Inserting in (i) we have (4) and if we put (5) we have the equation for u, (6) , The solution of this is (7) u = A cos fjLO) + B sin IJLX, where A and B are constants to be determined. From (2) we obtain — -•=- = R\u — K\ (A cos fjLX + B sin /JLX), (8) K\ w = — (B cos fjux — A sin fix). The functions (3) are solutions of the differential equations whatever the value of X. The values of X that are admissible are determined by the terminal conditions. We shall take as an example one of the simplest cases possible. Let us suppose that at one end, where x = I, the two wires are connected, while at the 552 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. other they are connected with the plates of a condenser of capacity K0. The conditions are then F=0, x = l, (9) Applying these to the solutions (3), (7), (8), we have A cos fi,l + B sin /zZ = 0, Eliminating A/B we obtain J£ (I i) /itan^=^-, Jtt-o a transcendental equation to determine /*. This has an infinite number of roots, which may be real or complex. When these are determined, X is determined for each root by the equation (5). Thus we find that there are an infinite number of possible periods for the free vibration, corresponding to the n periods for a system with n degrees of freedom. The equation (11) corresponds to the determinantal equation of § 241 (10). The ratio B/A is determined by (10). The determination of the absolute value of the co- efficients depends on the initial conditions. Having found an infinite number of particular solutions, any root fjL8 being distinguished by its suffix, the general solution is (12) F= 2«eV (As cos fax + B8 sin fax), where we sum for all the roots. If the potential is initially given by (13) V=F(x\ t = 0, we must have (14) F(x) = S« (A8 cos fax + Bs sin /JLSX). The problem to be solved is then that of developing an arbitrary function of a? in a trigonometric series of the form (14) where the fa's are the roots of a certain transcendental equation, namely (n). The problem is in general of considerable com- plexity, and we shall content ourselves with referring to Heaviside, who has treated it at great length. If there is no condenser at # = 0, but the circuit is open, equation (n) is tan jbl = oo 256] ELECTROMAGNETIC WAVES. 553 and the roots are (2S + 1)7T (14) /**= ~2~~T The series (14) then becomes a Fourier's series, with the even terms omitted. If R = 0 we see by (5) that X is a pure imaginary, so that all the oscillations are harmonic. The wave-length L is or the length of the wires is an odd number of quarter wave- lengths. If on the other hand the capacity KQ is infinite, we get (17) /*tan/zZ = 0, which is the same as if we had considered the circuit closed at the origin also, putting V= 0, for x = 0, from which (7) and the first of (10) would give sin pi = 0, (18) STT , 21 *~T' Z = 7' The length of the line is then a multiple of a half wave- length. The two cases correspond to the cases of an organ pipe open at one end and closed at the other, or closed at both ends. These conclusions have been verified by experiment. The above theory applies, for instance, to the experiments of Saunders cited above. The method employed in this example is typical of the general process, for the terminal conditions, of whatever nature, are given in the form of an ordinary linear differential equation in the time, involving the derivatives of V and /. Applying this to our assumed solutions (3) introduces algebraic functions of X, so that, eliminating by means of (5), we obtain an equation of the form (19) tan ^ =<£(/*), where is an algebraic function. The case we have considered is the simplest case of this transcendental equation. 554 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 257. Equations for Bodies in Motion. All the equations of this chapter have been deduced on the supposition that all the media were at rest. In deducing their extension to the case of media in motion we shall follow the method of Hertz, as given in the last and crowning paper of his remarkable researches*. We shall suppose the media to be moving at every point with velocities v whose components at any point are a, /3, 7. The medium is not supposed necessarily to be moving like a rigid body — it may be deformed in any manner. At the surface of separation between two media, although the velocity may be discontinuous its normal component must be continuous, in order to preclude the occurrence of vacant spaces. The fundamental assumption made by Hertz is that as the medium moves or is distorted, the lines of force are carried by the medium so as to pass through the same material points. That is, this would be the effect of the motion if it were the only influence at work to change the field. Besides this, we have the usual effects that appear in bodies at rest. Let 3£, §), 3» %> $R> $1, represent the field at any point at rest with respect to the coordinate-axes. The total change in $ at a point in motion will depend on several causes, the first being the change that is instantaneously taking place at the fixed point through which the material point happens to be passing. This we d¥ shall denote by ^- . Secondly the point is displaced to new parts ot of the field where the forces are different. The sum of these two parts we shall call , dt~~dt 8a; dt dydt dz dt~ 8* 8# By 7a* If a small element normal to the X-axis of area dS were displaced parallel to itself, the flux through it would vary as just stated. But if the element rotates, it takes in new amounts of flux* At the start the flux through it was $dS, but as it turns, * ^ it acquires a projection normal to the F-axis at the rate ^-, con- sequently its flux in the positive direction decreases from this * "Ueber die Grundgleichungen der Elektrodynamik fiir bewegte Korper." Wied. Ann. 41, p. 369, 1890 ; Trans, p. 241. 257] EQUATIONS OF ELECTROMAGNETIC FIELD. 555 cause at the rate g) r-, and in the same manner from its ^-projection at the rate 3 ^- • But the area of the X-projection of the element O /O O is also increasing, at the rate ^- in the F-direction and ~ in the ^-direction. From this cause the flux increases at the rate We have therefore to replace the term ^- in equations (A) by ut the sum , O , (2) ^r + a^- + £5- +7 — + ^- ^- dt dx dy ' dz (dy dz) ( dy dz We have thus added in virtue of the motion two parts, the first of which is the component of the curl of the vector whose components are 79-03, «3--y£ /3£-«g), that is the vector product of the induction of the field and the velocity. The last term is the component of the velocity times the divergence of the induction. We may therefore abbreviate our equations which replace both (A) and (B) thus (A") A |?f + curl Vgu + v div g + 4w?l = curl H, (ot ) h ^ L) L.D 1 A M O f\ 4A-A^i JAN 2 U 1963 . 22J«L'63jtt REC'D LD SEP 2 3 '64 -10 AM REC'D l. D i LD 21A-50w-12/60 (B6221slO)476B General Library University of California Berkeley UNIVERSITY OF CALIFORNIA LIBRARY ibt