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 THE LIBRARY 
 
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 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES
 
 in tftje City 0f ^jeitr ^0rk 
 
 ^jblicatio:n^ number one 
 
 OF THE 
 
 ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEABOH 
 
 Established December 17, 1904 
 
 FIELDS OF' FOBCE 
 
 A COURSE OF LECTURES IN MATHEMATICAL PHYSICS 
 DELIVERED DECEMBER 1 TO* 23, 1905 
 
 BY 
 
 VILHELM FRIMAN KOREN BJERKNES 
 
 PROFESSOR OF MECHANICS AND MATHEMATICAL PHYSICS IN THE UNIVERSITY OF STOCKHOLM 
 LECTURER IN MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY, 1905-6 
 
 Neto Yotk 
 THE COLUMBIA UNIVERSITY PRESS 
 
 THE MACMILLAN COMPANY/ gents 
 
 LONDON : MACMILLAN CO., lJ | 
 1906 . 1 ' 

 
 TJ- 
 
 M 
 
 Pisess or 
 Thc New En* PniNTiNQ company 
 
 Uf<C<9TCII, P*.
 
 in the City of ^txo ^arU 
 
 PUBLICATION NUMBER ONE 
 
 OF IHK 
 
 ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEAROH 
 
 Established Deceiniber 17, 1904 
 
 FIELDS OF FOECE 
 
 A COURSE OF LECTURES IN MATHEMATICAL PHYSICS 
 DELIVERED DECEMBER 1 TO 23, 1905 
 
 BY 
 
 VILHELM FRIMAN KOREN BJERKNES 
 
 PROFESSOR OF MECHANICS AND MATHEMATICAL PHYSICS IN THE UNIVERSITY OF STOCKHOLM 
 LECTURER IN MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY, 1905-6 
 
 THE COLUMBIA UNIVERSITY PRESS 
 
 THE MACMILLAN COMPANY, Agents 
 
 LONDON: MACMILLAN CO., Ltd. 
 
 1906
 
 y
 
 
 On the seventeenth day of December nineteen hundred and four, 
 Edward Dean Adams, of New York, established in Cohimbia 
 University " The Ernest Kempton Adams Fund for -Physical 
 Research" as a memorial to his son, Ernest Kempton Adams, 
 who received the degrees of Electrical Engineering in 1897 and 
 Master of Arts in 1898, and who devoted his life to scientific re- 
 search. The income of this fund is, by tlie terms of the deed of 
 gift, to be devoted to the maintenance of a research fellowship and 
 to the publication and distribution of the results of scientific re- 
 search on the part of the fellow. A generous interpretation of 
 the terms of the deed on the part of Mr. Adams and of the Trus- 
 tees of the University has made it possible to print and distribute 
 the following lectures as a publication of the Ernest Kempton 
 Adams Fund :
 
 FIELDS OF FORCE. 
 
 I. 
 
 ELEMENTARY INVESTIGATION OF THE GEO- 
 METRIC PROPERTIES OF HYDRO- 
 DYNAMIC FIELDS. 
 
 Introductory. 
 
 The idea of electric and magnetic fields of force was intro- 
 duced by Faraday to avoid the mysterious idea of an action at 
 a distance. After the victory which Maxwell's theory gained 
 through the experiments of Hertz, the idea of these fields took 
 its place among the most fruitful of theoretical physics. 
 
 And yet if we ask, what is an electric or a magnetic field of 
 force ? no one will be able to give a satisfactory answer. We have 
 theories relating to these fields, but we have no idea whatever of 
 what they are intrinsically, nor even the slightest idea of the path 
 to follow in order to discover their true nature. Above all other 
 problems which are related to fields of force, and which occupy 
 investigators daily, we have therefore the problem of fields of 
 force, viz., the problem of their true nature. 
 
 My lectures will not give the solution of this problem, but I 
 should be happy if they should contribute to a broadening of our 
 view of it. 
 
 AVhat I wish to insist upon especially, is this. Besides elec- 
 tric and magnetic fields there exist other fields which have strik- 
 ingly analogous properties, and which have, therefore, exactly the 
 same claim to be called fields of force. The investigation of these 
 other fields side by side with the electric and magnetic fields will 
 be advantageous, I think, in broadening our view of the problem, 
 especially as the true nature of these other fields of force is per- 
 1 1
 
 2 FIELDS OF FORCE. 
 
 fectly plain and intelligible, as intelligible, at least, as anything 
 can be in the limited state of onr power of nnderstanding. 
 
 These other fields of force exist in material media which are 
 in suitable states of motion. They are perfectly intelligible in 
 this sense, that their properties can be deduced from the principles 
 of dynamics. For the special case when the material medium is a 
 perfect fluid, the properties of these fields have been extensively 
 explored, and therefore our main subject will be the investigation 
 of the properties of hydrodynamic fields of force and their com- 
 parison with electric or magnetic fields. 
 
 The results which I shall bring before you were discovered 
 originally by my late father. Professor C. A. Bjerknes, of Chris- 
 tiania. But I will bring into application here new methods, which 
 allow us to find the results with much greater generality, and at the 
 same time with greater facility.* 
 
 My lectures will be divided into two parts. The first two lec- 
 tures will be devoted to the development of the properties of hy- 
 drodynamic fields by elementary reasoning and experiment ; the 
 following lectures will give the analytical investigation of the same 
 subject, based upon Euler's equations of motion for a perfect 
 fluid and Maxwell's equations for the electromagnetic field. 
 
 1. Field-x'edovs. — The electric field may be described in the 
 simplest case by either of two vectors, the electric jinx (or dis- 
 placement), or the electric field intensity (or force). In the same 
 way the magnetic field may be described by the magnetic flux (or 
 induction), or the magnetic ^le/fZ intensity (or force). The flux and 
 the field intensity will differ from each other only by a constant 
 factor, the electric or the magnetic inrJiirtivity of the medium which 
 supports the field, the flux being always the product of the field 
 intensity into the inductivity. 
 
 * For the historical development of C A. Bjerknes' ideas compare z'. Bjerknes: 
 C. A. Bjerknes, d'ediiehtnissrefle gehalten in der Geselhchnfl cler Wissenschuften zu 
 Christiania dm April, 1903. German translation, Leipzig, 1903. For the de- 
 velopment of the theory according to C. A. Bjerknes' methods, and for more 
 complete description of instruments and experiments, see v. Bjerknes, Vorlesungen 
 Ober hxjrhofhjnnmi»che Fernkriifte nach C A. Bjerknes' Theorie. Vols. I and II, 
 Leipzig, 1900-02.
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. 3 
 
 Ou the otlier hand, the fiehl of motion in any moving liquid 
 may also be described by either of two vectors, which are 
 related to each other in the same way as the flux and the field 
 intensity of electric or magnetic fields. The first of these 
 vectors is the velocity, and the other the j^roduct of the velocity 
 into the density. As to its dynamical significance, this vector is 
 the momentum per unit volume or the specific momentum in the 
 moving fluid. 
 
 In hydrodynamics we thus meet with two vectors which are 
 connected in a similar way as the flux and the field intensity in the 
 electric or magnetic field. This parallelism at once invites a com- 
 parison. There is only the question as to how the different vec- 
 tors should be paired, and this can be answered only by a closer 
 analysis of their properties. This brings us to the question of our 
 knowledge of the fields. 
 
 2. Geometric and Dynamic Properties of tlie Fields. — The extent 
 of our knowledge of the diiferent kinds of fields differs greatly. All 
 the properties of the hydrodynamic fields follow directly from the 
 most trustworthy laws of nature, that is from the principle of the 
 conservation of the mass, and from the principles of dynamics. 
 With reference to electric or magnetic fields, on the contrary, we 
 have only formal theories. First, we have an extensively devel- 
 oped geometric theory of the distribution in space of the vectors 
 which describe the field. And then, in a more or less superficial 
 connection to this geometric theory, we have a very much less 
 developed theory of the dynamic j^f'ojjerties of the fields. 
 
 Taking the facts as they lie before us, we shall be obliged, 
 therefore, to give to our theory a dualistic form, comparing 
 separately the geometric and the dynamic properties of the two 
 kinds of fields. It may be reserved for the future to penetrate to 
 the central point, where the geometry and the dynamics of the 
 question are perfectly united, and thus make the comparison of 
 the two kinds of fields perfectly easy. 
 
 In this lecture we will consider the geometric properties of the 
 fields.
 
 4 FIELDS OF FORCE. 
 
 3. Properties of the Field Vectors at a Surface of Separation. — 
 A characteristic geometrical property of the vectors at a surface of 
 separation of two media shows at once liow the fluxes and field 
 intensities should be paired with the hydrodynamic vectors. As 
 is well known, at a surfiice of separation the normal component 
 of the flux is always continuous, while the normal component of 
 the field intensity is necessarily discontinuous, if the inductivity 
 suddenly changes at the surface. On the other hand, at any sur- 
 face of separation in a moving liquid the normal component of the 
 velocity is continuous, corresponding to the normal component of 
 the flux. Otherwise we should have at the surface either creation 
 or annihilation of matter, or a break in the continuity, both of 
 which we consider excluded. From the continuity of the normal 
 component of the velocity follows the discontinuity of the specific 
 momentum for tiie case where the density of the fluid suddenly 
 changes at the surface. This vector, therefore, has a discontinuity 
 similar to that of the field intensity, and it follows at once that the 
 correspondence of the vectors is possible if the velocity correspond 
 to the flux and the specific.niomentum to the field intensity. Fur- 
 ther, as the flux is the product of the field intensity into the induc- 
 tivity, it also follows that not the density, but the reciprocal of the 
 density, or the specific volume, corresponds to the inductivity. 
 
 Just as the density gives the measure of the inert resistance of 
 the matter to the motion, the specific volume gives the measure of 
 the readiness of the matter to take inotion. The specific volume 
 may therefore also be termed the m.obilitij of the fluid. We thus 
 get the correspondence : 
 
 velocity flux, 
 
 sj)ecific momentum field intensity, 
 
 mobility inductivity. 
 
 We have now to examine more closely the content of this 
 correspondence. 
 
 4. Charged Particle — Expandiiu/ or Contracting Particle. — 
 TiCt us start with the simplest object met with in the first investi-
 
 INVESTIGATION OF GEOMETRIC TROPERTIES. 5 
 
 gations of electricity, namely, with an electrically charged particle. 
 In the field belonging to this particle the vectors are directed 
 radially outwards if the particle has a positive, and radially in- 
 wards if it has a negative charge, and their intensity decreases as 
 the inverse square of the distance. 
 
 It is seen at once that an expanding particle which is contained 
 in an incompressible fluid, such as water, will produce a field of 
 exactly the same geometrical nature as the field belonging to the 
 positively charged particle. It will produce a radial current 
 directed outwards, in which, as a consequence of the incompressi- 
 bility, the velocity, and therefore the specific momentum will de- 
 crease as the inverse square of the distance. In like manner a con- 
 tracting particle will be surrounded by a current directed radially 
 inwards, and will thus correspond to a negatively charged particle 
 (see Fig. 4, a and b, below). 
 
 This comparison of a radial electric and a radial hydrodynamic 
 field has one difficulty, however. The idea of an always expand- 
 ing, or of an always contracting particle, is impossible. Therefore 
 to make the comparison possible, not only for a moment, but for 
 any length of time, we are obliged to consider a motion of peri- 
 odic expansions and contractions, or a pulsating motion. In this 
 case there wnll be no absolute diiference between a positive and a 
 negative pulsating body. But two pulsating bodies may be in 
 exactly the same mutual relation to each other as an expanding 
 and a contracting body. For if they are pulsating in oppo- 
 site phase, the one will always be expanding while the other is 
 contracting, and vice versa. We can then distinguish these two 
 pulsating bodies from each other by opposite signs, just as w^e 
 do two oppositely charged particles, and we can represent the 
 mean state of motion in the surrounding radial field by a vector 
 directed outwards from the pulsating particle which we call, by 
 convention, positive, and inwards to the pulsating particle which 
 we call negative. 
 
 5. Complex Fields. — If we consider two charged particles 
 whose dimensions are sufficiently small in comparison to the dis-
 
 FIELDS OF FORCE. 
 
 tance between them, a field is produced which is the simple result 
 of the superposition of the two radial fields. The lines of force of 
 the complex fields may be found by the well-known constructions 
 of superposition. The results are the well known curves running 
 from one charged particle to the other for the case of opposite 
 charges, and the diverging curves with a neutral point between 
 the two charged particles for the case of charges of the same sign 
 (see Figs. 5 and 6 below). 
 
 In exactly the same way, if we consider two particles which 
 have a motion of expansion or contraction and which are suffi- 
 ciently small in comparison to the distance between them, the 
 radial currents produced by each will simply be superposed, and 
 the current lines, by which the complex field may be represented, 
 can be found by exactly the same construction as in the case of the 
 corresponding electric fields. And this result may be transferred 
 at once to the case of vibratory motion ; particles pulsating in the 
 same phase, expanding simultaneously and contracting simul- 
 taneously, will give a field corresponding geometrically to that 
 produced by particles carrying charges of the same sign, and 
 oppositely pulsating particles will produce a field corresponding 
 geometrically to that produced by particles which carry opposite 
 charges. 
 
 Just as we combine the fields of two charged particles, we can 
 combine the fields of any inunber of charged particles, and to a field 
 of any complexity obtained in this way we can construct a corre- 
 sponding hydrodyuamic field, obtained by the combination of the 
 fields of the corresponding system of expanding and contracting 
 particles, or of pulsating particles for the case of vibratory motion. 
 An extensive geometric analogy between hydrodyuamic and elec- 
 trostatic fields is thus found. 
 
 (i. hitv'nmralhi Polarized Bodies!. 0-'<cil/afinr/ Bodie>!. ^ What 
 we have said of electrified particles and tiie electric fields produced 
 by them may l)e repeated for magnetic poles and the correspond- 
 ing magnetic fields. But now tiie reservation must be made, that 
 magnetic ])oles are in reality mere fictions. For a distribution of
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. 7 
 
 magnetic poles we can, however, substitute a state of intrinsic 
 polarization, Avhich may be considered as the real origin of the 
 mao-netic field. Such states of intrinsic polarization are also met 
 with in electricity. Thus the pyro-electric crystal seems to give a 
 perfect electric analogy to the permanent magnet. 
 
 Let us now for the system of magnetic poles, by which a mag- 
 net can be represented symbolically, substitute the corresponding 
 system of expanding and contracting particles. In the region of 
 the fluid which corresponds to the magnet the total sum of ex- 
 pansions and contractions will be zero. But the field produced 
 in the exterior space by these expansions and contractions may 
 also be produced by quite another interior motion, involving no 
 expansion or contraction at all. For consider a closed surface 
 consisting of fluid particles, and surrounding the region of the 
 fluid which corresponds to the magnet. This material surface has 
 a certain motion ; it will advance on that side where the expand- 
 ing particles are situated, and recede on that side where the con- 
 tracting particles are situated. The result is a motion of the sur- 
 face as a whole, directed from the regions of contraction towards 
 the regions of expansion. And, as the sum of the expansions and 
 the contractions is zero, the volume within the surface will remain 
 unchanged during this motion. 
 
 Now the motion produced outside the surface will be entirely 
 independent of what goes on within it, provided only that the 
 motion of the surface itself remains unchanged. We can there- 
 fore do away with the expansions and contractions, and suppose 
 the volume within the surfiice filled with an incompressible fluid, 
 subject to the action of forces which give these fluid masses a 
 motion consistent with the required motion of the surface. 
 
 We have thus arrived at the following result : a motion of in- 
 compressible fluid masses, produced by suitable forces, can be 
 found, which will set up an exterior field similar to that set up by 
 a system of expanding and contracting particles, provided that the 
 sum of the expansions and contractions is zero. And this equivalence 
 corresponds exactly to the equivalence between the representation
 
 8 FIELDS OF FORCE. 
 
 of a magnet by a distribution of poles, and by a state of intrinsic 
 polarization. The hydrodynamic model of a body in a state of 
 intrinsic polarization is, therefore, a body consisting of incompressi- 
 ble fluid masses, moved through the surrounding fluid by suitable 
 exterior forces (see Fig. 8 below). 
 
 We have considered' here, for simplicity, only the instantaneous 
 state of motion. In the case of periodic motion we get an equiv- 
 alence between a system of oppositely pulsating particles and a 
 fluid body which takes forced oscillations under the influence of 
 suitable exterior forces. 
 
 7. Fields in Hcferoc/eneous Media. — The results already de- 
 veloped depend, essentially, upon the supposition that the fluid 
 surrounding the moving bodies is homogeneous and incompres- 
 sible. The case when it is heterogeneous must be examined 
 separately. That the heterogeneity has an influence upon the 
 geometric configuration of the field, is obvious. For only when 
 the fluid is perfectly homogeneous will there exist that perfect 
 symmetry in the space surrounding an expanding particle, which 
 entitles us to conclude that a perfectly symmetrical radial current 
 will arise. But if on one side of the expanding particle there ex- 
 ists a region where the fluid has a different density, the symmetry is 
 lost, and it is to be expected that the configuration of the field 
 will be influenced by tliis fact. On the other hand, as is well 
 known, any heterogeneity of the dielectric has a marked influence 
 upon the geometric configuration of the electric field, giving rise 
 to the ])henomena of electrification by influence. 
 
 Now, will the influence of the heterogeneity in the two cases be 
 of similar nature? To examine this question we shall have to 
 develop a very simple principle relating to the dynamics of fluids, 
 our considerations above having been based only on the principle 
 of the conservation of mass. 
 
 8. Principle of Kinetic Buoyancy. — Consider a cylinder, with axis 
 vertical, containing a body and, apart from the body, completely 
 filled with water. The condition of equilibrium will depend upon 
 the l)uoyancy, according to the Archimedian principle. If the body
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. 9 
 
 has exactly the density of the water, the buoyancy will balance the 
 weight of the body, and it will remain in equilibrium in any posi- 
 tion. If it be lighter, its buoyancy will be greater than its weight, 
 and it will tend to move upwards. If it be heavier, its buoyancy 
 will be less than its weight, and it will tend to move downwards. 
 Thus, if we have three cylinders, each containing one of three such 
 bodies, the light body will rise to the top, the heavy body will 
 sink to the bottom, and the body of the same density as the water 
 will remain in any position. 
 
 This static buoyancy depends upon the action of gravity. 
 But there exists a corresponding dynamic buoyancy, which is 
 easily observed as follows: To do away with the influence of 
 gravity, lay the cylinders with their axes horizontal, and let the 
 bodies be in the middle of the cylinders. Then give each cyl- 
 inder a blow, so that they move suddenly five or ten centimeters 
 in the direction of their axes. The following results will then be 
 observed : 
 
 1. The body which is lighter than the water has moved 
 towards the front end of its cylinder, and thus has had a motion 
 through the water in the direction of the motion of the water. 
 
 2. The body which has the same density as the water has 
 moved exactly the same distance as the water, and thus retained 
 its position relative to the water. 
 
 3. The body which is heavier than the water has moved a 
 shorter distance than its cylinder, and thus has had a motion 
 through the water against the direction of motion of the water. 
 
 If we give the cylinders a series of blows, the light body will 
 advance through the water nntil it stops against the front end. The 
 body of the same density as the water will retain its place, and the 
 heavy body will move backwards relatively to the cylinder, until 
 it stops against the end. The effect is strikingly analogous to 
 the effect of statical buoyancy for the case of the cylinders with 
 vertical axes, and this analogy exists even in the quantitative 
 laws of the phenomenon. 
 
 These quantitative laws are complicated in case the bodies are 
 2
 
 10 FIELDS OF FORCE. 
 
 free to move through the water, but exceedingly simple when they 
 are held in an invariable position relative to the water by the 
 application of suitable exterior forces. 
 
 This exterior force is nil in the case when the body has the 
 same density as the water. The body then follows the motion of 
 the surrounding water masses, subject only to the force resulting 
 from the pressure exerted by them. The motion of the body is 
 subject to the fundamental law of dynamics, 
 
 force = mass x acceleration. 
 
 As the l)ody has both the acceleration and the density of the 
 surrounding water masses, the force is equal to the product of the 
 acceleration into the mass of the water displaced by the body. 
 And this law evidently will be true even for the heavy or the 
 light body, provided only that they are held by suitable forces at 
 rest relatively to the moving water. For, the state of motion out- 
 side the body is then unchanged, and the pressure exerted by the 
 water against any surface does not at all depend upon the condi- 
 tions within the surface.- Thus we find this general result, which 
 is perfectly analogous to the Archimedian law : 
 
 Any hoiJy which 'participates in the translatory motion of a fluid 
 mass is subject to a kinetic buoyancy equal to the product of the 
 acceleration of the translatory motion multiplied by the mass of water 
 displaced by the body. 
 
 This law obviously gives also the value of the exterior force 
 which must be applied in order to make the body follow exactly 
 the motion of the fluid, just as the Archimedian law gives the 
 force which is necessary to prevent a body from rising or sinking. 
 This force is nil, if the body has the same density as the water, it 
 is directed against the direction of the acceleration, if the body is 
 lighter, and in the direction of the acceleration, if the body is 
 heavier. And, if no such force act, we get the result, illustrated 
 by the experiment, that the light body moves faster than the 
 water and tlic iieavy body shnver, and thus, relatively, against the 
 water.
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. H 
 
 9. Influence of Heterogeneitie.^ in the Electric or Magnetic and 
 in the Analogous Hydvodynamic Field. — From the principle of 
 kinetic buoyancy we thus find the obvious law, that, in a hetero- 
 geneous fluid, masses of greater mobility take greater velocities. 
 The mobility therefore influences the distribution of velocity, just 
 as the inductivity influences the distribution of the flux in the 
 electric, or magnetic field. For at places of greater inductivity we 
 have greater electric, or magnetic flux. 
 
 To consider a simple example, let us place in a bottle filled with 
 water a light sphere, a hollow celluloid bdl, for instance, attached 
 below with a fine string. And in another bottle let us suspend in a 
 similar manner a lead ball. If we shake the bottles, the celluloid 
 ball will take very lively oscillations, much greater than those of 
 the water, while the lead ball will remain almost at rest. With 
 respect to their induced oscillations, they behave, then, exactly as 
 magnetic or diamagnetic bodies behave with respect to the induced 
 magnetization when they are brought into a magnetic field ; the 
 light body takes greater oscillations than the water, just as the 
 magnetic body takes greater magnetization than the surrounding 
 medium. The heavy body, on the other hand, takes smaller oscil- 
 lations than the water, just as the diamagnetic body takes smaller 
 magnetization than the surrounding medium. And thus relatively, 
 the heavy body has oscillations opposite to those of the water, just 
 as the diamagnetic has a relative polarity opposite to that of the 
 surrounding medium. 
 
 10. Bef faction of the Lines of Flow. — The influence which the 
 greater velocity of the masses of greater mobility has upon the 
 course of the tubes of flow is obvious. At places of greater 
 velocity the tubes of flow narrow, and at places of smaller velocity 
 widen. They will thus be narrow at places of great, and wide at 
 places of small mobility, just as the tubes of flux in the electric or 
 magnetic field are narrow at places of great, and wide at places of 
 small inductivity. If we limit ourselves to the consideration ot 
 the most practical case, when the values of the mobility or of the 
 inductivity change abruptly at certain surfaces, we can easily prove
 
 12 FIELDS OF FORCE. 
 
 that tlie influence of the hete^'ogeneity in the two kinds of fields 
 corresponds not only qualitativity but quantitatively. 
 
 AVe suppose that the bodies which have other density than the 
 surrounding fluid are themselves fluid. It is only in experiments 
 that, for practical reasons, we must always use rigid bodies. At the 
 surface of separation between the surrounding fluid and the fluid 
 body the pressure must have the same value on both sides of the 
 surface. This is an immediate consequence of the principle of equal 
 action and reaction. From the equality of the pressure on both 
 sides of the surface it follows, that the rate of decrease of the pres- 
 sure in direction tangential to the surface is also equal at adjacent 
 points on each side of the surface. But this rate of decrease is the 
 gradient, or the force per unit volume, in the moving fluid. And, 
 as the acceleration produced by the force per unit v^olume is in- 
 versely proportional to the density, we find that the tangential ac- 
 celeration on tlie two sides of the surface of separation will be 
 inversely proportional to the density. Or, what is the same thing, 
 tlie product of tlie tnngeiitial acceleration into the density will have the 
 same value on both sides of the surface. 
 
 From this result there can not at once be drawn a general con- 
 clusion on the relation of the tangential components of the velocity, 
 or of the specific momentum. For two adjacent particles, which 
 are accelerated according to this law, will at the next moment no 
 longer be adjacent. If, however, the motion be periodic, so that 
 every particle has an invariable mean position, then adjacent par- 
 ticles will remain adjacent particles, and from the equality of the 
 tangential components of the products of the accelerations into the 
 densities at once follows the equality of the tangential components 
 of the products of the velocities into the densities. Thus, 
 
 //( the case of vibratory motion the specific momentum lias con- 
 tinuous tanf/ential components at the surface of separation of two 
 media of different mobility. 
 
 The law for the specific momentum is thus exactly the same 
 as for the electric or magnetic field intensities, which have con- 
 tinuous tangential components at the surface of separation of two
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. 13 
 
 media of different inductivity. As we have already found (3), 
 the law for the velocity is the same as for the electric or the mag- 
 netic flux. We see then, that the conditions fulfilled at a surface 
 of separation by the hydrodynamic vectors on the one hand, and 
 by the electric or magnetic vectors on the other, are identically 
 the same. The lines of flow and the lines of flux will show 
 exactly the same peculiarity in passing a surface of separation. 
 And, as is shown in all treatises on electricity, this peculiarity 
 consists in a refraction of the lines so that the tangents of the 
 angles of incidence and refraction are in the same ratio as the induc- 
 tivities on the two sides of the surface. In the hydrodynamic 
 case these tangents will be in the same ratio as the mobilities on 
 the two sides of the surface. This refraction gives to the tubes of 
 flow or of flux the sudden change of section which corresponds to 
 the increase or decrease of the velocity or of the flux in passing 
 from one medium into the other. 
 
 This refraction of the hydrodynamic lines of flow according to 
 the same law as that of the refraction of the electric or magnetic 
 lines of force is a phenomenon met with daily in the motion of super- 
 imposed liquids of different specific weights. If I suddenly move a 
 glass partly filled with mercury and partly with water, the mercury 
 rises along the rear wall of the glass, while the water sinks in front. 
 During the first instant of the motion, before we get the oscillations 
 due to gravity, the law of the refraction of the tubes of flow is ful- 
 filled at the surface of separation. Whatever be the course of the 
 tubes of flow at a distance from the surface, at the surface they 
 will be refracted so that the tangents of the angles of incidence 
 and of refraction are in the ratio of the mobilities of the mercury 
 and of the water, or in the inverse ratio of their densities, 1 : 13. 
 
 We get the same law of refraction at the surface of separation 
 of water and air, the tangents of the angles being then in the 
 ratio, 1 : 700. The accident of daily occurrence, in which a glass 
 of water flows over as the result of sudden motion, is thus the conse- 
 quence of a law strictly analogous to that of the refraction of the 
 electric or magnetic lines of force.
 
 14 FIELDS OF FOECE. 
 
 11. Experimental Verijieations. — We have been able from 
 kinematic and dynamic principles of the simplest nature to show 
 the existence of an extended analogy in the geometric properties of 
 the electric or magnetic, and hydrodyuamic lields. The dynamic 
 principles which form the basis of this analogy we have illus- 
 trated by experiments of the simplest possible nature. But even 
 though we have perfect faith in the truth of the results, it is 
 desirable to see direct verifications of them. Some experiments 
 have been made towards finding verifications, but not as many, 
 however, as might have been desirable. 
 
 These experiments were made with water motions of vibra- 
 tory nature, produced by pulsating or oscillating bodies, using 
 instruments constructed mainly for the investigation of the dyna- 
 mic properties of the field, which will be the subject of the next 
 lecture. Such pulsations and oscillations can easily be produced 
 by a pneumatic arrangement involving a generator which pro- 
 duces an alternating current of air. 
 
 12. The Generator. — A generator of this kind consists of two 
 small air pumps of the 'simplest possible construction, without 
 valves. To avoid metal work we can simply use drums, 
 covered with rubber membranes, which are alternately pressed in 
 and drawn out. These pumps should be arranged so that they 
 can work in either the same or in opposite phase, and so that the 
 am])litudes of the strokes of each pump can be varied indepen- 
 dently of the other. For convenience, it should be possible to 
 reverse the ])hase and vary the amplitudes without interrupting 
 the motion of the generator. 
 
 In Fig. 1 is shown a generator, arranged to fulfill these con- 
 ditions. In a wooden base are fixed two vertical steel or brass 
 sjjrings, s, which are joined by the horizontal connecting-rod, 6. 
 The upper ends of tiiese springs are connected by the piston-rods, 
 a, to the pistons of the air-pumps, which are supported on a 
 wooden frame in such a way that each is free to turn about a 
 iiorizontal axis, c, passing through the top of the corresponding 
 spring perpendicular to the piston-rod. Thus either pump can be
 
 INVESTIGATION OF GEOMETKIC PEOPEimES. 
 
 15 
 
 revolv'ed through 180°, or tlirough a smaller angle, without 
 stopping the pumps. The amplitude of the strokes in any posi- 
 tion is proportional to the cosine of this angle, since the compo- 
 nent of the motion of the top of the spring along the axis of the 
 cylinder is proportional to this cosine. At 90° the amplitude is 0, 
 and the phase changes, so that by a simple rotation we are able to 
 reverse the phase, or vary the amplitude of either, or both pumps- 
 The generator may be driven by a motor of suitable nature, 
 attached to the frame. As shown in the figure, we may use a 
 fly-wheel, d, carrying a crank which drives the springs, using an 
 
 Fig. 1. 
 
 electric motor, or any other suitable source, for motive power. 
 The use of the crank has the advantage that the amplitudes of the 
 oscillations of the springs are invariable and independent of the 
 resistance to the motion. It should be noted here, that, with the 
 crank, the springs may be used simply as rigid levers, by loosen- 
 ing the screws, m, which hold them in the base. The springs are 
 then free to turn about a pivot just below the screws. 
 
 A hydraulic motor might also be used to drive the generator. 
 Two coaxial brass cylinders, open at the same end, are so ar- 
 ranged that the inner projects slightly beyond the outer. A rubber 
 membrane is stretched over the open ends of the two tubes, so
 
 16 
 
 FIELDS OF FORCE. 
 
 that water admitted to the outer cylinder cannot pass into the 
 inner cylinder without pressing out the membrane. Under suita- 
 ble circumstances, tliis produces a vibration of the membrane, 
 
 Fig. 2. 
 
 Fkj. 3. 
 
 which can be communicated to the pumps by the connecting-rods. 
 The period will d('j)end upon the tension of the membrane, the
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. 17 
 
 stiffness of the springs, and the length and section of the dis- 
 charge-pipe. An electromagnetic vibrator is often convenient for 
 driving the generator. 
 
 13. Puhalor. Oscilhdor. — For a pulsating body we may use 
 an india-rubber balloon attached to one end of a metal tube, the 
 other end of which is connected by a rubber tube with one of 
 the pumps of the generator. As the balloon often takes irregular 
 forms and motions, it is usually more convenient to let the tube 
 end in a drum, which is covered on each side with a rubber 
 membrane. A diagram is given in Fig. 2. 
 
 A convenient form of oscillator is shown in Fig. 3. The oscil- 
 lating body is a hollow celluloid sphere, «, made in two halves, 
 and attached to a tube of the same material, b, which reaches 
 above the surface of the water. A metal tube, c, connected with 
 one pump of the generator, supports the sphere by pivots at h, and 
 terminates in a heavy drum, d, in the center of the sphere. The 
 rubber membrane, e, is connected with one side of the sphere by a 
 rod, f, so that. the alternating air current produces oscillations in 
 the sphere and in the drum. The sphere is made as light as pos- 
 sible and the drum heavy, so that, while the former takes large 
 oscillations, the latter will take very small oscillations because of 
 its greater mass. For convenience in recognizing the axis of 
 oscillation the two halves of the sphere may be painted in differ- 
 ent colors, so that, at any moment, the advancing hemisphere is 
 one color and the receding hemisphere another. Thus, two oscil- 
 lators connected with pumps in the same phase have hemispheres 
 of the same color advancing simultaneously. 
 
 14. Instrument for the Registering of ]rater Oseillations. — 
 When a pulsating or an oscillating body, like one of those just de- 
 scribed, is placed in the water, the motion produced by it cannot 
 be seen, as an obvious consequence of the transparency of the 
 water. This motion can, however, be observed indirectly in 
 several ways. For example, we can suspend small particles in 
 the water and observe their motions, and we might even succeed 
 in getting photographs of the paths of oscillation of the suspended 
 
 3
 
 18 FIELDS OF FORCE. 
 
 particles. This method has, however, never been nsed, and may 
 involve difficulties because of the small amplitudes of the oscilla- 
 tions. 
 
 A more mechanical method, depending upon the principle of 
 kinetic buoyancy, is preferable. A body which is situated in the 
 oscillating masses of fluid will be subject to a periodic kinetic 
 buoyancy which tries to set up in it oscillations of the same direc- 
 tion as those of the water. The amplitudes of the oscillations 
 produced will, however, generally be minute, but they may be in- 
 creased by resonance. The body is fixed upon an elastic wire, and 
 the period of the generator varied until it accords with the period 
 of the free vibrations of the body. The amplitude of the oscilla- 
 tions of the body is then greatly increased. 
 
 The body is made to carry a hair pencil, which reaches above the 
 surface of the water. One or two millimeters above the point of 
 the brush is placed a horizontal glass plate, resting upon springs. 
 When the body has acquired large oscillations, the glass plate may 
 be pressed down and the brush marks an ink line upon it. The 
 registering device is then, moved to another place in the fluid, and 
 the direction of the water oscillations at this place recorded on the 
 glass plate, and so on. In this way comj)lete diagrams of the lines 
 of oscillation in the fluid are obtained. 
 
 15. T)i<((/rani.s of lli/drodynamic and Corresijondmg Magnetic 
 Fields. — Figs. 4-8, «, give diagrams of hydrodynamic fields ob- 
 tiiined in this way, while Figs. 4-8, b, give the diagrams of the 
 corresponding magnetic fields, obtained in the well known way 
 with iron filings. 
 
 Fig. 4, a, gives the radial lines of oscillation obtained in the 
 space around a pulsating body, while Fig. 4, b, gives the corre- 
 sponding magnetic lines of force issuing from one pole of a long 
 bar magnet. 
 
 Fig. 0, a, gives the lines of oscillation produced in the fluid by 
 two bodies pulsating in the same phase. They represent the meet- 
 ing of two radial currents issuing from two centers. Fig. 5, 
 b, gives the perfectly analogous representation of the magnetic lines 
 of force issuing from two magnetic poles of the same sign.
 
 INV 
 
 .^ESTIGATION OF GEOMETRIC PROPERTIES. 
 
 19 
 
 - ^ I ; / / 
 
 \ 
 
 O 
 
 / 
 
 r-/ / 
 
 
 / / : \ \ \ 
 
 / / ! ' ^ 
 
 
 b 
 
 Fig. 4. 
 
 Fi. 6 «, gives the lines of oscillation produced iu the fluid by 
 two bodi;s iatiug iu opposite phase. The <^'^^2^^ 
 representation of a current which diverges from one pulsatn,g bodj
 
 20 
 
 FIELDS OF FORCE. 
 
 and converges toward the other. Fig. 6, 6, gives the perfectly 
 analogous representation of the magnetic lines of force produced 
 bv two magnetic poles of opposite sign. 
 
 i J ' I f M \ ( 
 
 I / 
 
 \ \ \ ; ; / / i-'ii \^ \ ; ' ' ' / / / 
 
 / /,/ 
 
 
 
 o 
 
 //;// \ \ \V:','//// 
 
 ^^:^^ 
 
 1 \ \ ^ 
 
 b 
 
 Fio. 5. 
 
 ^ 'g- "> ''> gives the more complicated representation of the line 
 of oscillation produced in tiio water l)ya combination of three pul-
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. 
 
 21 
 
 sating bodies, two ])ul.sating in the same phase, and one in the 
 opposite, and Fig. 7, h, gives the perfectly analogous representa- 
 tion of the magnetic lines of force produced by three magnetic 
 poles, of which two have the same sign, and one the opposite. 
 
 I / / . 
 
 / / 
 
 
 
 / / 
 
 o 
 
 .^^. 
 
 V'-:///,i,\A>:r-'f^!,\\ 
 
 t \ 
 
 Finally, Fig. 8, «, giv^es the lines of oscillation produced in the 
 fluid by an oscillating body, and Fig. 8, b, the corresponding lines 
 of magnetic force produced by a short magnet.
 
 22 
 
 X 
 
 \\M 
 
 FIELDS OF FORCE. 
 
 
 / 
 X 
 
 T--V/'" 
 
 / \ \ \\ \\\\V\ / / / / / » ^ 
 
 ■^ 
 
 / iW
 
 INVESTIGATION OF GEOMETRIC PROPERTIES. 
 
 23 
 
 These figures show very fully the analogy in the geometry of 
 the fields produced, on the one hand, by magnetic poles or magnets 
 in a surrounding homogeneous medium, and, on the other hand. 
 
 by pulsating or oscillating bodies in a surrounding homogeneous 
 fluid. The experimental demonstration of the analogy for the case
 
 24 FrKI.DS OK FOUCK. 
 
 wlieii tlic iiu'ditini siirrouiuling the magnets and the fluid siirround- 
 iiifi; thepidsatiiigor oscilhitinsi^ bodies con tain heterogeiu'itie.sis more 
 delicate. In the hydrodyntunic ease the heterogeneities should be 
 Ihiid, and it is |)ra(!tieally iinpossil)le, on account of the action of 
 t;ravity, to have a fluid mass of given shape flowing freely in a 
 fluid of other density. If for the fluid bodies we substitute rigid 
 bodies, suspended from above or anchored from below, according to 
 (heir density, it is easily seen, by means of our registering device, 
 tliat the lines of oscillation have a tendency to converge toward the 
 light, and to diverge from the heavy bodies. But this registering 
 device cannot be brought sufficiently near these bodies, to show 
 the curves in their immediate neighborhood. Here the observa- 
 tion of the oscillations of small suspended ])artieles would probably 
 be the best method to employ. Experiments which we shall per- 
 form later will give, however, indire(!t proofs that the fields have 
 exactly the expected character. 
 
 16. On PossiUe Kxfcnsioits of tlic Aualogy. — We have thus 
 found, by elementary reasoning, a very complete analogy between 
 the geometric propertiqs of hydrodynamic fields and electric or 
 magnetic flelds for the case of statical })henomena. And, to s'ome 
 extent, we have verified these results by experiments. 
 
 It is a natural (|uestion then, does the analogy extend to fields 
 of greater generality, or to flelds of electromagnetism of the most 
 general nature? In discussing this question further an introduc- 
 tory remark is important. The formal analogy which exists be- 
 tween electrostatic and magnetic fields has made it possible for 
 us to compare the hydrodynamic flelds considered with both elec- 
 trostatic and magnetic fields. If there exists a perfect hydro- 
 dynamic analogy to electromagnetic phenomena, the liydrodynamic 
 fields considered will, presiimal)ly, turn out to be analogous either 
 to electrostatic fields only, or to magnetic fields only, but not to 
 both at the same time. The (piestion therefore can now be raised, 
 would our hydrodynamic fields in an eventually extended analogy 
 correspond to the electrostatic or the magnetic flelds? To this it 
 must be answered, it is very probable that only the analogy to the
 
 IXVKSTKiATIOX OF C KOM irriMC I'UOl'Kiri'I KS. 25 
 
 electrostatic fields will hoUl. As an obvious argument, it may be 
 eiui)liasize(l that the hydrodynamic fields have exactly the gener- 
 ality of electrostatic fields, but greater generality than magnetic 
 fields. The analogy to magnetism will take the right form only 
 when the restriction is introduced, that changes oi' volume are to be 
 excluded. Otherwise, we should arrive at a theory of magnetism 
 where isolated magnetic poles could exist. To this argument 
 others may be added later. 
 
 But in spite of this, the formal analogy of the electric and mag- 
 netic fields makes it possible to formally compare hydrodynamic 
 fields with magnetic fields. And this will often be preferable, fi)r 
 practical reasons. This will be the case in the following discus- 
 sion, because the idea of the electric current is much more familiar 
 to us than the idea of the magnetic current, in spite of the formal 
 analogy of these two currents. 
 
 Let us compare, then, the hydrodynamic fields hitherto consid- 
 ered with magnetic fields produced by steel magnets. The lines of 
 force of these fields always pass through the magnets which produce 
 them, just as the corresponding hydrodynamic curves pass through 
 the moving bodies which produce the motion. The magnetic lines 
 of force produced by electric currents, on the other hand, are gener- 
 ally closed in the exterior space, and need not pass at all through 
 the conductors carrying the currents. To take a simple case, the 
 lines of force produced by au infinite rectilinear current are circles 
 around the current as an axis. 
 
 If it should be possible to extend the analogy so as to include 
 also the simplest electromagnetic fields, we would have to look for 
 hydrodynamic fields with closed lines of fiow which do not ])ass 
 through the bodies ]u-oducing the motion. It is easily precon- 
 ceived, that if the condition of the oscillatory nature of the fiuid 
 motion be insisted upon, the recpiircd motion cannot be pro- 
 duced by fluid pressure in a perfect fiuid. A cylinder, for 
 instance, making rotary oscillations around its axis will produce 
 no motion at all in a perfect fluid, (^uite the contrary is true, 
 if the fluid be viscous, or if it have a suitable transverse elasticity, 
 
 4
 
 26 
 
 FIELDS OF FORCE. 
 
 as does an aqueous solution of gelatine. But, as we shall limit 
 ourselves to the consideration of perfect fluids, we shall not con- 
 sider the phenomena in such media. 
 
 17. Detached Ui/drodynamic Ancdogy to the Fields of Stationary 
 Electromagnetism. — A direct continuation of our analogy is thus 
 made impossible. It is a very remarkable fact, however, that 
 there exist hydrodynamic fields which are geometrically analogous to 
 the fields of stationary electric currents. But to get these fields 
 we must give up the condition, usually insisted upon, that the 
 motion be of oscillatory nature. We thus arrive at an inde- 
 pendent analogy, which has a considerable interest in itself, but 
 which is no immediate continuation of that considered above. 
 
 Fi(i. 9. 
 
 Tliis analogy is that discovered by v. Helmholtz in his 
 research (»n the vortex motion of perfect fluids. According to his 
 celebrated results, a vortex can be compared with an electric cur- 
 rent, and the fluid field surrounding the vortex will then be in 
 exactly the same relation to the vortex as the magnetic field is 
 to the electric current which produces it. 
 
 To consider only the case of rectilinear vortices, the field of one 
 rectilinear vortex is represented by concentric circles. And this 
 field corresponds to the magnetic field of a rectilinear current. 
 The hydrodynamic field of two rectilinear parallel vortices which
 
 INVESTIGATION OF GEO]\[ETRIC PROPERTIES. 
 
 27 
 
 have the same direction of rotation is shown in Fig. 9, and this 
 field is strictly analogous to the magnetic field of two rectilinear 
 parallel currents in the same direction. Fig. 10 gives the hydro- 
 dynamic field of two rectilinear parallel vortices which have 
 opposite directions of rotation, and it is strictly analogous to the 
 magnetic field of two electric currents of opposite direction. 
 
 Fields of this nature can be easily produced in water by rotat- 
 ing rigid cylinders, and observed by the motion of suspended par- 
 ticles. At the same time, each cylinder forms an obstruction in 
 the field produced by the other. If only one cylinder be rotating, 
 
 Fig. 10. 
 
 the lines of flow produced by it will be deflected so that they run 
 tangentially to the surface of the other. The cylinder at rest thus 
 influences the field just as a cylinder of infinite diamagnetivity 
 would influence the magnetic field. The rotating cylinders there- 
 fore correspond to conductors for electric currents, which are con- 
 structed in a material of infinite diamagnetivity. 
 
 This analogy to electromagnetism is limited in itself, apart 
 from its divergence from the analogy considered previously. 
 The extreme diamagnetivity of the bodies is one limitation. An-
 
 "2S FIELDS OF FORCE. 
 
 other limitation follows from Helmholtz's celebrated theorem, 
 that vortices do not vary in intensity. Therefore phenomena 
 corresponding to those of electromagnetic induction are excluded. 
 Whichever view we take of the subject, the hydrodynamic 
 analogies to electric and magnetic phenomena are thus limited in 
 extent. To get analogies of greater extent it seems necessary to 
 pass to media witii other properties than those of perfect fluids. 
 But we will not try on this occasion to look for further exten- 
 sions of the geometric analogies. We prefer to pass to an exami- 
 nation of the dynamic properties of tiie fields whose geometric 
 properties we have investigated.
 
 II. 
 
 ELEMENTARY INVESTIGATION OF THE DYNAMI- 
 CAL PROPERTIES OF HYDRODYNAMIC FIELDS. 
 
 1. The Dynamics of the Electric or the Magnetic Field — Our 
 knowledge of the dynamics of the electric or magnetic field is 
 very incomplete, and will presumably remain so as long as the 
 true nature of the fields is unknown to us. 
 
 What we know empirically of the dynamics of the electric or 
 magnetic field is this — bodies in the fields are acted upon by 
 forces which may be calculated when we know the geometry of 
 the field. Under the influence of these forces the bodies may 
 take visible motions. But we have not the slightest idea of the 
 hidden dynamics upon which these visible dynamic phenomena 
 
 depend. 
 
 Faraday's idea, for instance, of a tension parallel to, and a 
 pressure perpendicular to the lines of force, as well as Maxwell's 
 mathematical translation of this idea, is merely hypothetical. 
 And even though this idea may contain more or less of the 
 truth, investigators have at all events not yet succeeded in mak- 
 ing this dynamical theory a central one, from which all the 
 properties of the fields, the geometric, as well as the dynamic, 
 naturally develop, just as, for example, all properties of hydro- 
 dynamic fields, the geometric, as well as' the dynamic, develop 
 from the hydrodynamic equations. Maxwell himself was very 
 well aware of this incompleteness of his theory, and he stated it 
 in the following words : 
 
 " It must be carefully born in mind that we liave only made 
 one step in the theory of the action of the medium. We have 
 supposed it to be in a state of stress but have not in any way ac- 
 counted for this stress, or explained how it is maintained. . . . 
 " I have not been able to make the next step, namely, to ac- 
 
 29
 
 30 FIELDS OF FORCE. 
 
 count by mechanical considerations for these stresses in the di- 
 electric." 
 
 In spite of all formal progress in the domain of Maxwell's 
 theory, these words are as true to-day as they were when Max- 
 AVELL wrote them. This circumstance makes it so much the more 
 interesting to enter into the dynamic properties of the hydrody- 
 namic fields, which have shown such remarkable analogy in their 
 geometric properties to the electric or magnetic fields, in order to 
 see if with the analogy in the geometric properties there will be 
 associated analogies in their dynamical properties. The question 
 is simply this : 
 
 Consider an electric, or magnetic field and the geometrically 
 corresponding hydrodynamic field. Will the bodies which pro- 
 duce the hydrodynamic field, namely, the pulsating or the oscillat- 
 ing bodies or the bodies which modify it, such as bodies of other 
 density than the surrounding fluid, be subject to forces similar 
 to those acting on the corresponding bodies in the electric or 
 magnetic fields ? 
 
 This question can be answered by a simple application of the 
 principle of kinetic buoyancy. 
 
 2. Jiesu/faiit Force against a Pidmting Body in a Si/nchronoudi/ 
 Oscillatiiir/ Cun'ent. — Let us consider a body in the current pro- 
 duced by any system of synchronously pulsating and oscillating 
 bodies. It will be continually subject to a kinetic buoyancy pro- 
 portional to the product of the acceleration of tlie fluid masses into 
 the mass of water displaced by it. If its volume be constant, so 
 that the displaced mass of water is constant, the force will be 
 strictly i)eriodic, with a mean value zero in the period. It will 
 then be brought only into oscillation, and no progressive motion 
 will result. 
 
 Jiut if the body has a variable volume, the mass of water dis- 
 placed by it will not be constant. If the changes of volume con- 
 sist in |)ulsation.s, synchronous with the pulsations, or oscillations, 
 of the distant bodies which produce the current, the displaced 
 mass of water will have a maximum when the acceleration has its
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 31 
 
 nuixiinum in one direction, and a minimum when the acceleration 
 lias its maximum in the opposite direction. As is seen at once, 
 the force can then no longer have the mean value zero in the period. 
 It will have a mean value in the direction of the acceleration at the 
 time when the pulsating body has its maximum volume. We thus 
 find the result : 
 
 A pulsating body in a si/nchronousli/ osciUating current is subject 
 to the action of a resultant force, the direction of which is that of the 
 acceleration in the current at the time ichen the pulsating body has its 
 maximum volume. 
 
 3. ITutual Attraction and Bepidsion between Two Pulsating 
 
 Bodies. As a first application of this result, we may consider the 
 
 case of two synchronously pulsating bodies. Each of them is in 
 the radial current produced by the other, and we have only to 
 examine the direction of the acceleration in this current. Evi- 
 dently, this acceleration is directed outwards when the body pro- 
 ducing it has its minimum volume, and is therefore about to expand, 
 and is directed inwards when the body producing it has its maxi- 
 mum volume, and is therefore about to contract. 
 
 Let us consider first the case of two bodies pulsating in the 
 same phase. They have then simultaneously their maximum vol- 
 umes, and the acceleration in the radial current produced by the 
 one body will thus be directed inwards, as regards itself, when the 
 other body has its maximum volume. The bodies will therefore 
 be driven towards each other ; there will be an apparent mutual 
 attraction. If, on the other hand, the bodies pulsate in opposite 
 phase, one will have its maximum volume when the other has its 
 minimum volume. And therefore one will have its maximum vol- 
 ume when the radial acceleration is directed outward from the 
 other. The result, therefore, will be an apparent mutual repulsion. 
 As the force is proportional to the acceleration in the radial cur- 
 rent,and as the acceleration will decrease exactly as the velocity, pro- 
 portionally to the inverse square of the distance, the force itself 
 will also vary according to this law. On the other hand, it is 
 easily seen that the force must also be proportional to two param-
 
 32 FIEI.DS OF FORCE. 
 
 eters, which measure in a proper way the intensities of the pulsa- 
 tions of each body. Calling these parameters the "intensities of 
 pulsation," we find the following law : 
 
 Between bodies puhating in the same phase there is an apparent 
 attraction; between bodies pulsating in the opposite phase there is an 
 apparent repulsion, the force being proportional to the product of the 
 two intensities of pulsation, and proportional to the inverse square of 
 the distance. 
 
 4. Discussion. — "We have thus deduced from the principle of 
 dynamic buoyancy, that is from our knowledge of the dynamics 
 of the hydrodynamic field, that there will be a force which moves 
 the pulsating bodies through the field, just as there exists, for 
 reasons unknown to us, a force which moves a charged body 
 through the electric field. And the analogy is not limited to the 
 mere existence of the force. For the law enunciated above has 
 exactly the form of Coulojik's law for the action between two 
 electrically charged particles, with one striking difference ; the 
 direction of the force in the hydrodynamic field is opposite to that 
 of the corresponding force in the electric or magnetic field. For 
 bodies pulsating in the same phase must be compared wdth bodies 
 charged with electricity of the same sign ; and bodies pulsating 
 in the opposite phase must be compared with bodies charged with 
 opposite electricities. This follows inevitably from the geometrical 
 analogy. For bodies pulsating in the same phase produce a field 
 of the same geometrical configuration as bodies charged with 
 the same electricity (Fig. 5, a and b) ; and bodies pulsating in 
 opjwsite i)hase produce the same field as bodies charged with 
 opposite electricities (Fig. 6, a and 6). 
 
 This exception in the otherwise complete analogy is most aston- 
 ishing, liut we cannot discover the reason for it in the present 
 limited state (»f our knowledge. We know very well why the 
 force in the hydrodynamic field must have the direction indicated 
 — this is a simple consequence of the dynamics of the fluid. But 
 in our total ignorance of the internal dynamics of the electric or 
 magnetic field we cannot tell at all why the force in the electric 
 field has the direction which it has, and not the reverse.
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 33 
 
 Thus, taking the focts as we find them, we arrive at the result 
 that with the geometrical analogy developed in the preceding lec- 
 ture there is associated an inverse dynamical analogy : 
 
 Falsating bodies act upon each other as if they were electrically 
 charged partides or magnetic jjoles, but with the difference that 
 charges or poles of the same sign attract, and charges or i^oles of 
 opposite sign repel each other. 
 
 5. Pulsation Balance. — In order to verify this result by experi- 
 ment an arrangement must be found by which a pulsating body 
 lias a certain freedom to move. This may be obtained in different 
 ways. Thus a pulsator may be suspended as a pendulum by a 
 long india-rubber tube through which the air from the generator 
 is brought. Or it may be inserted in a torsion balance, made of 
 glass or metal tubing, and suspended by an india-rubber tube which 
 brings the air from the generator and at the same time serves as a 
 torsion wire. These simple arrangements have at the same time 
 the advantage that they allow rough quantitative measurements of 
 the force to be made. For good qualitative demonstrations the 
 following arrangement will generally be found preferable. 
 
 The air from the generator comes through the horizontal metal 
 tube, a, (Fig. 11), which is fixed in a support. The air channel 
 continues vertically through the metal piece b, which has the form 
 of a cylinder with vertical axis. At the top of this metal piece 
 and in the axis there is a conical hole, and the lower surface 
 is spherical with this hole as center. The movable part of the 
 instrument rests on an adjustable screw, pivoted in this hole. This 
 screw carries, by means of the arm d, the little cylinder c, through 
 which the vertical air channel continues. The upper surface of this 
 cylinder is spherical, with the point of the screw as center. The 
 two spherical surfaces never touch each other, but by adjustment 
 of the screw they may be brought so near each other that no sensi- 
 ble loss of air takes place. To the part of the instrument c-d, 
 which gives freedom of motion, the pulsator may be connected by 
 the tube ef, the counter-weight maintaining the equilibrium. By 
 this arrangement, the pulsating body is free to move on a spherical 
 5
 
 34 
 
 FIEI.DS OF FORCE. 
 
 surface with the pivot as center, and the equilibrium will be neutral 
 for a horizontal motion, and* stable for a vertical motion. 
 
 G. ExperimenU with Pulsating Bodies. — Having one pulsator 
 in the pulsation balance, take another in the hand, and arrange the 
 
 Fig. 11. 
 
 generator for ])alsations of the same phase, and we see at once that 
 the two pulsating bodies attract each other (Fig. 12, a). This 
 attraction is easily seen with distances up to 10^15 cm., or more, 
 and it is observed that the intensity of the force increases rapidly 
 
 Fig. 12. 
 
 as the distance diminishes. The moment the relative phase of the 
 pulsations is changed, the attraction ceases, and an equally intense 
 rc|)ii]sion appears (Fig. 12, I>). With the torsion balance it may
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 35 
 
 be shown with tolerable accuracy, that the force varies as the in- 
 verse square of the distance, and is proportional to two parameters, 
 the intensities of pulsation. 
 
 In this experiment the mean value only of the force and the 
 progressive motion produced by it are observed. By using very 
 slow pulsations with great amplitudes, a closer analysis of the phe- 
 nomenon is possible. It is then seen that the motion is not a 
 simple progressive one, but a dissymmetric vibratory motion, in 
 which the oscillations in the one direction always exceed a little 
 the oscillations in the other, so that the result is the observed 
 progressive motion. 
 
 7. Actio)i of an Oscillating Body upon a Pulsating Body. — 
 Two oppositely pulsating bodies produce geometrically the same 
 field as two opposite magnetic poles. Geometrically, the field is 
 that of an elementary magnet. Into the field of these two oppo- 
 sitely pulsating bodies we can bring a third pulsating body. 
 Then, if we bring into application the law just found for the 
 action between two pulsating bodies, we see at once that the third 
 pulsating body will be acted upon by a force, opposite in direc- 
 tion to the corresponding force acting on a magnetic pole in 
 the field of an elementary magnet. In this result nothing will 
 be changed, if, for the two oppositely pulsating bodies, we substitute 
 an oscillating body. For both produce the same field, and the 
 action on the pulsating body will evidently depend only upon the 
 field produced, and not upon the manner in which it is produced. 
 We thus find : 
 
 An oscillating body will act upon a pulsating body as an ele- 
 mentary magnet upon a magnetic pole, but tcith the kmo of poles 
 7-eversed. 
 
 This result may be verified at once by experiment. If we take 
 an oscillator in the hand, and bring it near the pulsator which is 
 inserted in the pulsation-balance, we find attraction in the case 
 (Fig. 13, rt) when the oscillating body approaches the pulsating 
 body as it expands and recedes from it as it contracts. But as 
 soon as the oscillating body is turned around, so that it approaches
 
 36 
 
 FIELDS OF FORCE. 
 
 while the pulsating body is contracting and recedes while it is 
 expanding (Fig. 13,6), the attraction changes to repulsion. 
 
 To show how the analogy to magnetism goes even into the 
 smallest details the oscillating body may be placed in the })ro- 
 longation of the arm of the pulsation-balance, so that its axis 
 of oscillation is perpendicular to this arm. The pulsating body 
 will then move a little to one side and come into equilibrium in a 
 dissymmetric position on one side of the attracting pole (Fig. 
 13, f). If the oscillating body be turned around, the position of 
 equilibriiuii will be on the other side. Exactly the same small 
 
 Fig. 13. 
 
 lateral displacement is observed when a short magnet is brought 
 into the transverse position in the neighborhood of the pole of 
 a l(»ng bar magnet which has the same freedom to move as the 
 l)ulsating body. 
 
 8. Force mjaind an Omillatiinf/ Body. — If, in the preceding 
 experiment, we take the pulsating body in the hand and insert the 
 oscillating body in the balance, we cannot conclude a ■priori that 
 the motions of the oscillating body will prove the existence of 
 a force ccpial and opposite to that exerted by the oscillating body 
 upon the i)ulsating body. The principle of equal action and re- 
 action is empirically valid for the common actions at a distance 
 between two bodies. But for these apparcnf actions at a distance, 
 where not only the two bodies but also a third one, the fluid, are 
 engaged, no general coriclusion can l)e drawn.
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 
 
 37 
 
 To examine the action to which the oscillating body is subject 
 we must therefore go back to the principle of kinetic buoyancy. 
 The kinetic buoyancy will give no resultant force against a body 
 of invariable volume, which oscillates between two places in 
 the fluid where the motion is the same. For at both ends of 
 the path the body will be subject to the action of equal and oppo- 
 site forces. But if it oscillates between places where the motion 
 is somewhat different in direction and intensity, these two forces 
 will not be exactly equal and opposite. The direction of the 
 accelerations in the oscillating fluid masses is always tangential 
 to the lines of oscillation. If the field be represented by these 
 lines, and if the absolute value of the acceleration be known at 
 every point of the fluid at any 
 time, the force exerted on the 
 oscillating body at every point 
 of its path may be plotted, and 
 the average value found. As 
 we desire only qualitative re- 
 sults, it will be sufficient to 
 consider the body in the two 
 extreme positions only, where 
 we have to do with the ex- 
 treme values of the force. 
 
 Let, then, the continuous circle (Fig. 14) represent the oscillat- 
 ing body in one extreme position, and the dotted circle the same 
 body in the other extreme position, and let the two arrows be pro- 
 portional to the accelerations which the fluid has at these two places 
 at the corresponding times. The composition of these two alter- 
 nately acting forces gives the average resultant force. Let us now 
 substitute for the oscillating body a couple of oppositely pulsating 
 bodies, one in each extreme position of the oscillating body, and 
 let us draw arrows representing the average forces to which these 
 two pulsating bodies are subject. We then get arrows located 
 exactly as in the preceding case. And we conclude, therefore, 
 that if we only adjust the intensities of pulsation properly, the
 
 38 FIELDS OF FORCE. 
 
 two oppositely pulsating bodies will be acted upon by exactly the 
 same average resultant force as the oscillating body. From the 
 results found above for the action against pulsating bodies we can 
 then conclude at once : 
 
 An oscUlatiiif/ body in the hijdi-odynamic field vnll be subject to the 
 action of d force similar to that acting upon an elementary magnet 
 in the magnetic field, the onhj difference being the difference in the 
 signs of the forces which follows from the opposite joole-law. 
 
 9. Experimental Investigation of the Force exerted by a Pulsat- 
 ing Body upon an Oscillatiny Body. — Let us now insert the 
 oscillator in the balance, and turn it so that the axis of oscillation 
 is in the direction of its free movement. If a pulsator be taken 
 in the hand, it will be seen that attraction takes place when the 
 pulsiiting body is made to approach one pole of the oscillating 
 body (Fig. 13, a), and repulsion if it is made to approach the 
 other pole (Fig, 13, b). And, as is evident from comparison with 
 the preceding case, the force acting on the oscillating body is al- 
 ways opposite to that acting on the pulsating body. We have 
 equality of action and reaction, just as in the case of magnetism. 
 
 The analogy with magnetism can be followed further if the 
 pulsating i)ody be brought into the prolonged arm of the oscilla- 
 tion balance. The oscillating body will then take a short lateral 
 displacement, so that its attracting pole comes nearer to the pul- 
 sating body (Fig. 13, c). It is a lateral displacement correspond- 
 ing exactly to that take by an elementary magnet under the influ- 
 ence of a magnetic pole. 
 
 10. Experimental Investigation of the Mutual Actions betiveen 
 Two Oscillating llodicx. — The pulsator held in the hand may now 
 be replaced by an oscillator, while the oscillator inserted in the 
 balance is left unchanged, so that it is still free to move along its 
 axis of oscillation. We may first bring the oscillator held in the 
 hand into the position indicated by the figures 15, a and 6, so that 
 tlie axes of osciUation lie in the same line. The experiment will 
 then correspond to that with magnets in longitudinal position. We 
 get attraction in the case, (Fig. 15, a), when the oscillating bodies
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 
 
 39 
 
 are in opposite phase. This corresponds to the case in which the 
 magnets have poles of the same sign turned towards each other. 
 If the oscillator held in the hand be turned around, so that the 
 two bodies are in the same phase, the result will be repulsion (Fig. 
 15, b), while the corresponding magnets, which have opposite 
 poles facing each other, will attract each other. Finally, the oscil- 
 lator may be brought into the position (Fig. 15, e) in which it oscil- 
 lates in the direction of the prolonged arm of the oscillation- 
 
 FiG. 15. 
 
 balance. Then we shall again get the small lateral displacement, 
 which brings the attracting poles of the two oscillating bodies near 
 each other. 
 
 The oscillator in the balance may now be turned around 90°, so 
 that its oscillation is at right angles to the direction in which it is 
 free to move. If both bodies oscillate normally to the line join- 
 ing them, we get attraction when the bodies oscillate in the same 
 phase (Fig. 15, c), and repulsion when they oscillate in the oppo- 
 site phase (Fig. 15, d). This corresponds to the attraction and 
 repulsion between parallel magnets, except that the direction of the
 
 40 FIELDS OF FORCE. 
 
 force is, as usual, the reverse, the magnets repelling in the case of 
 similar, and attracting in the case of opposite parallelism. If, 
 finally, we place the oscillator in the prolonged arm of the bal- 
 ance with its axis of oscillation perpendicular to this arm (Fig. 
 15, /'), we again get the small lateral displacement described 
 above, exactly as with magnets in the corresponding positions, but 
 in the opposite direction. 
 
 AVe have considered here only the most important positions of 
 the two oscillating bodies and of the corresponding magnets. Be- 
 tween these principal positions, which are all distinguished by cer- 
 tain properties of symmetry, there is an infinite number of dis- 
 symmetric positions. In all of them it is easily shown that the 
 force inversely corresponds to that between two magnets in the 
 corresponding positions. 
 
 11. Rotations of the Oscillating Body. — We have con^^idered 
 hitherto only the resultant force on the oscillating body. But in 
 general the two forces acting at the two extreme positions also form 
 a couple, like the two forces acting on the two poles of a mag- 
 net. The first eifect of 'this couple is to rotate the axis of oscil- 
 lation of the l)ody. But if this axis of oscillation has a fixed 
 direction in the body, as is the case in our experiments, the body, 
 will be obliged to follow the rotation of the axis of oscillation. 
 
 To show the effect of this couple experimentally the oscillator 
 may be placed directly in the cylinder c (Fig. 11) of the pulsa- 
 tion-balance. It is then free to turn about a vertical axis passing 
 through the })ivot. If a pulsating body be brought into the neigh- 
 borhood of this oscillating body, it immediately turns about its 
 axis until the position of greatest attraction is reached, and as a 
 consequence of its inertia it will generally go through a series of 
 oscillations about this position of equilibrium. If the phase of 
 the pulsations be changed, the oscillating body will turn around 
 until its other pole comes as near as possible to the pulsating body. 
 Apart from the direction of the force, the phenomena is exactly 
 the analogue of a suspended needle acted upon by a magnetic pole. 
 
 The pulsating ImxIv may now be replaced by an oscillating body.
 
 INVESTIGATION OF DYNAMTCAI. PROPERTIES. 41 
 
 Except for the direction of the force, we shall get rotations corres- 
 ponding to those of a compass needle nnder the influence of a 
 magnet. The position of equilibrium is always the position of 
 greatest attraction (Fig. 15, «, c), the position of greatest repul- 
 sion being a position of unstable equilibrium. If the fixed oscil- 
 lating body oscillates parallel to the line drawn from its center to 
 that of the body in the balance, the position of stable equilibrium 
 will be that indicated in Fig. 16, h, and if it oscillates at right 
 angles to this line, it will be the position indicated in Fig. 16, d, 
 while the intermediate dissymmetric positions of the fixed oscil- 
 lator give intermediate dissymmetric positions of equilibrium of 
 the movable oscillating body. It is easily verified that the posi- 
 
 ■' (us': 
 
 c d 
 
 Fig. 16. 
 
 tions of equilibrium are exactly the same as for the case of two 
 magnets, except for the difference which is a consequence of the 
 opposite pole-law ; the position of stable equilibrium in the mag- 
 netic experiment is a position of unstable equilibrium in the 
 hydrodynamic experiment, and vice versa. 
 
 12. Forces Analogous to Those of Temporary Magnetism. — 
 We have already considered the forces between bodies wdiich are 
 themselves the primary cause of the field, namely the bodies 
 which have forced pulsations or oscillations. But, as we have 
 shown, bodies which are themselves neutral but which have 
 another density than that of the fluid also exert a marked influ- 
 ence upon the configuration of the field, exactly analogous to that 
 exerted by bodies of different inductivity upon the configuration 
 of the electric field. This action of the bodies upon the geomet- 
 rical configuration of the field is, in the case of electricity or mag- 
 6
 
 42 
 
 FIELDS OF FORCE. 
 
 netism, accompanied by a mechanical force exerted by the field 
 u}3on the bodies. AVe shall see how it is in this respect in the 
 hydrodyuamic field. 
 
 As we concluded from the principle of kinetic buoyancy, a body 
 which is lighter than the water is brought into oscillation with 
 greater amplitudes than those of the water ; a body of the same 
 density as the water will be brought into oscillation with exactly 
 the same amplitude as the water ; and a body which has greater 
 density than the water will be brought into oscillation with smaller 
 amplitudes than those of the water. From this we conclude 
 that during the oscillations the body of the same density as the 
 water will be always contained in the same mass of water. But 
 both the light and the heavy body will in the two extreme posi- 
 tions be in different masses of water, and if these have not exactly 
 
 Fjo. 17. 
 
 tiic same motion, it will be subject in these two positions to kinetic 
 buoyancies not exactly e(iual and not exactly opposite in direc- 
 tion. The motion cannot therefore be strictly periodic. As a 
 consequence of a feeble dissymmetry there will be superposed 
 ujx)n the oscillation a progressive motion. 
 
 That the average force which produces this progressive motion 
 is strictly analdgous to the force de])euding upon induced rqagnetism 
 or eloftriliciitioM by iuHuence, is easily seen. As we have already 
 shown in the ])receding lecture, the induced oscillations correspond 
 exactly to the induced states of polarization in the electric or the 
 magnetic field. Further, the forces acting in the two extreme posi-
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 
 
 43 
 
 tions of oscillation are in the same relation to the geometry of the 
 field as the forces acting on the poles of the induced magnets ; they 
 are directed along the lines of force of the field, and vary in inten- 
 sity from place to place according to the same law in the two kinds 
 of fields, except that the direction of the force is always opposite in 
 the two cases. Fig. 17, a shows these forces in the two extreme 
 positions of a light body, which oscillates with greater amplitudes 
 than the fluid, and Fig. 1 7, b shows the corresponding forces acting 
 on the two poles of a magnetic body. Therefore, in the hydro- 
 dynamic field, the light body will be subject to a force oppositely 
 equivalent to that to which the magnetic body in the corresponding 
 magnetic field is subject. Fig. 18, a shows the forces acting on 
 the heavy body in its two extreme positions, the oscillations repre- 
 sented in the figure being those which it makes relatively to the 
 
 Fig. 18. 
 
 fluid, which is the oscillation which brings it into water masses 
 with different motions. Fig. 18, 6 shows the corresponding forces 
 acting on the poles of an induced magnet of diaraagnetic polarity. 
 And, as is evident at once from the similarity of these figures, 
 the heavy body in the hydrodynaraic field will be acted upon by a 
 force which oppositely corresponds to the force to which a diamag- 
 netic body is subject in the magnetic field. 
 
 The well known laws for the motion of magnetic and diamag- 
 netic bodies in the magnetic field can, therefore, be transferred at 
 once to the motion of the light and heavy bodies in the hydro- 
 dynamic field. The most convenient of these laws is that of
 
 44 FIELDS OF FORCE. 
 
 Faraday, which connects the force with the absolute intensity, 
 or to the energy, of the field. Remembering the reversed direc- 
 tion of the force, we conclude that : 
 
 The lif/ht body rvUl move in the direction of decreasing, the heavy 
 body in the direction of increasing energy of the field. 
 
 13. Attraction and Repulsion of Light and Heavy Bodies by a 
 Pulsating or an Oscillating Body. — If the field be produced by 
 only one pulsating or one oscillating body, the result is very 
 simple. For the energy of the field has its maximum at the sur- 
 face of the pulsating or oscillating body, and will always decrease 
 witli increasing distance. Therefore, the light body will be re- 
 pelled, and tlie heavy body attracted by the pulsating or the oscil- 
 lating body. 
 
 To make this experiment we suspend in the water from a cork 
 floating on the surface a heavy body, say a ball of sealing wax. 
 In a similar manner we may attach a light body by a thread to a 
 sinker, which either slides with a minimum pressure along the 
 bottom of the tank, or which is itself held up in a suitable manner 
 by corks floating on the surface. It is important to remark- that 
 the light body should never be fastened directly to the sinker, but 
 by a thread of sufficient length to insure freedom of motion. 
 
 On bringing a pulsator up to the light body, it is seen at once 
 to be rej)clled. If one is sufficiently near, the small induced 
 oscillations of the light body may also be observed. If the pul- 
 sating body be brought near the heavy body, an attraction of simi- 
 lar intensity is observed. In both cases it is seen that the force 
 de<;reases much more rapidly with the distance than in all the 
 previous experiments, the force decreasing, as is easily proved, 
 as the inverse fifth power of the distance, which is the same law of 
 distance found for the action between a magnetic pole and a piece 
 of iron. 
 
 U for tlie ]>ulsating body we substitute an oscillating body, the 
 same attractions and repulsions are observed. Both poles of the 
 osci Hating body exert exactly the same attraction on the heavy 
 body, and exactly the same repulsion on the light body, and even
 
 IXVESTIGATION OF DYNAMICAL PROPERTIES. 45 
 
 the equatorial parts of the oscillating body exert the same attract- 
 ing or repelling force, though to a less degree. As is easily seen, 
 we have also in this respect a perfect analogy to the action of a mag- 
 net on a piece of soft iron, or on a piece of bismuth. 
 
 14. Simultaneous Permanent and Temporary Force. — As the 
 force depending upon the induced pulsations, oscillations, or mag- 
 netizations, decreases more rapidly with increasing distance than 
 the force depending upon the permanent pulsations, oscillations, or 
 magnetizations, very striking effects may be obtained as the result 
 of the simultaneous action of forces of both kinds. And these 
 effects offer good evidence of the true nature of the analogy. 
 
 For one of the simplest magnetic experiments we can take a 
 strong and a weak magnet, one of which is freely suspended. At a 
 distance, the poles of the same sign will repel each other. But 
 if they be brought sufficiently near each other, there will appear 
 an attraction depending upon the induced magnetization. This 
 induced magnetization is of a strictly temporary nature, for the 
 experiment may be repeated any number of times. 
 
 We can repeat the experiment using the pulsation -balance and 
 two pulsators, giving them opposite pulsations but with very dif- 
 ferent amplitudes. At a distance, they will repel each other, but 
 if they be brought sufficiently near together, they will attract. It 
 is the attraction of one body, considered as a neutral body heavier 
 than the water, by another which has intense pulsations. 
 
 Many experiments of this nature, with a force changing at a 
 critical point from attraction to repulsion, may be made, all show- 
 ing in the most striking way the analogy between the magnetic 
 and the hydrodynamic forces. 
 
 15. Orientation of Cylindrical Bodie.^. — The most common 
 method of testing a body with respect to magnetism or diamagnet- 
 ism is to suspend a long narrow cylindrical piece of the body in 
 the neighborhood of a sufficiently powerful electromagnet. The 
 cylinder of the magnetic body then takes the longitudinal, and 
 the cylinder of the diamagnetic body the transverse position. 
 
 The corresponding hydrodynamic experiment is easily made
 
 46 FIELDS OF FORCE. 
 
 The liglit cvlinder is attached from below and the heavy cylinder 
 from above, and on bringing near a pnlsating or an oscillating 
 body, it is seen at once that the light cylinder, which corresponds 
 to the magnetic body, takes the transverse, and the heavy cylinder, 
 which corres})onds to the diamagnetic body, the longitudinal position. 
 
 16. Xenfral Bodies Acted Upon by Two or 3fore Pulsating or 
 (hcUUdinr/ Bodies. — The force exerted by two magnets on a piece 
 of iron is generally not the resultant found according to the paral- 
 lelogram-law from the two forces which each magnet would exert 
 by itself if the other were removed. For the direction of the 
 greatest increase or decrease of the energy in the field due to both 
 magnets is in general altogether different from the ])arallelogram- 
 resultant of the two vectors which give the direction of this increase 
 or decrease in the fields of the two magnets separately. It is there- 
 fore not astonishing that we get results which are in the most 
 striking contrast to the principle of the parallelogram of forces, 
 considered, it must be emphasized, as a physical principle, not 
 merely as a mathematical ])rinciple ; /. e., as a means of the 
 abstract representation of one vector as the sum of two or more 
 other vectors. 
 
 In this way we may meet with very peculiar phenomena, which 
 have great interest here, because they are well suited to show 
 how the analogy between hydrodynamic and magnetic phenomena 
 goes even into the most minute details. We shall consider here 
 only the simplest instance of a phenomenon of this kind. 
 
 Let a piece of iron be attached to a cork floating on the surface 
 of the water. If a magnetic north pole be placed in the water a 
 little below the surfiice, the piece of iron will be attracted to a 
 point verti(!aliy above the pole. If a south pole be placed in the 
 same vertical symmetrically above the surface, nothing peculiar is 
 ol)serve<l; the |)iece of iron is held in its position of equilibrium 
 more strongly than before. But if the second pole be a north pole, 
 the iron will seem to be repelled from the point where it had pre- 
 viously stable e(juilil)rium. It will move out to some point on a 
 circle, the diainrtcr of which is about j7_ of the distance between the
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 47 
 
 poles. If the same experiment were made with a piece of bis- 
 muth aud sufficiently strong magnetic poles, the force would be 
 in every case the reverse. It would have unstable equilibrium in 
 the central point between two poles of opposite sign, and would 
 seem to be repelled from this point. But if the two poles were 
 of the same sign, the bismuth would seem to be attracted to the 
 point which previously repelled it, and it would be drawn to this 
 point from any point within the circle mentioned above. On 
 the circle itself it would have unstable equilibrium, and outside 
 it would be repelled. 
 
 These peculiar phenomena are at once understood if we re- 
 member that the central point between two poles of the same 
 sign is a neutral point, where the energy of the field has a mini- 
 mum (Fig. 5, b), and that the bismuth must move towards this 
 point, the iron from it. 
 
 To make the corresponding hydrodynamic experiment two pul- 
 sators may be placed one vertically above the other, and a light 
 body (Fig. 19, a) or a heavy body (Fig. 19, b) brought between 
 them. Then if they pulsate in opposite phase, the light body will 
 be repelled from, and the heavy body attracted to the central point 
 between the two pulsating bodies. But if the phase be changed, so 
 that the two bodies pulsate in the same phase, the light body will 
 be attracted to this central point from all points inside a circle 
 whose diameter is about Jq of the distance between the pulsating 
 bodies. At all points outside of this circle it will be repelled. The 
 heavy body, on the other hand, will be repelled from the center to 
 some point on the circle, but attracted from any point outside the 
 circle, so that it will be in stable ecpulibrium on the circle. 
 
 17. Mutudl Reactions between Bodies with Induced Magnetiza- 
 tiom or with induced Oscillations. — Besides the direct actions of 
 magnets on a piece of soft iron, we have also actions between any 
 two pieces of soft iron which are acted upon by a magnet. This is 
 of special interest, because it is upon this that the formation of the 
 representations of fields of force in the classical experiment with 
 iron filings depends.^ The iron filings lying in the same line of
 
 48 
 
 FIELDS OF FORCP:. 
 
 force acquire poles of opposite sign facing each other, and therefore 
 chain together. Iron filings lying near each other on a line nor- 
 mal to a line of force have, on the other hand, poles of the same 
 sign facing each other, and therefore the chains formed mutually 
 repel each other, so that they become separated by distinct inter- 
 vals. It is worth meutiouing that, if the same experiment could 
 be made with filings of a diamagnetic body, such as bismuth, the 
 chain would be formed in the same way. For wdien the poles of 
 
 Fig. 19. 
 
 all the filings are changed at the same time, the direction of the 
 forces between them will be unchanged. 
 
 Similar actions will be observed between particles which take in- 
 duced oscillations in the hydrodynamic field, except for the differ- 
 ence residting from the direction of the force, which is opposite in 
 every ca.se. The particles, therefore, will chain together normally 
 to the lines of flow in the fluid ; they will arrange themselves as 
 layers whici) follow the e(piip()tcntial surfaces, and which, as a con- 
 sequence of mutual repulsion, are separated from each other by 
 empty spaces. It is indifferent whether for the experiment we
 
 INVESTIGATION OF DYNAMICAL i'ROPERTIES. 
 
 41) 
 
 take a light powder, which would correspond to the iron filings, or 
 a heavy powder, which would correspond to the bismuth filings. 
 
 For practical reasons, it is ])referable to use a heavy powder, 
 which, in order that the experiment succeed nicely, must be 
 fairly homogeneous. A good powder may be obtained from com- 
 mon red lead, if both the finest and the coarsest particles be re- 
 moved by washing. This is distributed on a glass plate, directly 
 above which is placed for a few seconds a pulsating or an oscillat- 
 ing body with very intense pulsations or oscillations. The powder 
 immediately arranges itself along the expected curves. 
 
 
 Fig. 20. 
 Fig. 19 gives the circles of a section through the spherical equi- 
 potential surfaces surrounding a pulsating body, and Figs. 20 
 and 21 give the more complicated curves of plane sections through 
 the equipotential surfaces produced by two bodies pulsating in the 
 same and in opposite phase, respectively. In a similar way Fig. 
 22 gives a section through the system of equipotential surfaces 
 around an oscillating body. As is easily seen, the curves thus 
 obtained are normal to the lines of force or of flow represented by 
 
 Figs. 4-8. 
 7
 
 50 
 
 FIELDS OF FORCE. 
 
 It is worth remarking that the dynamical principle which ex- 
 plains the formation of these figures is the same as that which 
 explains the formation of Kundt's dust-figures in the classical 
 experiment for the measurement of the velocity of sound in gases. 
 Our figures also show a striking likeness to the ripple marks formed 
 in the sand along the shores by the waves. And even though the 
 dynamical princi})le developed here does not fully account for the 
 peculiarities of these ripple marks, especially when they have 
 great dimensions, it is certainly the principle which accounts for 
 
 Fig. 21. 
 
 the beginning of their formation. The fossil ripple marks, which 
 are well known to the geologists, then prove that the laws of hydro- 
 dynamic fiekls of force, which I develop before you in this lecture, 
 were the same in jirevious geological periods as they are to-day. 
 
 18. }oHi<r)i (did Electric Currents. — We have obtained the 
 most complete analogy possible of hydrodynamic phenomena to the 
 phenomena of electrostatics or of magnetism, the only difference 
 being that depending upon the inverse pole-law. 
 
 Our investigation of the geometry of the field showed us
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 
 
 51 
 
 that we meet with difficulties if we try to extend the analogy 
 beyond this point. The discovery of a complete dynamical analogy 
 to the phenomena of electromagnetism therefore necessarily sup- 
 poses a more or less complete modification of the views wiiich have 
 led us to the discovery of the partial analogy already developed. 
 To prepare for a discovery of this kind we can hardly do better 
 than to discuss the other conditions which lead to a partial 
 analogy which is related to the analogy which we have developed, 
 although it does not form an immediate continuation of it. 
 
 Ficx. 22. 
 
 As we remarked when we discussed the geometry of the fields, 
 there is an analogy, discovered by v. Hei.mholtz, between the 
 magnetic fields of electric currents and hydrodynamic fields de- 
 pending upon vortex motion. This geometric analogy very nearly 
 forms a continuation of the analogy with which we have been 
 mostly occupied, the only reason why it cannot form a perfect 
 continuation being that the fluid vortex must always go around 
 in the same direction, so that a vortex of vibratory nature is im- 
 possible. But taking this analogy as it is, detached from the pre-
 
 52 FIELDS OF FORCE. 
 
 ceding analogy, we will examine whether in this case also there 
 exists an analogy between the dynamics of the two systems. 
 
 Let us first consider a rectilinear cylindrical vortex in the 
 middle of a tank, which is itself at rest. The motion of circu- 
 lation around the vortex, which corresponds to the magnetic field 
 around the corresponding electric current, will be perfectly sym- 
 metrical. The distribution of the pressure will, therefore, also 
 be symmetrical, and there will be no resultant force against the 
 vortex. Nor will any such force appear if a common motion of 
 translation be communicated to the tank and to the vortex. 
 Otherwise it would be possible to discover by an experiment of 
 this kind the motion of the earth. 
 
 But now let us suppose the motion of translation to be given to 
 the tank only, while the vortex, or a rotating rigid cylinder sub- 
 stituted for the vortex, be held still. There will then be a dis- 
 symmetry in the distribution of the motion on the two sides of the 
 rotating cylinder; on one side, the motion of translation will be 
 added to, on the other sjde, subtracted from the motion of circulation 
 around the cylinder. As we have in this case a stationary motion 
 depending upon a potential, there will be in the fluid a diminutiou 
 of the j)ressure proportional to the kinetic energy in the fluid 
 motion, and therefore an excess of pressure on the side where 
 there is a neutralization of the two motions. The cylinder, there- 
 fore, is driven transversely through the field, in the direction 
 in which ihcrc is additi(m of the velocities. This corresponds 
 exactly to the transverse motion of an electric current throup-h a 
 homogeneous magnetic field, but with the same difference of sign 
 as befi)re ; the electric current is driven iu the direction in which 
 the field intensity due to the current is neutralized by that due to 
 the homogeneous field. 
 
 The rectilinear cylindrical vortex which we have considered may 
 now be an element of any vortex. Therefore we may draw this 
 general conclusion ; the elements of any vortex which is station- 
 ary in space, will, in the hydrodynamic field, be subject to a force 
 oppositely corresiwiiding to thattc» which the elements of the corre-
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 53 
 
 spending electric current are subject in the corresponding magnetic 
 field. As special consequences, we deduce, for example, that par- 
 allel vortices which rotate in the same sense, and which correspond 
 thus to currents of the same direction, will repel, while vortices 
 rotating in the opposite direction will attract each other. 
 
 As is seen from this deduction, the condition that the vortices 
 should be stationary in space is essential. If the elements of the 
 vortices participate in the motion of the surrounding field, we come 
 back to the case where the rectilinear vortex had the same motion 
 as the tank, and in this case there was no force. The analogy 
 which we have found is therefore strictly limited to the case of 
 stationary electromagnetism. Thus for two reasons this restric- 
 tion is imposed upon the analogy. As we saw in the investiga- 
 tion of the geometry of the analogy, the constancy of the vortices 
 makes liydrodynamic phenomena corresponding to the induction 
 of currents impossible. Now we see that the mechanical forces 
 have values analogous to those acting against the electric cur- 
 rents, only when the vortices which correspond to the electric 
 currents are perfectly stationary in space. The analogy, there- 
 fore, is a limited one ; but even in its limited state it may give us 
 suggestions. 
 
 19. Experiments with Rotating Ci/linders. — Simple cases of the 
 results developed may easily be tested experimentally. By means 
 of turbines driven by air-jets, we may set metal cylinders into ro- 
 tation, which in turn produce the required circulation of the sur- 
 rounding water masses in consequence of friction. One such 
 cylinder may be held in the hand by means of a suitable support. 
 Another may be introduced into the instrument previously used as 
 a pulsation- or oscillation-balance. It is necessary, hoAvever, in 
 order to prevent the cylinder in the balance from taking by itself 
 a translatory motion through the fluid, always to use two oppo- 
 sitely rotating cylinders which are arranged symmetrically about 
 a vertical axis through the pivot (Fig. 23). 
 
 With this instrument, it is easily shown that cylinders rotating 
 in the same direction repel, and that cylinders rotating in the 
 opposite direction attract.
 
 54 
 
 FIELDS OF FORCE. 
 
 AVe have observed also that a non-rotating cylinder effects the 
 configuration of the hydrodynamic field, just as a cylinder of 
 
 Fig. 23.
 
 INVESTIGATION OF DYNAMICAL PROPERTIES. 55 
 
 infinite diamagnetivity effects the magnetic field (I, 16). Even 
 this geometric analogy is accompanied with an inverse dynamic 
 analogy ; it is easily seen that the rotating and the resting cylin- 
 der attract each other, just as a wire, carrying an electric current, 
 and a diamagnetic body repel each other.
 
 III. 
 
 THE GEOMETRIC PROPERTIES OF ELECTRO- 
 MAGNETIC FIELDS ACCORDING TO 
 MAXWELL'S THEORY. 
 
 1. C. A. Bjerknes' Problems and Metlwds. — All the phenomena 
 investigated in the preceding lectures by elementary reasoning 
 and experiment were found originally through mathematical 
 analysis by the late Professor C. A. Bjerknes. While searching 
 for plienomena of hydrodynamics which should have the appear- 
 ance of actions at a distance, he solved the problem of the simul- 
 taneous motion of any number of spherical bodies in a liquid. 
 The discussion of the solution led him to results which he verified 
 later by a series of experiments, of which I have shown you the 
 most important, using, however, instruments of improved con- 
 struction. 
 
 We apparently deviate from the historical method in taking the 
 elementary reasoning and experiment first and then proceeding to 
 the mathematical theory. But this deviation may in some sense 
 be more ap])arent than real. For the phenomena to be examined 
 certainly had in the -mind of the discoverer the form of ideal 
 experiments long before their final mathematical solution was 
 obtained. And the exact calculations were, in part, at least, pre- 
 ceded by elementary reasoning, which was not always correct 
 perhaps and of which the greater part was lost after the exact 
 mathematical solution was found. We may therefore have good 
 reason to believe that, starting as we have done with elementary 
 reasoning and experiment, we have in some sense restored the 
 original method of the discoverer, improved according to our 
 present exact knowledge of the subject. 
 
 2. Tlir Proh/nii of Ati(f/o(/ics. — Proceeding now to the mathe- 
 matical theory we shall also, in one sense, deviate considerably from 
 
 56
 
 GEOMETRIC EQUATIONS OF ELECTROMAGNETIC FIELDS. 57 
 
 the original method followed by the discoverer. At the begin- 
 ning the solution of the problem of spheres was certainly the most 
 natural way of submitting the vague anticipations to a rigorous 
 test, for this was the time when the theory of the action at a dis- 
 tance was predominant, and the discovery of the simplest and most 
 striking instances of apparent actions at a distance was the most 
 fascinating result for a man opposing this theory to strive for. 
 
 But time has chano-ed. The doctrine of action at a distance 
 has been given up, and it is the aim of no natural philosopher to 
 oppose it. The time of fields of force has come, and it is our aim 
 now to widen and deepen our knowledge of these fields. The 
 hydrodynamic phenomena discovered by C. A. Bjerkxes were 
 field phenomena, and their analogy to electrical phenomena are 
 even still more striking according to our new views. But the 
 change of view also suggested a quite new method of developing 
 the results, with unexpected facility and generality. Of course, if 
 there exists a close analogy between hydrodynamic and electromag- 
 netic fields, this analogy must be contained implicitly in the funda- 
 mental equations of the two kinds of fields, namely in the hydro- 
 dynamic equations of motion on the one hand, and in Maxwell's 
 equations of the electromagnetic field on the other. And this is 
 exactly what I am going to show you, namely, that the analogy 
 may be developed directly from these two sets of equations. 
 
 The method thus indicated is, indeed, perfectly plain and easy. 
 There is no difficulty in finding the properties of hydrodynamic 
 fields, and the only real difficulty with which we meet arises from 
 the imperfection of our knowledge of electromagnetic fields. To 
 lay the safest possible foundation for our research we have first to 
 analyze carefully our knowledge of these fields. This will be the 
 object of the lectures of to-day and to-morrow. 
 
 3. MaxweWs Theory. — Our knowledge of electromagnetic fields 
 is contained in what is generally called Maxwell's theory. This 
 theory does not tell us what electromagnetic fields are in their true 
 nature. It is a formal theory, bearing upon two aspects of the 
 properties of the fields. What are generally called Maxwell's
 
 58 FIELDS OF FORCE. 
 
 equations give a very full description of the variation from time 
 to time of the geometric configuration of electromagnetic fields. 
 To this geometric theory is only feebly linked the much less devel- 
 oped theory of the dynamical properties of these fields. 
 
 INIaxwell's theory has a central core, generally called the 
 equations for the free ether, relating to which there is good 
 agreement among diflFerent writers. But this agreement ceases 
 when we pass to the equations for ponderable bodies and for mov- 
 ing media, and, as will be seen, the full discussion of the analogy 
 will depend upon certain details of the theory for this general 
 case. Proceeding to outline the theory, I shall follow principally 
 Oliver Heaviside,* whom I have found to be my safest guide 
 in this department of physics for several reasons, of which I will 
 emphasize two ; that he uses a perfectly rational system of units, 
 and that he takes into consideration more fully than other writers 
 the impressed forces, which play a great part, from a certain point 
 of view even the greatest part, in the theory to be developed. But 
 instead of Heaviside's I shall use my own notation, chosen partly 
 to economize letters, p.artly that analogies and contrasts in the 
 things shall be reflected in analogies and contrasts in ' the 
 notation. 
 
 In thus outlining Maxwell's theory I wish to emphasize that 
 I do not introduce anything new. AVhat I introduce I have found 
 in other authors, who were perfectly uninfluenced by the search 
 for the hydrodynamic analogy. The guarantee for an unpreju- 
 diced test of this analogy is, therefore, so far as I can see, perfect. 
 
 4. ImJudivHij. — To a material medium we attribute two con- 
 stants, defining its specific properties in relation to the two kinds 
 of fields. These two constants define, so to speak, the readiness 
 of the medium to let electric or magnetic lines of induction pass, 
 and may l)c called the electric inductivity, a, and the magnetic in- 
 durtivity, /3, 
 
 We do not know the exact nature of the properties defined by 
 these constants. They can, therefore, not be determined in abso- 
 
 * Oliver IleaviHide, Electromagnetic Tlieory. Vol. I. London, 1893.
 
 GEOMETRIC EQUATIONS OF ELECTRODYNAMIC FIELDS. 59 
 
 lute measure. What we can measure arc only their ratios for 
 any two media 
 
 a 
 
 If a, and /3^ be the constants of the free ether, these ratios are 
 called the specific inductive capacities, electric and magnetic 
 respectively, of the medium which has the inductivities a 
 and /9. 
 
 When we consider thus the properties of any medium in rela- 
 tion to the fields as defined by one electric and by one magnetic con- 
 stant only, we limit ourselves to the consideration of strictly iso- 
 tropic substances, which remain isotropic even when strained, as 
 is the case, for instance, with liquids. But any degree of hetero- 
 geneity may be allowed. These suppositions give to the fields 
 exactly the generality wanted for our purpose. 
 
 5. Electric and 3Iagnetic Vectors. — A^^e will consider in this 
 lecture the geometric description of electromagnetic fields. To give 
 this description, a series of special electric and special magnetic vec- 
 tors has been introduced. 
 
 We believe that these vectors represent real physical states exist- 
 ing in, or real physical processes going on in the medium whicli 
 is the seat of the field. But the nature of these states or processes 
 is perfectly unknown to us. What still gives them, relatively 
 speaking, a distinct physical meaning is, as we shall show more 
 completely in the next lecture, that certain expressions formed by 
 the use of these vectors represent quantities, such as energy, force, 
 activity, etc., in the common dynamical sense of these words. 
 These quantities can be measured in absolute measure. But their 
 expressions as functions of the electric or magnetic vectors contain 
 always two quantities of unknown physical nature. A\'hcn once 
 the discovery of a new law of nature allows us to write another 
 independent equation containing the same unknown quantities, we 
 shall be able to define perfectly the nature of the electric and mag- 
 netic vectors, and submit them to absolute measurements in the real 
 sense of this expression. Provisionally, we can only do exactly
 
 ()0 FIELDS OF FORCE. 
 
 the same as does the mathematician in problems where he has more 
 unknowns than e(|uations, viz., content ourselves with relative 
 determinations, considering provisionally one or other of the un- 
 known ((uantities as if it were known. But we retain the symbols 
 for the unknown quantities in all formulae bearing upon the pure 
 theory of electromagnetic phenomena, for this will be the best 
 preparation for the final solution of the problem. 
 
 This imperfect knowledge is, of course, also the reason why 
 our tlieory of electromagnetic fields is split into two different, 
 loosely connected, parts ; first, the geometric theory of the fields, 
 where the relation of the vectors to time and space is considered 
 independently of ev^ery question of the physical sense of the 
 vectors ; and second, the dynamical theory of the fields, where 
 the question of the nature of the vectors is taken up, but only 
 imperfectly solved. 
 
 G. C/asslJicatlon oj the Vectors. — The vectors introduced to 
 describe the fields may be divided into classes differing from each 
 other in their mathematical properties, or in the physical facts to 
 which they rehitc. 
 
 On the one hand, the electric as well as the magnetic vectors 
 are divided in two classes, designated generally ixs forces and fluxes. 
 As the forces cannot be proved to have anything to do with forces 
 in tlie classical dynamical sense of the word, a more neutral name 
 will be preferable. I will therefore use the words field intensities 
 and fluxes. Between field intensities and fluxes there is this re- 
 lation : by the multiplication of a field intensity by the induc- 
 tivity of the medium a corresponding flux is formed. 
 
 Field intensities and fluxes are vectors of different physical 
 nature. They cannot therefore be added together. This is an 
 im|)ortant remark. For, according to previous imperfect views of 
 the electromagnetic problem, this distinction was not made, and 
 mu(-Ii confusion was caused by the lumping together of hetero- 
 geneous ([uantities. But in the case of electricity, as well as in 
 magnetism, any two field intensities may be added together, like- 
 wise any two fluxes.
 
 GEOMETRIC EQUATIONS OF ELECTROMAGNETIC FIELD. 61 
 
 Taking now another point of view, we can divide the field in- 
 tensities into induced, and impressed or energetic field intensities, and 
 the fluxes likewise into induced, and iinpressed or energetic fiuxes. 
 
 Tiie theory of induced fluxes and field intensities we have to some 
 extent really mastered. Maxwell's equations are the laws princi- 
 pally obeyed by these vectors. But in order to complete the sys- 
 tem formally, the impressed or energetic fluxes and field intensities 
 are introduced. They represent certain states, or processes, under 
 certain circumstances existing in, or going on in, the matter, and 
 which are ultimately the origin of every electric or magnetic phe- 
 nomenon. The intrinsic polarization in the permanent magnet, 
 or in the pyroelectric crystal, is therefore represented by vectors 
 of this class. They are introduced further as auxiliary vectors 
 for the representation of the creation of electric energy by contact- 
 electricity, in the thermopile, or in the voltaic battery. As the 
 existence or the supply of electric or magnetic energy is related 
 ultimately to states or processes represented by these vectors, I 
 have termed them energetic vectors, a name given originally by C. 
 A. Bjerknes to the corresponding hydrodynamic vectors. 
 
 From the fundamental vectors thus defined we may form new 
 ones by the addition of vectors of the same kind. Thus the ad- 
 dition of the induced and the energetic field intensities gives the 
 total or actual field intensities, and the addition of the induced and 
 the energetic flux gives the total or actual fluxes. We have thus 
 introduced six electric and six magnetic vectors. But in each 
 group of six vectors only two are really independent of each other, 
 and thus only two are really needed for the full description of the 
 electric or the magnetic field. Which pair of vectors it will be 
 convenient to choose as independent will depend upon the nature 
 of the problem to be treated. But a certain pair of vectors seems 
 in the majority of cases to turn out as the most convenient ; this 
 is the actual fl.iLv and the induced field inten^^iti/. These we there- 
 fore distinguish beyond the others, calling them simply the flux, 
 and the field intensity, in every case when their qualities as actual 
 flux and induced field intensity need not be specially emphasized.
 
 62 
 
 FIELDS OF FORCE. 
 
 7. Notation. — It is very convenient for our purpose to intro- 
 duce such notation as to make it at once evident to which class or 
 group the vector belongs. To attain this I denote fluxes with capi- 
 tals and field intensities with the corresponding small letters. On 
 the other hand, actual, induced, and energetic vectors are desig- 
 nated by the subscripts a, i, e, but with the exception that the letters 
 designating the flux and the field intensity, according to the defini- 
 tions above, are distinguished by the omission of subscripts. 
 Finally, for the electric vectors I use the first, and for the magnetic 
 vectors the second letter of the Latin alphabet, corresponding to 
 the first and second letter of the Greek alphabet introduced above 
 to represent the inductivities. 
 
 The system of notation is contained in the following scheme : 
 
 (^) 
 
 
 Electric. 
 
 Magnetic. 
 
 
 Flux. 
 
 Field iutensity. 
 
 Flux. 
 
 Field intensity. 
 
 Actual 
 
 Induced 
 
 Energetic 
 
 A 
 
 A; 
 
 A, 
 
 a.a 
 
 a 
 
 a.e 
 
 B 
 B. 
 
 Be 
 
 b 
 be 
 
 Electric inductivity a. 
 
 Magnetic inductivity /?. 
 
 Between each group of six vectors there are, according to what 
 is stated above, four relations, namely : 
 
 (a) 
 
 A = A. + A^, 
 a, = a + a , 
 A. = aa, 
 
 K = Ota , 
 
 I ' e' 
 
 b = b + b , 
 B, = /3b, 
 
 B, = /3b^. 
 
 By different eliminations we can of cour.se give different forms 
 to these equations of connection. AVlien we agree to use the flux 
 and the field intensity as the fundamental vectors, we need the 
 efpiations of connection, especially if vectors of the energetic group 
 have to be introduced. As we prefer generally in such cases
 
 GEOMETRIC EQUATIONS OF ELECTROMAGNETIC FIELDS. 63 
 
 to introduce the energetic flux, we shall usually have to employ 
 the following form of the equations of connection, 
 
 (6) A = aa + A , B = /3b + B^. 
 
 I am aware, of course, that the multiplication of systems of 
 notation, already too numerous, may be objectionable. But it will 
 serve for my excuse, I hope, that suggestive notations are perhaps 
 nowhere of greater importance than in researches of a comparative 
 nature. The question of a system of notation, at the same time 
 simple and suggestive, with reference to the whole of theoretical 
 physics, will, I think, necessarily arise sooner or later. 
 
 8. Conductivlti/, Time of Relaxation. — Besides their electric 
 and magnetic inductivities, some or most media have still an in- 
 trinsic property, their electric conductivity. The constant best 
 suited to represent this property in the fundamental equations is 
 the time of relaxation, introduced first by E. Cohn. If an elec- 
 tric field in a conducting medium be left to itself, its electric 
 energy will be transformed into heat, and the electric field will 
 disappear. This may happen so that the configuration of the field 
 is left unaltered during this process of relaxation. Tlie time in 
 which the electric vector, during this process, diminishes to the 
 fraction l/e of its initial value {e being the base of the natural 
 logarithms) is the relaxation time T. This is a real intrinsic con- 
 stant of the medium, measurable moreover in absolute measure, 
 and therefore in theoretical researches to be preferred to the con- 
 ductivity 7, to which it is related by the equation 
 
 a 
 («) ^= y- 
 
 A corresponding magnetic conductivity and time of relaxation 
 is not known. It is convenient, however, in order to obtain a 
 perfect symmetry of the formulae, to introduce symbols even for 
 these fictitious quantities, say k for magnetic conductivity and T' 
 for the corresponding time of relaxation 
 
 w ^' = f •
 
 (54 FIELDS OF FORCE. 
 
 9. Integral Form of the Fundamental Laws. — Using induced 
 field intensity, actual, and energetic flux, avb have always to re- 
 member first the equations of connection 
 
 A = aa + A^, 
 ^"^ B = ^b + B^. 
 
 A set of cross connections between electric flux and magnetic 
 field intensity, and vice versa, between magnetic flux and electric 
 field intensity, is contained in the two " circuital " laws, to use 
 Heaviside's expression. To find the mathematical expressions 
 of these laws we consider a surface bordered by a closed curve. In 
 case the medium is moving, the surface should also move, follow- 
 ing exactly the material particles with which it coincides at the 
 beginning. To coordinate the positive side of the surface with 
 the positive direction of circulation on the bordering curve we 
 utilize the positive screw-rule. Denoting by r the radius vector 
 from a fixed origin to a point of the closed curve, by <h the vector- 
 line element of the curve, by ds the vector-surface element of the 
 surface, the two circuital laws may be written 
 
 If* ■'^= + /r *■''= = /" ■*' 
 
 The sum of the surface integrals of the first equation is generally 
 called the electric current through the surface, the first represent- 
 ing the displacement-current in the widest sense of this word, and 
 the second the conduction current. In the same way the surface- 
 integrals of the second equation represent the magnetic current, 
 the second term, which represents the magnetic conduction-cur- 
 r<'nt, being merely fictitious. Utilizing these expressions, the core 
 of our knowledge of the properties of electro-magnetic fields in 
 relation to time and space may be expressed in the following 
 propositions. 
 
 TJic r/ectrlc current tlirourjh a moving material surface equals the
 
 CxEOMETRIC EQUATIONS OF ELECTROMAGNETIC FIELDS. G5 
 
 'po.ntive line integral of the magnetic field intensity round the border 
 of the surface. 
 
 The magnetic current through a moving material surface equals 
 the negative line integral of the electric field intensity round the 
 border of the surface. 
 
 To these equations, containing the two fundamental laws, we 
 have to add two equations containing the definition of two im- 
 portant auxiliary quantities, the electric and the magnetic mass, or 
 equivalently, the electric and the magnetic density. Calling E the 
 electric and M the magnetic density, then the electric or the mag- 
 netic mass contained within a closed surtace is the volume integral 
 respectively of E or M within the surtace. These masses are de- 
 fined as functions of the fluxes by the equations 
 
 fEdT= fAds 
 
 fMdr = Jb • ds, 
 
 dr being the element of volume. Thus the mass within a surface 
 is defined by the total flux through the surface. 
 
 It must be emphasized that these equations are, from our point 
 of view, only equations of definition, not laws of nature. This 
 assertion is not contradicted by the historical fact that the notion 
 of masses was first introduced, and later the vectors defined by 
 use of the masses, while we now consider the vectors as the 
 fundamental quantities, and define the masses by the vectors. It 
 seems to be an empirical fact, however, that no magnetic mass ex- 
 ists, and this assertion then contains a law of nature to which the 
 magnetic flux is subject, and which limits the generality of the 
 magnetic field. But for the sake of analytical generality and the 
 advantages of a complete symmetry, it will be convenient to retain 
 the symbol M for magnetic density in our formulae. 
 
 To these fundamental equations a system of supplementary con- 
 ditions is usually to be added. Thus, it is generally considered 
 that the values of each inductivity, « and fi, and the relaxation 
 time T are known at all points of the field. The same supposition 
 9
 
 66 FIELDS OF FORCE. 
 
 is generally made for the energetic fluxes, and for the electric 
 and magnetic densities. In the case of conductors a knowledge of 
 the total electric mass only for each conductor is wanted. This 
 sort of special knowledge is wanted only for material bodies, and 
 not for the free ether. For it is generally admitted that here the 
 inductivities have constant values, a,,, and '^^, that the relaxation- 
 time is infinite, l/T = 0, and that energetic vectors and electric or 
 magnetic densities do not exist, A^ = B^ = 0, jE'= 31 = 0. These 
 conditions very much simplify the problems. 
 
 10. Differential Form of the Fundamental Equations. — From 
 the integral forms above we can, by a well known mathematical 
 process, pass to the differential form of the same equations, and 
 thus arrive at the form of the system of electromagnetic equations 
 generally most convenient for practical use. 
 
 The equations of connection of course retain their form 
 
 A = aa + A , 
 
 («) 
 
 B = ySb + B^. 
 
 The equations expressing the two circuital laws may be written 
 in the following simple forms, 
 
 (6,) ' = ""> "' 
 
 k = — curl a, 
 
 where the auxiliary quantities c and k are the electric and the 
 
 magnetic current densities respectively, the full expressions for 
 
 which are 
 
 rA , 1 
 
 c = .^ + curl (A X V) + (div A)V + ^ A, 
 
 iK) 
 
 k = ',. + curl (B X V) + (div B)V -f ^,B, 
 
 V being the velocity of the moving medium, and djdt the local time 
 differentiator, which is related to the individual time differentiator 
 used above by the Eulerian relation 
 
 /7 \ d d 
 
 <'^=> * = 5l + ^'^-
 
 GEOMETRIC EQUATIONS OF ELECTROMAGNETIC FIELDS. 
 
 67 
 
 The second equation contains two terms which represent merely 
 fictitious quantities, namely, (div B) V, which represents the mag- 
 netic convection-current, and \ / T B, which represents the mag- 
 netic conduction current. 
 
 The equations of definition of the electric and magnetic densities 
 
 finally take the form 
 
 ^ E = div A, 
 
 (^) M= div B. 
 
 To these fundamental relations we add the equations which give 
 the special features of the free ether, namely, 
 
 .=.„ J. = 0, ^=0, A.= 0, 
 
 {d) . 
 
 ^=^. J.-0, J/=0, B. = 0, 
 
 which are satisfied in all space outside the bodies. 
 
 1 1 . SMionary State. - The principal feature of electromagnetic 
 fields, as expressed by the equations above, is this : every varia- 
 tion in time of an electric field is connected with the existence of 
 a magnetic field of a certain geometric quality; and vicev^rsa, 
 every variation in time of a magnetic field is connected with the 
 existence of an electric field of a certain geometric quality. 
 
 This close cross connection of electric and magnetic phenomena 
 is reduced to a feeble link in the case of stationary phenomena, 
 and disappears completely when we pass to static phenomena. 
 To consider stationary fields, that is, fields which do not vary in 
 time, let the medium be at rest, V = 0, and let the v-ectors A and 
 B have values which are at every point of space independent of 
 the time. The expressions (10,6,) for the two current densities 
 reduce to 
 
 (a) 
 
 C = m A, 
 
 k= j,B.
 
 68 FIELDS OF FORCE. 
 
 The first of these equations is the most general expression for 
 Ohm's law for the conduction-current, which is thus the only- 
 current existing under stationary conditions. The second equa- 
 tion gives the corresponding law for the fictitious magnetic cur- 
 rent. The currents are the quantities which connect the elec- 
 tric fields with the magnetic fields, and vice versa. But utilizing 
 the invariability of the current, we can now simply consider the 
 distribution of the currents in the conducting bodies as given, 
 and thus treat the two stationary fields separately, without any 
 reference to each other. 
 
 Writing the equations of the two stationary fields, we have 
 
 A = aa -f A , B = ySb -f B^, 
 
 (i,) curl a = — k, curl b = c, 
 
 div k= E, div B = Jf, 
 
 where the current densities c and k are now among tjie quanti- 
 ties generally considered as given. To these fundamental equa- 
 tions the conditions for the free ether must be added. The 
 condition that the free ether has no conductivity implies now 
 that no current whatever exists in it ; these conditions can be 
 written 
 
 ^ = /3o, 
 
 (/.) ^^=^' ^^ = ^' • 
 
 ^=0, Jf=0, 
 
 k = 0, c = 0, 
 
 for the two fields respectively. 
 
 Each of the two systems of equations contains one fictitious 
 (juantity. The equations for the electric field contain the sta- 
 tionary magnetic current density k, and the equations for the elec- 
 tric field contain the density of magnetism M, both of which are 
 fictitious. 
 
 12. static Slate.— jr, in the equations for stationary fields, we 
 suppose the current density to be everywhere nil, we get the
 
 GEOMETRIC EQUATIONS OF ELECTROMAGNETIC FIELDS. G9 
 
 equations for static fields, 
 
 («l) 
 
 A = aa + A^, 
 curl a = 0, 
 div A = B, 
 
 B = ^b + B^, 
 curl b = 0, 
 div B = M, 
 
 with 
 
 the conditions for the free ether, 
 
 
 a = a^, 
 
 ^ = /3o, 
 
 {\) 
 
 A=0, 
 
 Jf=0. 
 
 These static fields exist independently of each other, the links 
 which, in the general case, connect the one kind of field to the 
 other, namely, the currents, being nil. 
 
 13. The Energy Integral. — A research relating to the com- 
 pleteness of the description which the preceding equations give of 
 the geometry of the fields will be of fundamental importance in 
 the search for the analogy of these fields to other fields. As an 
 introduction to this research, we will examine from an analytical 
 point of view an integral, the physical significance of which will 
 occupy us in the next lecture, namely, the integral expressing the 
 electric or the magnetic energy of the field. 
 
 The expression for the electric energy can always be written 
 
 (a) ^ = */A-a//T, 
 
 where the integration is extended to all space. Now in the case 
 of perfect isotropy the actual field intensity is related to the flux 
 simply by the relation 
 (6) A = aa ,, 
 
 and, therefore, we have the equivalent expressions for the energy
 
 70 FIELDS OF FORCE. 
 
 Now let us write the vector-factor A, of the scalar product, iu the 
 
 form 
 
 A = — aS7(f) + curl G, 
 
 expressing it thus by a scalar potential cf) and a vector potential 
 G, as is possible with any vector. The integral may then be 
 written 
 
 4> = — hfA ■ V(f)dT + J*ia,^ • curl Gch. 
 
 To avoid circumlocution we shall suppose that there exists in 
 the field no real discontinuity, every apparent surface of discon- 
 tinuity being in reality an extremely thi-n sheet, in which the 
 scalars or the vectors of the field chauo-e their values at an exceed- 
 ingly rapid rate, but always continuously. Further, we suppose 
 that the field disappears at infinity. Both integrals can be trans- 
 formed then according to well known formulae, giving for the en- 
 ergy the new expression 
 
 (d) ^ = hf<j) div Adr + i Jg • curl a //t. 
 
 Now div A is the density of true electrification, which e.xists 
 only in material bodies. It will be sufficient, therefore, to apply 
 the first integral to material bodies only, and not to the surround- 
 ing ether. If we split the actual field intensity a^ into its induced 
 and energetic parts, we get 
 
 curl a, = curl a + curl a.. 
 
 Here, according to the fundamental equations, — curl a repre- 
 sents the magnetic current k. By analogy, — curl a^ can also be 
 said to represent a magnetic current k . By this current the in- 
 trinsic polarization, say in a turmaline-crystal, can be represented, 
 in the same way as the intrinsic magnetization can be represented, 
 according to Ampere's theory, by a distribution of electric cur- 
 rents. 
 
 Now in the case of a stationary field the current — curl a can 
 only exist in material bodies, not in the surrounding ether. And 
 the current— curl a^, or the vector a^=l/aA^from which it is
 
 GEOMETRIC EQUATIONS OF ELECTROMAGNETIC FIELDS. 71 
 
 derived, never exists outside material bodies (10, d). Therefore, 
 in the case of a stationary field it will be sufficient to apply the 
 second integral in (d) to material bodies only. 
 
 From the symmetry of the two sets of equations it is seen at 
 once that the integral expressing the energy of the magnetic field, 
 namely, 
 
 ^ = ^ Jb • h/h, 
 
 can be transformed to a form corresponding to (d), namely, 
 
 ^ = ij-v/r div Bdr + i/H • curl hdr, 
 
 involving thus the true density of magnetism, div B, and the 
 electric current, curl B^, which is made up of the true electric cur- 
 rent, curl b, and the fictitious current, curl b^, by which, accord- 
 ing to Ampere's theory, the intrinsic magnetization can be repre- 
 sented. Now under stationary conditions the true current, curl 
 a, only exists in material bodies, not in the ether, and the quantities 
 div B and curl b^ never exist except in material bodies. 
 
 We can therefore assert that the energy of the stationary field, 
 whether it be electric or magnetic, can be expressed by integrals 
 which apply to material bodies only, not to the surrounding ether. 
 
 15. Conditions for the Vanishing of tlie Stationary Field. — Let 
 
 us consider now the stationary electric field in the case where 
 
 there exists no true electrification, div A = 0, no energetic field 
 
 intensity, a^ = 0, and no true current, curl a = 0. We shall then 
 
 have 
 
 div A = 0, curl a^ = 0, 
 
 and under these circumstances the energy of the field disappears 
 completely, as is seen from the expression (13, d). But accord- 
 ing to the expression (13, c) for the same energy, which is the 
 sum of only positive elements, the flux A must disappear in every 
 part of the field. But when both the flux A and the energetic 
 field intensity disappear, it is seen from the equations of connec- 
 tion that the induced field intensity will also disappear, and there
 
 72 FIELDS OF FORCE. 
 
 will exist no electric field at all. lu the case of the magnetic 
 field perfisctly parallel conclusions can be drawn. Thus : 
 
 If there exists no true electrification, no energetic fiiix, and no 
 magnetic current, there ivill exist no stationary electric field. 
 
 If there exists no true magnetism, no energetic flux, and no elec- 
 tric current, there will exist no stationary magnetic field. 
 
 10. Unique Deter mi nateness of the Stationary Field. — From this 
 result a new one can be drawn at once. Let us consider two 
 fields, represented by the vectors A, a, and A', a', both subject to 
 the condition of having the same distribution of the energetic 
 flux A^, of the magnetic current k, and of the true electrification 
 E. The equations of the two fields will then be 
 
 A = Ota + A^, A' = aa' + A^, 
 
 curl a = — k, curl a' = — k, 
 
 div k = E, div A' = E. 
 
 T>et us consider next the field represented by the difference of the 
 vectors of the two fields, i. e., the field 
 
 A" == A' - A, 
 a" = a' — a. 
 
 As is seen at once, this field will be subject to the conditions 
 
 A" = aa", 
 curl a" = 0, 
 div A" = 0. 
 
 It will thus be a field having no energetic flux, no magnetic cur- 
 rent, and no true electrification, and it will disappear completely 
 according to the result above. Thus the fields A, a, and A', a', 
 cannot differ from each other. 
 
 Perfectly ])aralU'l developments can be given for the magnetic 
 field, aud we arrive; thus at the following parallel results :
 
 GEOMETRIC EQUATIONS OK ELECrKOMAONBTK; FIELDS, 73 
 
 According to our system of equations, Ihe Miomry declrk field 
 ;., nn^nch deU-nninM by the didribuHon of true eleeinfieal^on, of 
 eneJio Lrh fl«x, and of rnagneUc curraU ; and (Ae «ta(.on«,-y 
 „umeth field is nniquehi determined by the dMr,bufiO» of true 
 Jejnellm, of enerejeile .uujnefu; fl,„; and of eleMe cun-ent. 
 
 Tliese the'orems show tlie amount of knowledge of the geometry 
 of the stationary fields which is laid down in the equations 11, 6. 
 They contain iu the most condensed form possible onr whole knowl- 
 edjof this geometry. And the importance of these theorems 
 for our purpose is perfectly clear: if we succeed later ,n represent- 
 g the hydrodynamic field by a similar system of equations, there 
 will, u„d r similar conditions, be no chance for ditference in the 
 geometric properties of the hydrodynamic field, and the stationary 
 
 electric or magnetic field. , , t i 
 
 But before we proceed t« the investigation of the hydrodynamic 
 field we have to consider the dynamic properties of the electric and 
 the magnetic field. 
 
 10
 
 IV. 
 
 THE DYNAMIC PROPERTIES OF ELECTROMAG- 
 NETIC FIELDS ACCORDING TO 
 MAXWELL'S THEORY. 
 
 1. Electvio and Magnct'iG Etiergi/. — The Maxwell equations 
 give, as I have emphasized, only a geometric theory, bearing upon 
 tlie distribution in space of a series of vectors whose physical 
 meaning is perfectly unknown to us. To give this theory a phys- 
 ical content an additional knowledge is wanted, and this is afforded 
 by our experience relating to the transformations of energy in the 
 electromagnetic field. 
 
 The safest way, in our present state of knowledge, of establish- 
 ing this dynamical theory of the electromagnetic field, seems to 
 be this ; start with the expression which is believed to represent 
 the energy of the electric and of the magnetic field, and bring 
 into api^lication the universal principle of the conservation of 
 energy. 
 
 The general feature of the method to be used is thus perfectly 
 clear; nevertheless, the details will be open to discussion. First 
 of all, there is no perfect accordance between the different writers 
 with regard to the true expression of the energy of the fields. 
 All authors agree that it is a volume integral in which the func- 
 tion to be integrated is the half scalar product of a flux and a field 
 intensity. ]3nt opinions seem to differ as to whetlier it should be 
 the actual fluxes and field intensities or only the induced ones. Fol- 
 lowing Heavtside, I suppose that the «c^wa^ fluxes and field inten- 
 sities are the proper vectors for expressing the energy, and thus 
 write the expression for the total energy of the electromagnetic field 
 
 cl> + >p = J 1 A • slJt + / iB ■ hdr. 
 
 Here, the first integral gives the amount of Hie electric, and the 
 
 74
 
 DYNAMIC EQUATIONS OF ELECTROMAGNETIC FIELDS. 75 
 
 second the amount of the magnetic energy, the integrations being 
 extended over the whole field. 
 
 2. Localization and Continuity of Energy. — Starting with this 
 expression for the energy of the field and bringing into application 
 the principle of the conservation of energy, we can of course de- 
 duce only results strictly in accordance with the experience which 
 led us to tliis form of expression for the energy. We are able 
 then to derive the amount of mechanical work done, and conse- 
 quently the forces doing it, for the case when the different bodies 
 in the field are displaced relatively to each other. 
 
 But for the sake of the problem before us, it is very desirable 
 to go a step further, to determine not only the resultant forces 
 acting against the bodies as a whole, but also the system of ele- 
 mentary forces, which act upon the elements of volume of the 
 bodies, and of which the resultant forces are composed. Of these 
 elementary forces we have only a very limited experimental 
 knowledge, and to derive them, additional knowledge is needed, 
 which is not contained in the mere statements of the form of the 
 energy integral and of the principle of the conservation of energy. 
 We do not possess this in universally accepted form, but we admit 
 as working hypotheses the following two principles : 
 
 First, we suppose that it is allowable to speak not only of 
 amounts of energy, but also of a distribution of energy in space. 
 That this should be so is, a priori, not at all clear. The uni- 
 versal principle of the conservation of energy relates only to 
 amounts of energy. And in the model science relating to energy, 
 abstract dynamics, the notion of a certain distribution of en- 
 ergy in space seems to be often of rather questionable clearness 
 and utility. But still it may have a more or less limited useful- 
 ness. Assuming this, we admit as a working hypothesis, that 
 the energy integral not only gives the total amount of electric 
 and magnetic energy, but also the distribution of this energy 
 in space, the amount of energy per unit volume in the field being 
 
 lA a -f- IB ■ b .
 
 76 FIELDS OF FORCE. 
 
 To this princii)le of the localization of energy we add the second, 
 the principle of the continuity of energy, which is this : energy can- 
 not enter a space Avithout passing through the surface surrounding 
 this space. This principle forces us to admit a more or less de- 
 termined motion of the energy, which in connection with the trans- 
 formations of the energy regulates the distribution of the electro- 
 magnetic energy in space. To this principle we may make sim- 
 ilar objections as to the previous one. The idea of a determinate 
 motion of the energy does not in abstract dynamics seem to be 
 always very clear or useful, even though it may seem to have 
 in this branch of physics also a certain limited meaning. And 
 even though considerable doubt may fall upon these two supposi- 
 tions considered as universal principles, no deciding argument can 
 be given at present against their use to a limited extent as work- 
 ing hypotheses. 
 
 3. Electric and Magnetic Activity. — To these abstract and 
 general principles we have to add definite suppositions suggested 
 more or less by experiment. The first is this : the rate at which 
 the electric or magnetic energy is created by the foreign sources 
 of energy is given per unit volume by the scalar product of the 
 energetic field intensity into the corresponding current. This 
 princij)le was originally suggested by the observation that the rate 
 of doing work by the voltaic battery was the product of its in- 
 trinsic electromotive force and the current produced by it. And 
 it is generalized by inductive reasoning so that it is made to in- 
 clude every impressed or energetic force and field intensity, every 
 furrent, electric or magnetic, conduction current, or displacement 
 current. 
 
 Starting thus with Maxwell's equations for the general case 
 of a moving medium 
 
 c = curl b, 
 
 k = — curl a, 
 
 we can at once find the rate at which energy is supplied per unit 
 volume by the foreign sources of energy. For, multiplying
 
 DY.NAMIC EQUATIONS OF ELECTROMAGNETIC FIELDS. 77 
 
 these equations by the energetic field intensities and adding, we get 
 (^a) a^ • c + b • k = a^ • curl b - b^ • curl a. 
 
 The left hand member gives the rate at which this energy is sup- 
 plied. The discussion of the right hand member therefore will 
 show how the energy supplied is stored, transformed, or moved to 
 other places. In this discussion we shall follow the method mdi- 
 cated by Heaviside.* 
 
 4. Storage, Transformation, and Motion of the Energy. — io 
 examine the right hand member of the equation we express the 
 energetic field intensities as the differences of the actual and the 
 induced field intensities, 
 
 a^ = a„ - a, b^ = b,^ - b. 
 The equation of activity then takes the form 
 ^a) a . c + b^ • k = a , curl b - b„ • curl a - a ■ curl b + b • curl a. 
 For the last two forms we write, according to a well known vec- 
 tor formula, 
 (■J) _ a • curl b + b • curl a = div (a x b). 
 
 In the first term on the right hand side of equation {a) we in- 
 troduce for curl b the developed expression for the electric current, 
 
 (III., 10, h.). Thus 
 
 ^. 1 
 
 (c) a„curlb=a,,/^ + a„ curl(AxV)+a„VdivA+ya,A. 
 
 Remembering that A = ota,, we find easily, 
 
 K-Tt = ^^'^^r = ''^-dt-^^^'dt 
 
 8 da ., da 
 
 or finally 
 
 *0 Heavlide : On the forces, stresses and fluxes of energy in the electro- 
 n^agneti! field. Transactions of the Koyal Society, London, 1892. Electrical 
 papers, Vol. 11, p. 52L
 
 78 FIELDS OF FORCE. 
 
 cA ^ / 2 ^^ 
 
 Now we have in general (III., 10, h^) 
 da da 
 
 And if we suppose that the moving individual element does not 
 change the value of its inductivity as a consequence of the mo- 
 tion, we have dajdt = 0, and 
 
 da 
 
 And therefore 
 
 dA d 
 (c^ a ■ "^- = ^, (iA ■ a ) - V • ia- Va. 
 
 Passing to the next term in (c), we can transform it by the vec- 
 tor formula (i), writing a^ for a and A x V for b. Thus 
 
 a^ • curl (Ax V) = A x V ■ curl a^ — div [a_ x (A x V)] . 
 
 In the first right hand term we interchange cross and dot, and 
 change the order of factors by cyclic permutation. In the second 
 term we develope the triple vector product according to the well- 
 Uiiowii foi-miilii; we have then 
 
 (cj a, •curl(Ax V) = V (cuvlajx Afdiv [(a, A)V-(a^ • V)A] . 
 
 Substituting (r-.) and (c^) in (c) we get 
 
 d 1 
 
 {d\ ^" ' "''""^ ^ ^ c}t ^^^ ■ ^"^ + T^A • a« + V • [ (div A)a„ 
 
 - Kvo! + (curl aj X A] -f div {(A • ajV - (a, • V)A] . 
 
 In exactly the same way, introducing the full expression for the 
 magnetic current, we have 
 
 (,) -b,curla = ^(P-bJ+ ^,Bb, + V{(divB)b„ 
 
 - ib;^V/3 + (curl bj X B} + div [(B • bjV - (b„ • V)B}.
 
 DYNAMIC EQUATIONS OF ELECTROMAGNpync FIELDS. 79 
 
 Tlie developments (h), {d), and (r) are now introduced in (^n). 
 Suitably distributing the terms, we get 
 
 d 
 ac+bk=^{iAa +iBb} 
 
 1 1 
 
 + ^A a, + ^,B b„ 
 
 ^'^ ) +V • {(div A)a^ — ia;v^ + (curl aj x A} 
 
 + V • { (div B)b^ - ib; v/3 + (curl bj x B] 
 + div [ax b + |(A ■ a, + B • bjV} 
 + div {_(a,^- V)A + J(A - aJV - (b^- V)B + K^'^J^ }, 
 
 which is the completely developed form of the equation of activity. 
 
 The first member gives, as we have said, the rate of supply of 
 
 electromagnetic energy per unit volume, and the second member 
 
 shows how the energy supplied is used. Taking one term after 
 
 the other in each line, the common interpretation of them is this : 
 
 The first term 
 
 d 
 
 ~dt 
 
 {iA-a^+ p-b, 
 
 gives the part of the energy supplied which is simply stored as 
 electric and magnetic energy in the unit volume. The second term 
 
 1 1 
 
 ^A a„+ ^,B b„ 
 
 gives the part of the energy supplied which is wasted as heat, 
 according to Joule's law, the waste due to the fictitious magnetic 
 conduction current being also formally included. 
 
 The following two terms contain the velocity V of the moving 
 material element of volume as a scalar factor. As the equation is 
 an equation of activity, the other factor must necessarily be a 
 force, in the common dynamic sense of this word, referred to 
 unit volume of the moving particle. These factors are then the 
 forces exerted by the electromagnetic system against the exterior
 
 80 FIELDS OF FORCE. 
 
 forces, the factor of the first terra being the mechanical force de- 
 pending on the electric field, and the factor of the second term 
 being the force depending upon the magnetic field, 
 
 f = (div A)a^ — i^f Va + Ccurl a J x A, 
 ^^^ f,„ = (div B)b„ - i-bf V/3 + (curl bj x B. 
 
 The first of the two terms of (/) which have the form of a di- 
 vergence gives, according to the common interpretation, that part 
 of the energy supplied which moves away. There are two reasons 
 for this uiotion of energy, first, the radiation of energy, given by 
 
 the Poynting-flux 
 
 a X b, 
 
 and second, the pure convection of electromagnetic energy, given 
 
 by the vector 
 
 ^(A-a„ + B-bjV, 
 
 which is simply the product of the energy per unit volume into 
 the velocity. 
 
 Finally, the last term gives, according to the common interpre- 
 tation, that part of the energy supplied which, in terms of the 
 theory of the motion of energy, moves away in consequence of the 
 stress in the medium which is the seat of the field, the flux of 
 energy depending upon this stress being given by the vector 
 
 - iK • V)A + 1 ( A • aJV - (b„ • V)B + i(B • bJV, 
 
 whose divergence appears in the equation of activity. For this 
 flux of energy may be considered as that due to a stress, the com- 
 poticnt of which against a plane whose orientation is given by 
 the unit normal N is 
 
 a„(A ■ N) - (1 A ■ aJN + b^B • N) - (iB ■ bJN. 
 
 This stress splits up into an electric and a magnetic stress. And, 
 in the case of isotropy, which we assume, the first of these 
 stresses consists of a tension parallel to, and a pressure perpen-
 
 DYNAMIC EQUATIONS OF ELECTRO^IAGNETIC FIELDS. 81 
 
 dicular to the lines of electric force, in amount equal to the elec- 
 tric energy per unit volume ; the second consists of a tension and 
 pressure bearing the same relation to the magnetic lines of force 
 and magnetic energy per unit volume. This is seen when the unit 
 normal N is drawn first parallel to, and then normal to the corre- 
 sponding lines of force. 
 
 The theory thus developed may be given with somewhat greater 
 generality and with greater care in the details. Thus the aniso- 
 tropy of the medium, already existing, or produced as a conse- 
 quence of the motion, can be fully taken into account, as well as 
 the changes produced by the motion in the values of the induc- 
 tivities and in the values of the energetic vectors. On the other 
 hand, there exist differences of opinion with regard to the detads 
 of the theory. But setting these aside and considering the ques- 
 tion from the point of view of principles, is the theory safely 
 founded ? If we knew the real physical significance of the electric 
 and magnetic vectors, should we then in the developements above 
 meet no contradictions ? 
 
 This question may be difficult to answer. The theory must 
 necessarily contain a core of truth. The results which we can 
 derive from it, and which depend solely upon the pnnciple ot the 
 conservation of energy and upon the expression of the electro- 
 magnetic energy, so far as this expression is empirically tested, 
 must of course be true. But for the rest of the theory we can 
 only say, that it is the best theory of the dynamic properties ot 
 the electromagnetic field that we possess. 
 
 5 The Forces in the Electromagnetic Fiehl-^yhai particularly 
 interests us is the expression for the mechanical forces in the held, 
 (4 a) As the expressions for the electric and the magnetic force 
 have exactly the same form, it will be sufficient to consider one 
 of them. Let us take the magnetic force, 
 
 f = (div B)b„ - IKV^ + (curl b.) x B. 
 This is a force per unit volume, and if our theory is correct, this 
 expression should give the true distribution of the force acting upon 
 
 11
 
 82 FIELDS OF FORCE. 
 
 the elements of volume, and not merely the true value of the re- 
 sultant force upon the whole body. The significance of each term 
 is obvious. The first term gives the force upon the true magnet- 
 ism, if this exists. It has the direction of the actual field intensity, 
 and is equal to this vector multiplied by the magnetism. The 
 second term depends upon the heterogeneity of the bodies, and 
 gives, therefore, the force depending upon the induced magnetism. 
 The elementary force which underlies the resultant forces observed 
 in the experiments of induced magnetism should therefore be a 
 force which has the direction of the gradient, — v/3, of the induc- 
 tivity /3, and which is equal in amount to the product of this gra- 
 dient into the magnetic energy per unit volume. When we consider 
 a body as a whole, the gradient of energy will exist principally in the 
 layer between the body and the surrounding medium. It will point 
 outwards if the body has greater inductivity than the medium, 
 but its average value for the whole body will be nil in every direc- 
 tion. But the force, which is the product of this vector into half 
 the square of the field intensity, will therefore have greater aver- 
 age values at the places of great absolute field intensity, quite 
 irrespective of its direction. Hence, the body will move in the 
 direction which the inductivity gradient has at the places of the 
 greatest absolute strength of the field, i. e., the body will move 
 in the direction of increasing absolute strength of the field. And, 
 in the same way, it is seen that a body which has smaller induc- 
 tivity than the surrounding field will move in the direction of 
 decreasing absolute strength of the field. The expression thus 
 contains Faraday's well known qualitative law for the motion of 
 magnetic or diamagnetic bodies in the magnetic field. 
 
 The third term of the equation contains two distinct forces, 
 which, having the same form, are coml)ined into one. Splitting 
 the actual field intensity into its induced and energetic parts and 
 treating the curl of the vector in the same way, we get 
 
 curl b = curl b -}- curl b = c 4- c , 
 
 where c is f/ic (rue electric current, and c^ the fictitious current, by
 
 DYNAMIC EQUATIONS OF EI.ECTRO MAGNETIC FIELDS. 83 
 
 which, according to Ampere's theory, the permanent magnetism 
 may be represented. The last term of the expression for the force 
 therefore splits into two, 
 
 (curl hj X B = c X B + c^ x B, 
 
 where the first term is the well known expression for the force per 
 unit volume in a body carrying an electric current of density c. 
 The second terra gives the force upon permanent magnetization, 
 and according to the theory developed, this force should be the 
 same as the force upon the equivalent distribution of electric 
 current. 
 
 6. The Resultant Force. — As we have remarked, our develop- 
 ments may possibly contain errors which we cannf)t detect in the 
 present state of our knowledge. The value found for the elemen- 
 tary forces may be wrong. But however this may be, we know th'n^ 
 with perfect certainty ; if we integrate the elementary forces for 
 the whole volume of a body, we shall arrive at the true value of 
 the resultant force to which the body as a whole is subject. For 
 calculating this resultant force, we come back to the results of 
 the observations which form the empirical foundation of our 
 knowledge of the dynamic properties of the electromagnetic field. 
 A perfectly safe result of our theory will therefore consist in the 
 fact that the expression 
 
 (rt) F = /(div B)h Jt - /^b^V/Sc/r + /(curl bj x Bch, 
 
 where the integration is extended over a whole body, gives the true 
 value of the resultant force upon the body. By a whole body, 
 we understand any body surrounded by a perfectly homogeneous 
 gaseous or fluid dielectric of the constant inductivity ^„, which is 
 itself not the seat of any magnetism M, of any energetic mag- 
 netic flux B^, or of any electric current c. To avoid mathe- 
 matical prolixity we suppose that the properties of the body 
 change continuously into those of the ether, the layer in which 
 these changes take place being always considered as belonging to 
 the body. Thus at its surface the body has all the properties of
 
 84 FIELDS OF FORCE. 
 
 the ether. By this supposition, we shall avoid the introduction 
 of surface integrals, which usually appear when transformations of 
 volume integrals are made. 
 
 By transformations of the integrals we can pass from the above 
 expression for the resultant force to a series of equivalent ex- 
 pressions. To find one of these new expressions we split the 
 actual field intensity into its two parts, 
 
 b = b + b , 
 and we get 
 
 (6) F = /(div B)b(ZT - / Jb- VySr/r + /(curl b) x Bdr + J, 
 where 
 
 J = r(div B)b (h - f(h ■ b ) V/SJt 
 
 - f^hlv^ch + /(curlb) X Bdr. 
 
 To reduce the expression for J we consider the first term. 
 Transforming according to well known formulse, we get 
 
 /(div B)b/?T = - /Bvb//T = - jBh^vdr - /(curl b ) x- Bdr. 
 
 Substituting, we get J reduced to three terms, 
 
 {h") ]=- fB\vdT - /(b • bj v/3r7T - /lb; v/3f7T. 
 
 Introducing in the first of these integrals B = /Sb + /3b , we get 
 
 - /Bb^vr/r = - //3(bb v)fZT - f^(hh^v)dT, 
 
 in which we have to remember that the operator v works only 
 upon the vector immediately preceding it. In the first of the 
 two integrals of the right hand member we join the scalar factor /3 
 with the vector b , upon which v works, remembering /8b^ = B^. 
 A term containing VyS must then be subtracted. The second 
 integral we can change, letting the operator v work upon both 
 factors. Then 
 
 - /Bb.V'/T = - /bB^vr/r + /(bb,)v/3r7T - /|/3vbV7T.
 
 DYNAMIC EQUATIONS OF ELECTROMAGNETIC FIELDS. 8') 
 
 Finally, integrating the last term by parts and remembering that 
 b^ disappears at the surface of the body, 
 
 - fBh^vdr = - /bB, VfZr + J"(b bj V^f/r + /^b; V A?t. 
 Substituting this in (6"), we get simply 
 (6'") J = -fhBydr. 
 
 This leads to the expression 
 (c) F =/(div B)bdr -fWv/Sdr +f[cm'\ b) x Bdr -JhB^ V dr 
 
 for the resultant force. The four terras give the forces depending 
 upon the true magnetism, the induced magnetism, the electric cur- 
 rent distribution, and the i^erraanent magnetization respectively. 
 The resultant force is represented here by a system of elementary 
 forces, given by 
 
 1, = (div B)b - WvB -bB^V + (curl b) x B. 
 These elementary forces must be considered as fictitious if the 
 expression found above represents the true values of the ele- 
 mentary forces. But if our developments have not been altogether 
 trustworthv, the reverse might also be the case, or else none of 
 them may give the true values of the elementary forces, while both 
 of them give the true values of the resultant forces. 
 
 7. Other Forms for the Resultant Force. — In writing the ex- 
 pression for the resultant force we have hitherto used scalars and 
 vectors of a fundamental nature. By the introduction of certani 
 auxiliary scalars or vectors the expression for the resultant force 
 may be brought to forms of remarkable simplicity. But as this 
 is obtained at the cost of the introduction of artificial quantities, 
 the possibility that the expressions under the integral signs repre- 
 sent the real elementary forces is lost. 
 
 The transformation to these simple forms of the expression for 
 the resultant force depends upon the introduction of a vector B , 
 defined by the equation 
 (a) B = /3„b + B,
 
 86 FIELDS OF FORCE. 
 
 This has the form of the true equation of connection, excej3t that 
 the constant inductivity /3^^ of the ether is introduced instead of 
 the true inductivity of the body. B^ is therefore a virtual ener- 
 getic flux, to compensate for our leaving out of consideration the 
 variations of the inductivity. This is the well known artifice of 
 Poisson's theory of induced magnetism, which enables us to treat 
 the induced magnetism as if it were permanent. 
 
 To introduce this vector into the expression for the resultant 
 force we first remark that in the second integral of the expression 
 (6, c) we can write ^ — ^^ instead of yS. Performing the integra- 
 tion by parts throughout the whole volume of the body and remem- 
 bering tliat /3 — /Sjj disappears at the surface of the body, we get 
 
 - /AbV/Sf?T = _/ibV(/3 - y8,>?T 
 
 = /(^-^JbbvrfT. 
 
 In like manner, the transformation by parts of the integral in the 
 expression (0, e) expressing the force upon permanent magnetism 
 gives 
 
 — JbB^ V (It = jBhsydr. 
 
 The integrals for the temporary and the permanent force may now 
 be added, and remarking that equation («), in connection with 
 tlie fundamental equation of connection, gives B = (/3 — /S,,)b -|- B^, 
 we get 
 
 - JWv^dr-JhBydT = jB^bVf?T. 
 
 The substitution of tliis in ((J, c) gives the following more com- 
 pact form of tlie expression for the resultant force 
 
 (h) F = /(div B)b^/T -h /b ,bV(/T + J(curl b) x B(?t. 
 
 Here, the resultant force seen)s to come from an elementary force 
 fg = (div Bp + B^bv -f (curl b) x B.
 
 DYNAMIC EQL^ATIONS OF ELECTROMAGNETIC FIELDS. 87 
 
 A still shorter form of the resultant force and of the corre- 
 sponding fictitious elementary force may be found as follows. 
 According to a well known vector formula, we can write 
 
 jB^hvdr = Jb, vb(7T - /(curl b) x B/h. 
 
 Transforming the first integral of the second member according 
 to a well known formula and remembering that B^. = at the 
 surface of the body, we get 
 
 fB.hvdr = — /(div B,)bf/T — /{curl b) x B.dr. 
 
 Introducing this expression and remarking that, according to (c), 
 div B = /3^ div b + div B^., we get 
 
 F = ^^ /(div b)bf/T + /3^, /(curl b) x bdr, 
 
 which is the most concise form of the expression for the resultant 
 force. It is expressed here by a fictitious elementary force 
 
 f^ = yS„ (div b)b + yS, (curl b)x b. 
 
 The divergence of the field intensity, which appears here, is 
 called the free density of magnetism. The force upon true mag- 
 netism, upon permanent magnetic polarization, and upon induced 
 magnetism can be condensed into one expression, and the whole 
 force is expressed in an exceedingly simple way by the field in- 
 tensity, its divergence, its curl, and the inductivity of the sur- 
 rounding medium. 
 
 8. Eesimie — It will be convenient on account of the following 
 lectures to sum up the fundamental equations for the stationary 
 electric, and the stationary magnetic field. Using for the descrip- 
 tion of the fields the vectors of scheme III., and in some cases 
 even the artificial vectors A^. or B^. (lY., 7, a), we have first a set 
 of equations of connection, by use of which we introduce in the 
 fundamental equations the vector wanted for any special purpose. 
 Of these equations of connection we note the following, referring 
 for more special cases to the complete system (III., 7, a).
 
 88 FIELDS OF FORCE. 
 
 Electric Magnftic 
 
 A = aa„, B = ^b„, 
 
 (A) = aa + A , = ^b + B^, 
 
 = a„a + A,, = ^,b + B,. 
 
 Then we have the proper equations of the fields, which express 
 the relation between the field intensity and the current density, 
 
 (^B) curl a = — k, curl b = c. 
 
 Finally, we have the equations of definition for the density of 
 electrification, or of magnetism, 
 
 ( C) div A = B, div B = M. 
 
 To complete the geometric description of the field w^e have finally 
 a number of special conditions which are fulfilled in the free ether, 
 namely, 
 
 (A) « = «o' /3 = ^,„ 
 
 {JD,) -^=0, 31=0, 
 
 (A) k=0, c = 0, • 
 
 (A) A^=0. B^ = 0- 
 
 This set of equations gives, in the sense of the theorems (HI., 16), 
 a complete description of the geometry of the fields. 
 
 Our knowledge of the dynamics of the field is less complete. 
 According to the analysis of Heaviside, we have reason to believe 
 that the elementary force in the field per unit volume is given by 
 the expression 
 
 f = (div A)a — ia^ V^ + (curl a ) x A, 
 f„, , = (div B)b. - lb; v/3 + (curl bj X B. 
 But other forms are not excluded, and we may have 
 
 f, ., = (div A)a — ?,a- v^: + (curl a) x A — aA v, 
 (E.) .- V y - V J 
 
 f,„^ ^ = (div B)b - 1 b-VyS + (eurl b) x B - bB^v.
 
 DYNAMIC EQUATIONS OF ELECTROM AGNPync FIELDS, S!l 
 
 Our reliable knowledge is reduced to this — we get on integrat- 
 ing any of these forces for a whole body the resultant force which 
 produces the motion of the whole body. The same value of the 
 resultant force may also be found from other purely artificial dis- 
 tributions of the elementary force, for example, 
 
 or 
 
 f^ ^ = (div A)a -f- A^ av -I- (curl a) x A, 
 4 3 = (div B)b + B,.b V + (curl b) x B, 
 
 t, 4 = ^o(f^^v ^)^ + a^(curl a) X a, 
 f „, ^ = /3^(div b)b + /3,(curl b) X b. 
 
 12
 
 y. 
 
 GEOMETRIC AND DYNAMIC PliOPERTIES OF THE 
 HYDEODYNAMIC FIELD. GENERAL DEM- 
 ONSTRATION OF THE ANALOGY TO 
 THE STATIONARY ELECTRO- 
 MAGNETIC FIELDS. • 
 
 1 . Prelhiuii(u-y Remarks. — Our preliminary investigations, 
 based on elementary reasoning and experiment, have already 
 given the general feature of the analogy, which we are now going 
 to examine more closely. According to these preliminary results, 
 we have no reason to look for an analogy extending beyond the 
 phenomena termed stationary. The main feature of the analogy 
 is given by the correspondence : 
 
 flux velocity, 
 
 field intensity specific momentum, 
 
 inductivity mobility (specific volume). 
 
 To facilitate the comparison of the fields I shall denote the 
 hydrodynamic quantities by the same letters as the corresponding 
 electrical quantities. The symmetry in the properties of the elec- 
 tric and magnetic fields will make it possible to pass at once from 
 the comparison with the electric field to the comparison with the 
 magnetic field. 
 
 2. The I Iijdr adynamic Equations. — The basis of our investi- 
 gation will be the hydrodynamic equations, of which there are 
 two; the scalar ecpiation for the conservation of the mass, generally 
 called the equation of continuity, and the vector equation of motion. 
 
 a being the specific volume of the fluid, A the vector velocity, 
 and fJ'df representing the individual time-differentiation, the equa- 
 tion of continuity may be written 
 
 (a) = div A. 
 
 ^ ^ a at 
 
 90
 
 PROPERTIES OF THE HYDRODYNAMIC FIELD. 91 
 
 The first member is the velocity of expansion per unit volume 
 of the moving fluid particle, expressed through the effect of this 
 expansion upon the specific volume, or the volume of unit mass. 
 The second member is the same velocity of expansion expressed 
 through the distribution of velocity in the fluid. The equality 
 of these two expressions of the same velocity of ex])ansion insures 
 the conservation of the mass during the motion of the fluid. 
 
 Now f being the exterior force acting per unit volume of the 
 moving fluid masses, and p the pressure in the fluid, the vector 
 equation of motion may be written 
 
 (P) adt='-^^- 
 
 The first member is the product of the density, l/ot, of the moving 
 particle into its acceleration, dkjdt, and the second member gives the 
 vector sum of the forces per unit volume acting upon it. These 
 forces are the exterior force f, and the force due to the pressure, 
 — Vp, generally called the gradient. 
 
 In the use of these equations it is always to be remembered that 
 the individual differentiating symbol djdt is related to the local 
 differentiating symbol djdt by the Eulerian expansion 
 
 d d 
 
 W J^ = a^" + ^^- 
 
 These equations do not give the geometry and the dynamics of 
 the hydrodynamic field as separate theories. They contain the 
 properties of the fields viewed from one central point, from which 
 their geometric and dynamic properties seem perfectly united. It 
 will be our problem to artificially separate from one another cer- 
 tain geometric and certain dynamic properties, m order to be 
 able to carry out the comparison with those other fields which we 
 know onlv as the result of an inspection from without, an inspec- 
 tion which has allowed us only to recogni/e two separate sides 
 of their properties, without any deeper insight into their true 
 relations.
 
 92 FIELDS OF FORCE. 
 
 3. Equation of Continuity — Equation for the Density of Electri- 
 fication. — The equation of continuity has the form of one of the 
 fundamental equations of the electric field. To show this we have 
 only to represent the v'elocity of expansion per unit volume, 
 \ja dajclt, by a single letter E, and obtain the equation corre- 
 sponding to (IV., 8, C), 
 
 div A= E, 
 
 which, in the interpretation of the symbols for the electrical case, is 
 the equation which gives the density of electrification in the elec- 
 tric field. 
 
 4. Transformation of the Dynamic Equation. — The dynamic 
 equation does not in its original form show any resemblance to 
 any of the equations of the electric field. Some simple transforma- 
 tions will, however, bring out terms of the same form as appear in 
 the dynamic equations of the electric field. 
 
 To show this let us first introduce instead of the velocity A the 
 actual specific momentum, a , according to the equation 
 
 (a) ' A = aa . 
 
 The equation of motion then takes the form 
 
 da 1 da 
 
 -di + a dt *" = * - '^1'' 
 or, according to the equation of continuity (2, a), 
 
 ^«-f (divA)a„ = f-v;^. 
 
 In the left hand member we have the term (div A)a^^, the analogue 
 of which appears in the expression f^ for the elementary forces in 
 the electric field (IV., 8, A\). It is the elementary force acting 
 upon the true electrification, div A. 
 
 Further simple transformations bring in the other corresponding 
 terms appearing in the expression for f, for the elementary forces 
 in the electric field. Using the Eulerian expansion, we first get 
 
 da 
 
 ^'^ + Ava,, + (div A)a^ = f — v;^,
 
 PROPERTIES OF THE HYDRODYNAMIC FIELD. 
 
 93 
 
 and then transforming the second left hand term according to a 
 well known vector formula, we have 
 
 ^+ Aa V + (curl a ) x A + (div A)a, = f - Vy>. 
 
 Now, the term (curl aj x A has appeared, which correspondingly 
 appears in the expression (lY, 8, E,) for the force in the electric 
 field, representing in one term the force exerted upon permanent 
 polarization and upon magnetic current. 
 
 According to (a), the second term in the left hand member may 
 
 be written ^ 
 
 Aa,v = ^a,a,v = J'^va;, 
 
 or finally, 
 
 Aa„v = V(j^a;) - iK^a. 
 
 Substituting this above, we have 
 
 ^^« + V (i«a;;) - |a;Vot + (curl aj x A + (c^v A)a, = f - VyJ, 
 
 giving us all the corresponding terms contained in the expression 
 for the force (IV., 8, E;) in the electric field. 
 
 5. Separation of the Equation of Motion.--\Ne thus seem to have 
 found some relation between the hydrodynamic equation and the 
 equation giving the dynamics of the electric field. But we still 
 have the geometry and the dynamics of the hydrodynamic field 
 united in one set of equations. To make the first step to^^^rds 
 the separation of certain geometric and dynamic properties from 
 one another we have to consider the hydrodynamic field as the sum 
 of two partial fields, just as we consider the electric field as the sum 
 of two partial fields, the induced and the energetic field. Let us 
 represent the vector a„, the actual specific momentum, as the sum 
 of two vectors a and a^, thus 
 
 (a) a, = a + a. 
 
 The equation then develops into 
 
 -' '' ^* +(divA)a. = f-vp.
 
 94 FIELDS OF FORCE. 
 
 Now we have the right to submit one of the auxiliary vectors, 
 say a, to a condition. Let this condition be that it shall satisfy 
 the equation 
 
 The other vector will then have to satisfy the equation 
 
 aa 
 
 (c) — " = f — (div A)a^^ + |a; v« — (curl a,) x A. 
 
 6. Geo'indric Properfy of the Induced Motion. — We have thus 
 introduced the consideration of two fields, which superimposed 
 upon each other represent the actual hydrodynamic field. But the 
 equations of both partial fields are still dynamic equations. How- 
 ever, from one of them we can at once proceed to a purely 
 geometric equation. For taking the curl of e([uation (b) and 
 changing the order of the operations d/dt and curl, we get 
 
 d , 
 
 ^^curl a= 0. 
 ' ct 
 
 To complete the nomenclature I will call the curl of the velocity 
 the kinevKific, and the curl of the specific momentum the dynamic 
 vortex demiti/. The dynamic vortex density is thus invariable at 
 every point of space. Integrating with respect to the time and 
 writing — k for the constant of integration, we get 
 
 (d) curl a = — k, 
 
 which expresses the local conservation of the dynamic vortex den- 
 sity. As regards its form, this is the same equation which in 
 the electric interpretation of the symbols expresses the relation be- 
 tween the electric field intensity a and the magnetic current k (IV., 
 8, B). And, as the conservation of k is local, equation (d) cor- 
 responds exactly to the equation for the electric field for the cases 
 of magnetic currents which are stationary both in space and in time. 
 
 7. Fnndamental (k'ometric Properties of the Jli/drodi/namic Field. 
 — We have thus succeeded in representing the hydrodynamic field
 
 PROPERTIES OF THE HYDRODYNAMIC FIELD. 95 
 
 as the sum of two partial fields. Writing A^ = ota^, we have for the 
 vectors introduced the equation of connection 
 
 (J.) A = aa^ r= aa -f A^. 
 
 Then the induced field described by a has the property of local 
 conservation of the dynamic vortex, 
 
 (i?) curl a = — k, 
 
 while from the field of the actual velocity we calculate the veloc- 
 ity of expansion per unit volume, E, from the equation 
 
 (C) divA = ^. 
 
 In form, these equations are precisely the fundamental eijuations 
 for the geometric properties of tlie stationary electric field. 
 
 8. Bodies and Fundamental Fluid. — To complete the investiga- 
 tion of the geometric properties we shall have to examine whether 
 we can introduce conditions corresponding to the supplementary 
 conditions (IV., 8, D). The introduction of conditions of this 
 nature for the fluid system evidently involves the distinction be- 
 tween certain limited parts of the fluid, which we have to com- 
 pare with material bodies, and an exterior unlimited part of the 
 fluid, which we have to compare with the free ether. The part of 
 the fluid surrounding the fluid 6ot?«w we shall call the fundameufal 
 fluid. 
 
 Introducing the condition 
 
 (A) ^ = "'«' 
 
 where a^ is constant, we simply require the fundamental fluid 
 
 to be homogeneous. Introducing the condition 
 
 (A) ^=«' 
 
 we require it to be incompressible. There is nothing which pre- 
 vents us from introducing the additional condition 
 
 (A) ^ = ^' 
 
 for, at every point of space the dynamic vortex has, according to
 
 96 FIELDS OF FORCE. 
 
 the fundamental equation (B), a constant value. We are there- 
 fore free to impose the condition that in the parts of space occu- 
 pied by the fundamental fluid this constant shall have the value 
 zero. This, in connection with the general condition {B), of course 
 involves also a restriction upon the generality of the motion of 
 the fluid bodies. The nature and consequence of this restriction 
 will be discussed later, but for the present it is sufficient for us to 
 know that nothing prevents us from introducing it. 
 . The question now arises : are we also entitled to introduce for 
 the hydrodynamic system a condition corresponding to the condi- 
 dition (7)J for the ether? To answer this we must refer to the 
 dynamic equation (5, c). On account of the restriction (Dj), we 
 shall have vot= in the fundamental fluid. On account of con- 
 dition (Dg), we shall have div A = 0, so that two of the right hand 
 terms of the equation for the energetic motion disappear. Writing 
 a J = a 4- a^ and remembering the condition (^^3), just introduced, 
 we find curl a^ = curl a^, and the equation therefore reduces to 
 
 da. 
 
 Furthermore, we are free to introduce the condition that the ex- 
 terior force f shall be zero for every point in the fundamental fluid, 
 so that the equation becomes 
 
 "dt ^ ~~ ^ ^«^ ^ ^' 
 Now if at any point in space a^ = 0, we shall also have 
 
 ^a ^ 
 dt "' 
 
 /. e., under the given conditions there can be no energetic field 
 intensity a^ unless it existed previously. The same will be true 
 of the energetic velocity A^, which is simply proportional to the 
 corresjwnding field intensity a^. Nothing prevents us, conse- 
 quently, from requiring that in the space occupied by the funda- 
 mental fluid we shall have the condition
 
 PROPERTIES OF THE IIYDKODYNAMIC FIELD. 07 
 
 (A) A=0 
 
 always fulfilled. For evidently we have the right to introduce 
 the condition (Z)J as an initial condition. And, as we have seen, 
 if it is fulfilled once, it will always be fulfilled. 
 
 Summing up the contents of (/>>,)• • ■{T)^) we find that we have 
 introduced the following conditions defining the difference between 
 the fluid bodies and the surrounding fundamental fluid, which is 
 analagous to the difference between the bodies and the surround- 
 ing ether in the electromagnetic field. The fundamental fluid has 
 constant mobility (specific volume), just as the ether has constant 
 inductivity ; the fluid bodies may have a mobility varying from 
 point to point and differing from that of the fundamental fluid ; 
 just as the bodies in the magnetic field may have an inductivity 
 varying from point to point and differing from that of the ether. 
 The fundamental fluid never has velocity of expansion or con- 
 traction, E, while this velocity may exist in the fluid bodies; just 
 as in the free ether we have do distribution of true electrification 
 or magnetism, while such distribution may exist in material bodies. 
 The fundamental fluid never has a distribution of dynamic vortices, 
 while such distributions may exist in the fluid bodies ; just as the 
 ether in the case of stationary fields never has a distribution of 
 currents, electric or magnetic, while such distributions may exist 
 in material bodies. The fundamental fluid never has an energetic 
 velocity, while this velocity may exist in the fluid bodies ; just as 
 the ether never has an energetic (impressed) polarization, while such 
 polarization may exist in material bodies. 
 
 Under these conditions the geometric properties of the hydro- 
 dynamic field and the stationary electric or magnetic field are de- 
 scribed by equations of exactly the same form. Thus, under the 
 given conditions, whose physical content we shall consider more 
 closely later, there exists a perfect geometric analogy between the 
 two kinds of fields. 
 
 9. Dynamic Properties of the Hydrodynamie Field. — It is 
 easily seen that under certain conditions an inverse dynamic 
 13
 
 98 FIELDS OF FORCE. 
 
 analogy will be joined to this geometric analogy. For let us im- 
 pose the condition that 
 
 shall always be satisfied, /. e., that the energetic specific momentum 
 shall be conserved locally. When this condition is fulfilled, the 
 equation of the energetic motion, which we will now have to use 
 for the bodies only, reduces to 
 
 {IS) f = (div A)a^ — |a;^vot + (curl aj x A, 
 
 /. e., if the condition of the local conservation of the energetic 
 specific momentum must be fulfilled, there must act upon the system 
 an exterior force f, whose distribution per unit volume is given 
 by (b). According to the principle of equal action and reaction, 
 this force thus balances a force fj, exerted under the given condi- 
 tions by the fluid system. The fluid system therefore exerts the 
 force 
 (^,) fi = — (div A)a^^ + ^a;v« — (curl a J x A, 
 
 which, in form, oppositely corresponds to the force which is exerted, 
 according to Heaviside's investigation, by the electric or the 
 magnetic field in the corresponding case. 
 
 10. Second Form of the Analogy. — The physical feature of the 
 analogy thus found is determined mainly by the condition (9, a) for 
 the local conservation of the energetic specific momentum. The 
 physical content of this condition we will discuss later. But first 
 we will show that even other conditions may lead to an analogy, in 
 which we do not arrive at Heaviside's, but at some one of the 
 other expressions for the distribution of force. 
 
 We start again with the equation of motion, 
 
 IdA 
 
 Now, instead of introducing the actual specific momentum a^, I 
 introduce at once the induced sj)ecific momentum a and the ener- 
 getic velocity A, according to the equation of connection 
 
 (^) A = aa +A .
 
 PROPERTIES OF THE HYDRODYNAMIC FIELD. 
 
 99 
 
 Performing the differentiation and making use of the eciuation of 
 continuity (2, a), we have 
 
 da. 1 dA 
 
 -^ + (divA)a+^^=f-Vy. 
 
 Introducing in the first left hand term the U)cal time-derivation, 
 da. 1 ^^K 
 
 or, transforming the second left hand member according to the 
 vector formula, 
 
 (c) /^ + Aa V + (curl a) x A + (^iv A)a + ^ ^^ = f - Vy>. 
 
 Using the equation of connection (6) and performing simple trans- 
 formations, we get for the second term in the left hand member 
 
 Aav = '^aav + A av 
 = iava^ + A^av 
 = V(j-^a- + A^ • a) - |a- v« - aA V- 
 
 Introducing this in {<■), 
 
 da ., , . . _x , 1 ^^A 
 
 (J) 
 
 + (curl a)xA-aA^V = f- V/>. 
 
 Now, we can split the equation in two, requiring that the vector 
 a satisfy the equation 
 
 (e) |^=_v(i. + ^^a^ + A-a), 
 
 and we find that the other vector A must satisfy the equation 
 
 ^ + V(iaa^ + A -a) + ^ ^^^^+ (div A)a- ^aVot 
 
 (/) 
 
 ^ ^ = f _ (div A)a + H'va - (curl a) x A -h aA^V. 
 
 a dt
 
 100 FIELDS OF FORCE. 
 
 Both equations are different from the corresponding equations (5, 
 6) and (5, e). But, as is seen at once, the new equation for the in- 
 duced motion involves the same geometric property as the previous 
 one, namely, the local conservation of the dynamic vortex, expressed 
 by (i>). We arrive thus at the same set of fundamental geomet- 
 ric equations as before, (A) ■ ■ ■ (C). Furthermore, we have evi- 
 dently the same right as before to introduce the restrictive condi- 
 tions (Dy), {D.^, {D^. A discussion of equation {f), similar to 
 that given above for equation (5, d), shows us that we are entitled 
 in this case also to impose the condition (D^) upon the fundamental 
 fluid, since in a fluid having the properties (D,) • • • (Dg) a moving 
 fluid particle cannot have an energetic velocity if this did not 
 exist previously. 
 
 The geometric analogy therefore exists exactly as before, the 
 conditions for its existence being changed only with respect to this 
 one point, that the condition (DJ now refers to the material parti- 
 cles belonging to the fundamental fluid, and not to the points in 
 space occupied by this fluid. The consequence of this difference 
 will be discussed later. 
 
 Finally, we see that to this geometric analogy we can add a 
 dynamic analogy. Requiring that the energetic velocity be con- 
 served individuaUy, we have 
 
 and, reasoning as before, we find that under this condition the fluid 
 system will exert per unit volume the force 
 
 (^2) ^2= — (^iv ^)^ + ia-va — (curl a) x A + aA^v, 
 
 which, in form, oppositely corresponds to the forces in the electric 
 or magnetic field, according to the expression (IV., 8 E^). 
 
 11. We have thus arrived in two different ways at an analogy 
 between the e([uations of hydrodyuamic fields and those of the 
 stationary electric or magnetic field. And, from an analytical 
 point of view, this analogy seems as complete as possible, apart 
 from the ojijwsite sign of the forces exerted by the fields.
 
 PROPERTIES OF THE IIYDRODYNAMIC FIELD. l^l 
 
 In regard to the closeness of this analytical anology, we have to 
 remark that we do not know /vith perfect certainty which of 
 the expressions {E^) or (E.^), if either, represents the true distribu- 
 tion of the elementary forces in the electric or the magnetic field, 
 while the corresponding distribution of forces in the hydrodynamic 
 field are real distributions of forces which are exerted by the field 
 and which have to be counteracted by exterior forces, if the condi- 
 tions imposed upon the motion of the system are to be fulfilled. A\ e 
 cannot, therefore, decide which of the two forms that we have found 
 for the analogy is the most fundamental. But we kno^ with per- 
 fect certainty that, if we integrate this system of elementary forces 
 for a whole body, we get the true value of the resultant force in the 
 electric or magnetic field. When we limit ourself to the considera- 
 tion of the resultant force only, the two forms of the analogy are 
 therefore equivalent. And from the integration performed in the 
 preceding lecture we conclude at once, that the resultant forces 
 upon the bodies in the hydrodynamic field can also be repre- 
 sented as resulting from the fictitious distributions 
 
 (^^) fs = — (div A)a - A,av - (curl a) x A, 
 
 and 
 
 (^j f^ = - a^, (div a)a - a, (curl a) x a. 
 
 The fact, which we have just i)roved, that the laws of the elec- 
 tric or magnetic fields and of the hydrodynamic fields can be rep- 
 resented by the same set of formuke, undoubtedly shows that there 
 is a close relation between the laws of hydrodynamics and the laws 
 of electricity and magnetism. But the formal analogy between the 
 laws does not necessarilv imply also a real analogy between the 
 things to which they relate. Or, as Maxwell expressed it: the 
 analogy of the relations of things does not necessarily unply an 
 analogy of the things related. 
 
 The subiect of our next investigation will be, to consider to 
 what extent we can pass from this formal analogy between the 
 hydrodynamic formulae and the electric or magnetic formula^ to an 
 analogy of perfectly concrete nature, such as that represented by 
 our experiments.
 
 YI. 
 
 FURTHER DEVELOPMENTS AND DISCUSSIONS OF 
 THE ANALOGY. 
 
 1. According to the systems of formulae which we have de- 
 veloped, the hydrodyiiamic analogy seems to extend to the whole 
 domain of stationary electric, or stationary magnetic fields. But 
 according to our elementary and experimental investigation, we 
 arrived at two different analogies which were wholly detached 
 from each other. There is no contradiction involved in these re- 
 sults. In our analytical investigation we have hitherto taken only 
 a formal point of view, investigating the analogy between the for- 
 mal laws of hydrodynamics and of electromagnetism. If, from the 
 analogy between the formal laws, we try to proceed further to an 
 analogy between the different physical phenomena obeying them, 
 we shall arrive at the two detached fragments of the analogy 
 which we have studied experimentally. 
 
 2. Between the hydrodynamic and the electric or magnetic 
 systems there is generally this important difference. The hydro- 
 dynamic system is moving, and therefore generally changing its 
 configuration. But apparently, at least, the electric or magnetic 
 systems with which we compare them are at rest. The corre- 
 spondence developed between hydrodynamic and electromagnetic 
 formulae therefore gives only a momentary analogy between the 
 two kinds of fields, which exist under different conditions. 
 
 To get an analogy, not only in formulae but in experiments, 
 we must therefore introduce the condition that the bodies in the 
 hydrodynamic system should appear stationary in space. This 
 can be done in two ways. First, the fluid system can be in a 
 steady state of motion, so that the bodies are limited by sur- 
 faces of invariable shapes and position in space. Second, the 
 fluid can be in a state of vibratory motion, so that the bodies per- 
 form small vibrations about invariable mean positions. 
 
 102
 
 DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. lOo 
 
 3. Steadij State of Motion. — Tlie first form of the analytical 
 analogy, in which we supposed local conservation of the energetic 
 specific momentum, 
 
 («) -aj=o, 
 
 immediately leads us to the consideration of a perfectly steady 
 state of motion, at which we arrive, if we assume besides (a) also 
 the local conservation of the induced specific momentum, 
 
 which is perfectly consistent with («). But in the case of a steady 
 state of motion the generality of the field is very limited, on ac- 
 count of the condition that the fluid, both outside and inside, 
 moves tangentially to the stationary surface which limits the 
 bodies. 
 
 4. Ir rotational Circulation Outside the Bodies. — As the motion 
 outside the bodies fulfills the condition curl a = 0, and, in conse- 
 quence of the constancy of the specific volume, a,,, also the con- 
 dition curl A = 0, the motion in the exterior space will be the 
 well known motion of irrotational circulation, which is only possible 
 if the space be multiply connected. If,- then, there is to be any 
 motion of the exterior fluid at all, one or more of the bodies must 
 be pierced by channels through which the fluid can circulate. 
 Bodies which have no channels act only as obstructions in the 
 current, which exists because of the channels through the other 
 bodies. The velocity or the specific momentum by which this 
 motion is described has a non-uniform scalar potential. The 
 stream-lines are all closed and never penetrate into the interior 
 of the bodies, but run tangentially to the surfaces. The corre- 
 sponding electrodynamic field, with closed lines of force running 
 tangentially to the bodies and having a non-uniform potential, 
 is also a well known field. 
 
 5. Corresponding Field Inside the Bodies. — This exterior field 
 can correspond, in the hydrodynamic, as well as in the electro-
 
 104 FIELDS OF FORCE. 
 
 magnetic case, to different arrangements in the interior of the 
 bodies. The most striking restriction on the exterior field is the 
 condition that the lines of force or of flow shall never penetrate 
 into the bodies. In the magnetic case this condition will always 
 be fulfilled if the bodies consist of an infinitely diamagnetic 
 material, and a field with these properties will be set up by any 
 distribution of electric currents in these infinitely diamagnetic 
 bodies. The hydrodynamic condition corresponding to zero in- 
 ductivity is zero mobility. The bodies then retain their forms 
 and their positions in space as a consequence of an infinite 
 density and the accompanying infinite inertia. Now in the case 
 of infinite density an infinitely small velocity will correspond to a 
 finite specific momentum. We can then have in these infinitely 
 heavy bodies any finite distribution of specific momentum and of 
 the dynamic vortex, which corresponds to the electric current, and 
 yet to this specific momentum there will correspond no visible 
 motion which can interfere with the condition of the immobility 
 of the bodies. 
 
 Other interior arrangements can also be conceived which, pro- 
 duce the same exterior field. The condition of infinite diamag- 
 netivity may be replaced by the condition that a special system 
 of electric currents be introduced to make bodies appear to be 
 infinitely diamagnetic. The corresponding hydrodynamic case will 
 exist if we abandon the infinite inertia as the cause of the immo- 
 bility of the bodies and also dispense with the creation of any gen- 
 eral distribution of dynamic vortices in the bodies, and if we in- 
 troduce instead, special distributions of vortices, subject to the 
 condition that they be the vortices of a motion which does not 
 change the form of the bodies or their position in space. This 
 distribution of the dynamic vortices will, from a geometric point 
 of view, be exactly the same as the distribution of electric current 
 which makes bodies appear infinitely diamagnetic. 
 
 Finally, a third arrangement is possible. In bodies of any in- 
 ductivity we can set up any distribution of electric currents, and 
 simultaneously introduce a special intrinsic magnetic polarization
 
 DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. lOo 
 
 which makes the bodies appear to be infinitely diamagnetic. Cor- 
 respondingly, we can give to bodies of any mobility any distribu- 
 tion of dynamic vortices under the condition that we fix the 
 bodies in space by a suitable distribution of energetic velocities 
 produced by external forces. 
 
 6. The Dynamic Analogy. — In the cases thus indicated the 
 geometric analogy between the fields will be perfect. And with 
 this direct geometric analogy we have an inverse dynamic analogy. 
 The system of elementary forces, by which the field tends to pro- 
 duce visible motions of the bodies, and which must be counter- 
 acted by exterior forces, oppositely corresponds in the two systems. 
 
 The simplest experiments demonstrating these theoretical results 
 are those showing the attraction and the repulsion of rotating cylin- 
 ders, and the attraction of a non-rotating, by a rotating cylinder, 
 corresponding to the repulsion of a diamagnetic body by an elec- 
 tric current. 
 
 As the analogy thus developed holds for any arrangement of 
 electric currents in infinitely diamagnetic bodies, it will also hold 
 for the arrangement by which magnets can be represented accord- 
 ing to Ampere's theory. We can thus also get an analogy to 
 magnetism, but in a peculiarly restricted way, since it refers only 
 to permanent magnets constructed of an infinitely diamagnetic 
 material. The hydrodynamic representation of a magnet is there- 
 fore a body pierced by a multitude of channels through which the 
 exterior fluid circulates irrotationally. Such bodies will then exert 
 apparent actions at a distance upon each other, corresponding in- 
 versely to those exerted by permanent magnets which have the 
 peculiar property of being constructed of an infinitely diamagnetic 
 material. This peculiar analogy was discovered by Lord Kelvin 
 in 1870, but by a method which differs completely from that 
 which we have followed here. 
 
 7. Besfricted Generality of the Field for the Case of Vibratory 
 Motion. — The hypothesis of a vibratory motion also restricts the 
 generality of the field, but in another way than does the condition of 
 steady motion. For, when the specific momentum is vibratory, its 
 
 14
 
 106 FIELDS OF FORCE. 
 
 curl, if it has any, must also be vibratory. But we have found that 
 this curl, or the dynamic vortex density, is a constant at every 
 point in space, and is thus independent of the time. The dynamic 
 vortex therefore must be everywhere zero, and the equations ex- 
 pressing the geometric analogy reduce to 
 
 A = aa + A^, 
 
 (a) curl a = 0, 
 
 div A= B, 
 
 with the conditions for the surrounding fluid, 
 
 (6) a=a^, JE=0, A=0. 
 
 The equations thus take the form of the equations for the static 
 electric, or the static magnetic field, so that the analogy will not 
 extend beyond the limits of static fields. To establish the cor- 
 responding dynamic analogy we may use neither of the conditions 
 (V., 9, a or 10, </). For both are contradictory to the condition 
 for vibratory motion. We have to return to the unrestricted 
 equation for the energetic motion, and the form which in this case 
 leads to the most general results is (10, f), which according to (a) 
 reduces to 
 
 (c) ^ '-^^ = f - (div A)a + la^Va + aA V. 
 
 This system of equations is valid for any single moment during the 
 vibratory motion. We shall have to try to deduce from it another 
 system of equations which represents the invariable mean state of 
 the system. 
 
 8. Periodic Inunctions. — To describe the vibratory motion we 
 shall employ only one periodic function of the time, and therefore 
 the different particles of the fluid will not have vibratory motions 
 independent of each other. The motion of the fluid will have 
 the character of a fundamental mode of an elastic system. To 
 describe this fundamental mode we use a periodic function, y, of the 
 period t ; thus 
 
 («) At + r) =f{t).
 
 DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. 107 
 
 The values of the function /shoukl be contained between finite 
 limits, but the period t should be a small quantity of the first 
 order. Further, the function / must be subject to the following 
 conditions : during a period it shall have a linear mean value 0, 
 and a quadratic mean value 1, thus 
 
 Evidently these conditions do not restrict the nature of the func- 
 tion, provided it be periodic. Any periodic function may be made 
 to fulfil them by the proper adjustment of an additive constant and 
 of a constant factor. An instance of a function which fulfils the 
 conditions is 
 
 (d) f{t) = V2, sin 27r ( ^ + h j • 
 
 From the conditions that the period is a small quantity of the 
 first order and that the mean linear value of the function for 
 a period is zero, it is deduced at once, that the time integral of the 
 function over any interval of time multiplied by any finite factor 
 n will never exceed a certain small quantity of the first order. AVe 
 may thus write 
 
 (e) Jl'nf{t)dt<h, 
 
 where n is a finite factor, and h a certain small quantity of the 
 first order. 
 
 9. Representation of the Vibratory State of 3fotion by Quantities 
 Independent of the Time. — To get equations which define uni- 
 formly the vibratory motion we can now make use of the property 
 of the field, that it is determined uniquely by the energetic veloc- 
 ity A^ in connection with the velocity of expansion E. The motion 
 will thus be definitely determined by the two equations, 
 
 («.) A = A,, Jit),
 
 108 FIELDS OF FORCE. 
 
 where A,„, aud E^^ are quantities independent of the time, but 
 varying of course from particle to particle. As to their absolute 
 values, these constants are the quadratic mean values of the ener- 
 getic velocity A^ and of the velocity of expansion E. For, from 
 equation (8, c), we get 
 
 1 /*t+T 
 
 F' = - i Ehlt. 
 
 The constants A,,„ and £",„ for different particles in space may 
 have different signs. These are always given by the equations 
 (Oj) and (a^), and the rule of signs may be expressed thus ; the 
 quantities A,„, and E„, have respectively always the same sign as 
 the variable quantities A and E had at a certain initial time. The 
 absolute signs thus attributed to A,^ and E„^ have no great im- 
 portance, but it is important that this rule determines perfectly the 
 signs which the different quantities A,„, and E,,^ have relatively to 
 each other. 
 
 With regard to the motions determined by («), we can conclude 
 from the property (8, e) of the function/, that the energetic velocity 
 produces displacements from the mean position of the particle, 
 which never exceed a certain small quantity of the first order. 
 And in the same way we conclude, that the change of volume pro- 
 duced by the periodic velocity of expansion and contraction never 
 exceeds a small quantity of the same order. This has the impor- 
 tant consequence that, neglecting small quantities of the first order, 
 we can consider the specific volume, oc^ of the fluid as constant, ex- 
 cept, of course, in cases where it has to undergo a differentiation 
 with respect to the time. 
 
 According to this, it is easy to write the explicit expressions of 
 the actual velocity A and of the specific momentum a. Doing 
 this, 
 {\) A = A„,/(0,
 
 DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. 109 
 
 For the substitution of these expressions and the expressions (a) 
 in the equations (7, a) shows that they satisfy them, if the quanti- 
 ties independent of the time satisfy the equations 
 
 (c) curl a„^ ^ 0, 
 div A = E , 
 
 in connection with the conditions for the exterior fluid 
 
 (d) a = a^, A,„, = 0, E„^ = 0. 
 
 If these equations be satisfied, (6) will satisfy the equations and 
 represent the solution, as there exists but one. 
 
 The equations (c), which the quantities A,„, a„„ A,,„„ E,„, satisfy, 
 have exactly the same form as the equations (7, a). They give, 
 therefore, for all times the same analogy to an invariable electro- 
 magnetic field as the corresponding variable quantities give for a 
 single moment. The similarity is so great that it is not even 
 necessary to introduce two sets of notation. To pass from the 
 one form of the analogy to the other it is sufficient to change the 
 signification of the letters in the equations (7, a) ; if these quan- 
 tities are interpreted, not as the velocities and the specific mo- 
 menta themselves, but as representing in the indicated manner the 
 mean intensities of these quantities, they give the geometric 
 analogy existing at any time between the electric or magnetic 
 field and the case of vibratory motion in the hydrodynamic field. 
 
 1 0. The Mean Value of the Force in the Vibratory Field. — Fi nally , 
 to examine the dynamics of the field we have to substitute the 
 expressions (9, a) and (9, b) in the equation of energetic motion 
 (7, c) and perform the integration over a period of the oscilla- 
 tions. Using the property (8, b) of the function /, we find that 
 the left hand member of the equation disappears. Designating by 
 f^l the mean value of the exterior force f and using the property 
 (8, e) of the function/, we find 
 
 = f,; - (div AJa,, + iaf. v« + a,„A.„, v.
 
 110 FIELDS OF FORCE. 
 
 This equation shows that during the vibratory state of motion 
 the external force will have to balance a mean force exerted by 
 the system, which has the value 
 
 f = — (div A )a + |a- v^ + a,„A,,„, v. 
 
 The expression has again exactly the same form as the expres- 
 sion for the force in the case of the momentary analogy, except that 
 the varying quantities are replaced by quantities independent of 
 the time. The similarity of the expressions makes it unnecessary 
 to use two systems of notation. We can write the expression for 
 the force 
 
 f^ = — (div A)a + |a"V« + aA^V, 
 
 and interpret, according to the circumstances, the quantities a and 
 A as the momentary values of specific momentum and velocity, re- 
 spectively or as the quantities which represent in the way indicated 
 the mean intensities of these quantities. In one case we arrive at 
 the analogy which exists for a moment only, in the other case at 
 the analogy which exists independent of the time. Both analogies 
 have the same degree of exactness, the geometric analogy being 
 direct, and the dynamic analogy being inverse. 
 
 11. We have thus arrived at this result, that in the case of 
 vibratory motion the hydrodynamic field can be described with 
 reference to geometric properties by the following formulae, 
 
 A = aa + A^, 
 (a) curl a = 0, 
 
 div A = B, 
 
 together with the conditions for the fundamental fluid, 
 
 (6) ^=%, B=0, A = 0. 
 
 And this fluid system, in the supposed vibratory state of motion, 
 will exert exterior forces tending to produce visible motions, which 
 are given by 
 (c) f ^ = — (div A)a + |a'-vst + aA^v.
 
 DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. Ill 
 
 In these equations all quantities arc independent of the time. 
 
 But these equations are also the fundamental equations for an 
 electrostatic or for a magnetic system, except for the diiference that 
 the force f^ has the opposite direction. It is an open question 
 whether this expression for the elementary forces in the case of the 
 electric or magnetic field is fundamental, or only a fictitious force 
 which gives the right value of the resultant force upon the whole 
 body. 
 
 We have succeeded in proving this : the vibratory hydrodynamic 
 field has the same geometric configuration as an electrostatic or a 
 magnetic field. In tJie hydrodynamic field there are forces whose 
 resultant upon finite bodies oppositely corresponds to the correspond- 
 ing resultant forces in the electric or magnetic field. 
 
 To show that this result gives the full explanation of all our 
 experiments with the pulsating and oscillating bodies we have 
 only to add one remark. In our experiments we used pulsating 
 and oscillatinsr bodies constructed of solid material. On the 
 other hand, in our mathematical developments we have consid- 
 ered the bodies as fluid. But these fluid bodies are subject to the 
 action of forces which give the prescribed state of vibration, and 
 which are subject to no restrictive conditions. Nothing prevents 
 us, therefore, from adjusting these forces so as to give the fluid 
 bodies the same motion as they would have if they were con- 
 structed of solid material. The reactions exerted upon them by 
 the surrounding fluid will then of course be exactly the same as 
 if they were constructed of solid material. 
 
 12. We have nothing to add to the demonstration of the anal- 
 ogy. But, to make ourselves better acquainted with it, we may 
 make a simple application of it. In the analogy, for instance, 
 pulsating particles produce fields of the same geometric configura- 
 tion as electrically charged particles, and are acted upon by forces 
 oppositely corresponding to those acting upon the latter. Pulsating 
 particles will therefore act upon each other according to a law 
 analogous to that of Coulomb, except for the reversed sign of the 
 force. Introducing for the charges, or the intensities of pulsation 
 of the two particles
 
 112 FIELDS OF FORCE. 
 
 e = fEch, e = jE'dr, 
 and using the rational system of units, we get for this law 
 
 a^ 47rr" 
 
 r being the distance between the two particles, and a^ the induc- 
 tivity, or the mobility of the medium. 
 
 Let us now imagine an investigator who observes the attraction 
 and the repulsion of the pulsating bodies, but who is not capable 
 of observing the water which transfers the action, or the pulsa- 
 tions which set up the field in the water. He will then believe 
 that he sees an action at a distance, following a law having the 
 same form as that governing the action at a distance between elec- 
 trified particles. 
 
 Let us imagine that, as he proceeds in his further investigations, 
 he moves one pulsating body, e', from point to point in the space 
 surrounding the other, measures at each point the force F, and 
 draws an arrow representing the value of F/e'. He then arri.ves 
 at the formal disposition of a field which is associated with the 
 pulsating body e. He has, no more than in the electrical case, 
 a formal right to attribute to this field a physical significance, or 
 to attribute to the recorded vector a physical existence. His ex- 
 periments give him evidence only of this, that there is a force act- 
 ing at the point where he places his second charge, e'. But he 
 has no evidence of the existence of a physical vector at this point 
 after he has removed the charge e'. 
 
 But in spite of this, he may try to change his view. He may 
 imagine the existence of a medium which he does not see, and 
 make the hyj)othcsis that the vector represents some state exist- 
 ing, or some process going on, in this medium. In the electrical 
 case we have no direct evidence that this hypothesis is correct, 
 although thus far, the development of our knowledge of electricity 
 makes it extremely probable that there must be some truth in it. 
 But in the hydrodynamic case we have the full evidence: the
 
 DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. 113 
 
 medium exists ; it is an incompressible fluid. And the vector re- 
 corded represents tlie specific momentum in the field set up in the 
 fluid by the pulsating body. Thus we get a verification by analogy 
 of the hypothesis which forms the basis of the whole modern 
 theory of electricity. 
 
 13. But now let our hydrodynamic investigator proceed still 
 further. Let him conclude with Maxwell, that the attraction 
 and repulsion between the pulsating bodies must depend upon 
 a stress in the medium. Following Maxwell's developments 
 he will arrive at the expression of Maxwell's stresses, with 
 the reversed sign. But his conclusion in this case, that Max- 
 w- ell's stresses exist in the fluid and produce the attraction of the 
 pulsating body, is wrong. The stress that exists in the fluid and 
 produces the apparent actions at a distance is not Maxavell's 
 stress, but the isotropic stress or pressure in the fluid. We can- 
 not conclude from this that Maxwell's developments are also 
 wrong for the electric field. But we have full evidence that they 
 may be wrong, even in this case. 
 
 To return to the hydrodynamic case, it is easy to point out 
 where the error comes in. Maxwell only introduces his stresses 
 to account for the forces which produce the visible motions. But 
 in the hydrodynamic field the stress or pressure has a double 
 task; first, to maintain the field, and second, to produce the visible 
 motions. And it is extremely remarkable that the stress which 
 has this double eifect is a stress of much simpler nature than the 
 stress imagined by Maxwell, which produces only one of the 
 two effects. 
 
 When we developed the electromagnetic equation of activity 
 according to Heaviside, we also met with the more general 
 stresses introduced by him, which reduce in simple cases to 
 Maxwell's stresses. We cannot test Heaviside's develop- 
 ments in the same way as Max\\-ell's. For we have no 
 hydrodynamic analogy extending to the electromagnetic phe- 
 nomena of the most general type, from which he starts when he 
 forms the equation of activity. But the fact remains that the 
 15
 
 114 FIELDS OB" FORCE. 
 
 srtesses, even in Heaviside's theory, are introduced only to explain 
 the visible, motion observed in the field, not the formation or main- 
 tenance of the field itself. And even Heaviside gives, while em- 
 phasizing the im])ortance of the stress-problem, in different forms 
 expression to the unsatisfactory nature of our present solution of 
 it. Thus 
 
 '^ Our attitude towards the general application of the special 
 form of the stress theory obtained should, therefore, be one of 
 scientific scepticism. This should, however, be carefully distin- 
 guished from an obstinate prejudice founded upon ignorance, such 
 as displayed by some anti-Maxwellians, • • • * 
 
 " It is natural to ask Avhat part do the stresses play in the prop- 
 agation of disturbances ? The stresses and accompanying strains 
 in an elastic body are materially concerned in the transmission of 
 motion through them, and it might be thought that it might be 
 the same here. But it does not appeur to be so from the electro- 
 magnetic equations and their dynamical consequences — that is to 
 say, we represent the propagation of disturbances by particular 
 relations between the space- and the time-variations of E and H ; 
 and the electromagnetic stress and possible motions seem to be 
 accompaniments rather than the main theme."t 
 
 It may, therefore, be a question whether this will not be the 
 great problem in the theory of electricity, to find a stress which 
 accounts for both the formation and propagation of the electro- 
 magnetic field and for the visible motions of the charged or polar- 
 ized bodies, just as the pressure in the fluid accounts for both the 
 formation of the hydrodynamic field and for the visible motions 
 of the pulsating or oscillating bodies. 
 
 * Electromagnetic Theory, Vol. I, p. 87. 
 t Loc. cit., p. 110.
 
 VII. 
 
 GENERAL CONCLUSIONS. 
 
 Remarks on Methods of Research and of Instruction 
 IN Theoretical Physics. 
 1. The Problem of Fields of Force.— ^Ng have iu the pre- 
 ceding lectures taken the terra "field of force" in a more general 
 sense than usual. From the electric or magnetic fields we have 
 extended this term also to the fields of motion in a perfect fluid. 
 And this has been perfectly justified by the results obtained, 
 the most striking of which is the extraordinary analogy in the 
 properties of the two kinds of fields. So fiir as the analogy ex- 
 tends, there is one, and only one, diiference, the reversed sign 
 of the energetic forces. The relation of the electromagnetic and 
 the hydrodynamic fields may be compared to the relation be- 
 tween an object and its image in a mirror ; every characteristic 
 detail of the object is recognized in the image, but at the same 
 time there is the characteristic difference that left and right 
 are interchanged. But, however peculiar this diiference may be, 
 it cannot hide the common structure of the object and its image. 
 The discovery of this extraordinary analogy gives rise to sev- 
 eral considerations, and one of the first is this : Has our research 
 been exhaustive? Are the phenomena investigated by us the 
 only phenomena which have the same general structure as the 
 electromagnetic phenomena, or can still other phenomena with 
 corresponding fundamental properties be discovered ? ^ 
 
 I think that it is very improbable that our investigation has 
 been exhaustive. Even within the domain of hydrodynamics our 
 investigation has probably been incomplete. There are, indeed, 
 very strong indications that an analogy between electromagnetic 
 and hydrodynamic fields may be found with quite another cor- 
 respondence between the electric and the hydrodynamic quanti- 
 
 115
 
 116 FIELDS OF FOKCE. 
 
 ties. And if we no longer limit ourselves to the consideration of 
 fluids, but pass to media of other and more general properties, we 
 may hope to find still other forms of the analogy, perhaps of even 
 greater generality. 
 
 2. Fields in Other Media than Fluids. — The question now 
 arises: Are not the laws which we have found so entirely depen- 
 dent upon the fluid properties that it will be useless to look for 
 similar laws when we pass to other media? To answer this ques- 
 tion we have to look for the origin of the hydrodynamic analogy. 
 We then see that the geometric analogy had its origin to a great 
 extent in the equation of continuity. And, as this equation ex- 
 presses the principle of the conservation of mass, it holds for any 
 material medium, and furnishes the same basis for a possible geo- 
 metric analogy to electromagnetic fields. 
 
 On the other hand, the dynamic properties of the hydrodynamic 
 fields had their origin principally in the inertia of the fluid masses. 
 This is seen equally well in the elementary development of the 
 forces by the principle of kinetic buoyancy and in the mathematical 
 developments of Lecture Y, where it is seen that the complete 
 expression of the energetic force develops from the inertia term of 
 the hydrodynamic equation of motion. 
 
 A brief consideration thus shows that the principal conditions 
 from which the hydrodynamic analogy to the electromagnetic fields 
 developed, exist in any material medium, not alone in fluids. 
 But the special form which the analogy will take, its accuracy, and 
 its extent, will depend upon the special properties of the different 
 media. Thus the special properties of fluids admitted the exist- 
 ence of an analogy whicli is perfectly accurate, if we except the 
 inverse nature of the forces, but limited in extent. It will there- 
 fore be a most fascinating subject for research to examine whether 
 there exist media in which the accuracy of the analogy is pre- 
 served, while its scope is widened. Or, in other words, to deter- 
 mine the dynamic conditions of a medium in which the analogy, 
 with unaltered precision, has the greatest possible extent. 
 
 3. The Fields in a Transverse Elastic Medium. — To examine
 
 GENERAL CONCLUSIONS. 117 
 
 the chances of progress along this line it will bo advantageous to 
 consider briefly the fields in a medium with the common elastic 
 solid properties. Now it is well known that there is an exten- 
 sive geometric analogy between the fields of motion in an elastic 
 medium with properly adjusted constants and the electromagnetic 
 fields of the most general type. The coexistence and equivalence 
 of the two theories of light, the elastic and the electromagnetic, 
 proves this perfectly. Indeed, the electromagnetic theory of light 
 originated from the analogy which Maxwell succeeded in stating 
 between the equations for optical phenomena, developed by FiiES- 
 NEL and his successors from the hypothesis of the transverse elas- 
 tic ether, and the equations which he had himself developed to 
 describe electromagnetic fields. 
 
 We will consider this analogy in the simplest possible case. Let 
 the medium be homogeneous and isotropic, and, furthermore, in- 
 compressible and subject to the action of no exterior force. U being 
 the vector displacement, a the specific volume, and fi the constant 
 of transverse elasticity, the equation of motion of the nicdiimi is 
 
 generally written 
 
 1 d^V 
 
 («) -^ = /.v-u. 
 
 As a and fi are constants, this may be written 
 
 ^-^ = vVU. 
 
 On the right hand side of the equation we can now introduce 
 the velocity 
 
 W ^ = -et' 
 
 This member may at the same time be written in a modified fi)rni, 
 the operation V"^ being, for the solenoidal vector U, equivalent to 
 — curF. The equation may then be written 
 
 (c) — = — curl- a/LtU = — curl aV I curl U j.
 
 118 FIELDS OF FORCE. 
 
 Let us introduce now 
 
 , 1 
 {d) B = - curl - U, 
 
 from which we get 
 
 dB 1 d\J ,1 
 
 ^rr = — curl ^ = — curl - A, 
 
 dt a at a ' 
 
 or, if we introduce the specific momentum a according to the 
 equation 
 
 (e) A = aa, 
 
 we have 
 
 ^B 
 
 On the other hand, the introduction of {(l) in (c) gives 
 
 dA 
 
 {g) ^ = curl fia'B. 
 
 If we introduce 
 
 b = /^a-B, 
 ((j) finally takes the form . 
 
 (A) -^ = cind b. 
 
 Thus we can substitute for equation (a) the following system of 
 
 equations 
 
 dA 
 
 -,- = curl b, 
 ot 
 
 clB 
 
 -— = — curl a, 
 
 dt 
 
 where the vectors A and a, B and b are connected by the equa- 
 tions 
 
 A = aa, 
 
 B = /Sb, 
 where /S has the signification
 
 GENERAL CONCLUSIONS. 119 
 
 But this system is the system of Maxwell's equations for a 
 medium which is electrically and magnetically homogeneous and 
 isotropic, and which is the seat of no intrinsic electromotive or 
 magnetomotive forces. And we get the following correspondence : 
 
 A electric flux velocity 
 
 a electric field intensity .. .specific momentum 
 
 B magnetic flux. curl of specific mass-displacement 
 
 b magnetic field intensity . . (curl of sp. mass-displacement) tin?' 
 
 a electric inductivity specific volume 
 
 /S magnetic inductivity . . . . density^ /coeff. of elasticity 
 
 As is well known, we are free to give different forms to this 
 geometric analogy. We have used this freedom to choose a form 
 which makes the analogy a direct continuation of the hydrody- 
 namic analogy. 
 
 The extent of this geometric analogy is very great even though 
 we have avoided full generality by neglecting heterogeneities and 
 intrinsic forces. For it extends now to that point where the cross- 
 ing of electric and magnetic phenomena takes place, the point at 
 which the hydrodynamic analogy ceased. 
 
 4. Dynamics of the Field in the Transverse Elastic Medium. — 
 These well known developments, which lead to the geometric 
 analogy of electromagnetic and elastic fields, apparently give not 
 the faintest indication of the existence also of a dynamic analogy, 
 corresponding to that which we know from the investigation of 
 the hydrodynamic field, which is quite the opposite of what we 
 should expect from our preceding considerations. 
 
 The explanation of this apparent contradiction is, however, im- 
 mediate. As we have remarked, the energetic force in the hydro- 
 dynamic fields originated in the inertia term of the hydrody- 
 namic equation. But the equation of motion of the elastic medium, 
 as it is generally written (3, a), contains this term incompletely, 
 the local time derivation djdt being used as a first approxima- 
 tion for the individual derivation djdt, which would give to the 
 left member of the equation its proper form.
 
 120 FIELDS OF FORCE. 
 
 Let us repeat, therefore, the preceding development, but start- 
 ing with the equation 
 
 («) ^ ^^- = /.v-u + f, 
 
 in which the left member has its exact form, and in which we have 
 added on the right hand side the exterior force f, which we sup- 
 pose, however, small in comparison to the elastic forces. The left 
 hand member of this equation is identical with the left hand member 
 of the hydrodynamic equation, and may be developed in exactly 
 the same way. We may thus write, as in (V, 10), 
 
 A = aa -f A^, 
 and then equation («) in the form 
 
 da. 1 f/A 
 
 -^j + V(Aa!a- + A a) + ^ ^^ + (div A)a - laVa, 
 
 + (curl a) X A — aA^v = ftV'U + f, 
 
 corresponding to (V, 10, d). As the medium is supposed homo- 
 genous and incompressible, this equation reduces to 
 
 <5a , , 1 c?A , , , 
 
 ^^ + V (laa^ + A • a) -h - ^f -h (curl a) x A 
 
 — aA^V = f^v'^U + f. 
 
 This may now be introduced in equation («), and the equation 
 then separated into two equations, just as in the case of the cor- 
 responding hydrodynamic equation. We thus arrive at the sys- 
 tem of equations 
 
 t'^a 
 
 (h) ^^ = /^v'U - v(aA -f i^a-), 
 
 1 clA 
 
 (c) ^ 'df^^'^ ^^'^ ~ ^"""'^ ^^ ^ ^' 
 
 where the first is that of tlie " induced," the second that of the 
 " energetic " motion. 
 
 The first of these equations differs from equation («) only by
 
 GENERAL CONCLUSIONS. 121 
 
 quantities of the order generally neglected in the theory of elas- 
 ticity. If we agree to neglect these quantities, we may still de- 
 scribe the geometry of the field by the system of equations 
 
 dA 
 
 ~- = curl b, 
 
 dB 
 
 where now 
 
 --r = — curl a, 
 ot 
 
 A = eta 4- A^, 
 B= /3b. 
 
 But if we proceed to the second approximation, we have, besides 
 these equations describing the geometric configuration from time 
 to time, to consider another partial motion, given by equation (c). 
 And if we demand here that the energetic velocity be conserved 
 individually, dAJdt = 0, we find that an exterior force f must 
 be applied, which has the value 
 
 f = aA^v — (curl a) x A. 
 
 This force inversely corresponds to the exterior force which had to 
 be applied in the corrfesponding electromagnetic system, in order 
 to prevent the production of visible motions due to the forces 
 exerted by the system upon intrinsic electric polarization, cor- 
 responding to A, and upon magnetic current, — curl a. 
 
 5. This result thus gives a new and remarkable extension of 
 the analogy. And the fact that continued research leads to further 
 extension of the analogy between the formal laws of the phenom- 
 ena, if not between the phenomena themselves, seems to indicate 
 that there exists a common set of laws, the laws of the fields of 
 force, where the expression fields of force is taken in a suitably 
 extended sense. If this be true, the investigation of this common 
 set of laws and the discovery of all i)henomena obeying them will 
 be one of the great problems of theoretical physics. And investi- 
 16
 
 122 FIELDS OF FORCE. 
 
 gations suggested by this idea may perhaps, sooner or later, lead 
 even to the discovery of the true nature of the electric or mag- 
 netic fields. 
 
 6. But investigations of this kind can be considered as only 
 just begun. And if we return to our result relating to the 
 elastic field, it is easy to point out its incompleteness. In this 
 field we have not only the well known geometric analogy, but 
 also a dynamic analogy to the electrodynamic field, at least so 
 long as we confine our attention to the analogy between the 
 formal laws of the phenomena, and not to the phenomena them- 
 selves. And this dynamic analogy has exactly the same inverse 
 nature as in the case of the hydrodynamic field. But it should 
 be emphasized that this dynamic analogy, in the form in which 
 we have found it, has not the same degree of completeness as 
 the geometric analogy. I pass over here the fact that we have 
 given to our development only a restricted form, by supposing 
 the medium to be homogeneous and incompressible, and thus ex- 
 cluding beforehand heterogeneities and changes of volume. Most 
 likely this gap can be filled. But the great drawback is this : the 
 dynamics of the electromagnetic field relates to two classes of forces, 
 the electric forces and the magnetic forces, while our analysis of 
 the elastic field has led us to the discovery of only one class of 
 forces, namely, forces which correspond to the electric forces, ac- 
 cording to our interpretation of them ; but we have discovered 
 no trace of forces corresponding to the magnetic forces of the elec- 
 tromagnetic field. It is true that, making use of the symmetry, 
 we can change the interpretation, comparing from the beginning 
 the velocity with the magnetic, instead of the electric flux. The 
 elastic field will then, according to our analysis, give forces cor- 
 responding to the magnetic forces of the electromagnetic field, but 
 at the cost of the complete disappearance of the forces which pre- 
 viously corresponded to the electric forces. 
 
 7. Final Remarks on the Problem of Fields of Force. — It is too 
 early of course to consider tliis incompleteness as a decisive failure 
 of the analogy in the elastic media. From the beginning there
 
 GENERAL CONCLUSIONS. 123 
 
 seemed to exist no dynamic analogy at all. However, writing the 
 inertia-term of the elastic equation in its correct form, we found 
 at once forces corresponding to one class of forces in the elec- 
 tromagnetic field. But even in this form, the elastic equations 
 will generally be only approximations. For the expression of the 
 elastic forces is based upon Hooke's law of the proportionality of 
 the stresses to the deformations, and this law is an approxima- 
 tion only. Will the addition of the neglected terms, under cer- 
 tain conditions, bring full harmony between the electromagnetic 
 and the elastic field ? I put this question only to emphasize a 
 problem which is certainly worth attention. If the research be 
 carried out, it will certainly lead to valuable results, whether the 
 answer turns out to be positive or negative. And even if the 
 answer be negative, the investigation of the fields of force will 
 not therefore be completed. It is not at all to be expected that 
 the intrinsic dynamics of the electromagnetic field should corre- 
 spond to that of one of the simple media of which we have a 
 direct empirical knowledge. When the fields of these simple 
 media are thoroughly explored, so that we know how far the 
 analogy of their fields to those of electromagnetism goes, the time 
 will then have come, I think, to put the prol)lem in another form : 
 What should be the properties of a medium, whose fields shall give 
 the completest possible analogy to electromagnetic fields ? 
 
 Even when the problem is put in this form, we have the advan- 
 tage that preparatory work of great value has already been done. 
 The gyrostatic ether, which was introduced by MacCullagh and 
 Lord Kelvin, is a medium with very remarkable properties. As 
 is well known, the fields in this medium give as perfect a geo- 
 metric analogy to the electromagnetic field as the elastic medium. 
 And the form of the expression for the energy in this medium 
 seems to indicate the possibility of a dynamic analogy of greater 
 extent than that which is likely to be found in the case of the 
 common elastic medium. 
 
 It will be clear after these few remarks, that the problem of fields 
 of force is of vast extent. We are only at the beginning of it.
 
 124 FIELDS OF FORCE. 
 
 8. Kinetic Theories. — The problem of fields of force iu this 
 general sense evidently belongs to a class of problems which has 
 been present in the minds of the natnral philosophers from the 
 very beginning of our speculations with regard to nature ; but 
 the method of stating the problem has changed. 
 
 From the very first of human speculations on the phenomena 
 of nature strong efforts have been made to construct dynamic 
 models of these phenomena. These dynamic models seem to be 
 the natural way to render the phenomena of nature intelligible 
 to the human mind. I need only remind you of the efforts 
 of the old philosophers of the atomistic school, such as Demok- 
 RiTOS or Epicurus, or of philosophers of later time, like Des- 
 cartes. Or I may mention a long series of theories of special 
 physical phenomena, for instance Huyghen's, and Newton's 
 theories of light, theories opposed to each other, but both of them 
 dynamic theories. Or I may remind you of the kinetic theory of 
 gases of Bernoulli, Kronig, Clausius, and Maxwell, or of 
 Maxwell's ingenious ideas of " physical lines of force." 
 
 But most of these speculations have broken down more or less 
 completely. Of the universal constructions of the atomists nothing 
 is left except the building stones themselves, the atoms, whichj 
 however, have remained to this day an indispensable idea to the 
 natural philosopher. Descartes' theory of universal vortices 
 had the same fate. But though it fell, it left germs of fruitful 
 ideas, leading in the direction of the fields of force. Newton's 
 theory of light also broke down. But it did not exist in vain. 
 For the fact that phenomena of radiation could be explained ac- 
 cording to his principle immensely facilitated the interpretation of 
 the new phenomena of radiation, discovered in vacuum tubes and 
 in radioactive substances. The theory of light of Huyghens and 
 Fresnel is still unshaken, if it is considered merely as an abstract 
 undulation theory. But it is open to doubt whether it still exists 
 in its original form as a theory which explains the phenomena of 
 light on dynamic principles. For a dynamic theory of light will 
 hardly be satisfactory before we have a dynamic theory of electro- 
 magnetism.
 
 GENERAL CONCLUSIONS. 125 
 
 This fate of dynamic theories which have had the unanimous 
 support of all physicists may also bring into a dubious light dy- 
 namic theories which are still highly appreciated, as, for instance, 
 the kinetic theory of gases. As a matter of fact, a strong reaction 
 against dynamic theories has appeared. 
 
 9. The Relation of Kinetic Theories to the Phenomenological 
 Principles of Research. — Reactions against exaggerations are 
 always wholesome. On the other hand, it is a law of nature that 
 reactions usually go to exaggerations. In accordance with this 
 law, the energetic school developed. I will not enter upon the 
 exaggerations of this school. But it has done good by em- 
 phasizing phenomenological research, the principles of w'hich 
 were developed especially by Professor Mach at Vienna, pre- 
 vious to the formation of the energetic school, and without its 
 exaggerations. 
 
 The leading principle of Professor Mach is, that the phenomena 
 of nature should be investigated with perfect impartiality and free- 
 dom from prejudice that the research should lead ultimately to a 
 kinetic theory, or to any other preconceived view of natural phe- 
 nomena. If this idea be carried out with perfect consistency, it is 
 necessary, of course, not only that we should avoid the positive 
 prejudice that the physical phenomena are ultimately phenomena 
 of pure kinetics, but that we should also avoid the negative preju- 
 dice that the phenomena of nature are not ultimately kinetic. 
 The principles of phenomenological research are therefore, rightly 
 understood, not hostile to kinetic research, if this be only con- 
 ducted with perfect impartiality. 
 
 If this be admitted, the extreme importance of kinetic research 
 will not be denied by the adherents of the phenomenological 
 principles of research. For no unprejudiced observer will deny 
 that physical phenomena are inextricably interwoven with kinetic 
 phenomena. Neither will he deny that our power of kinetic 
 research exceeds by far our power of every other kind of physical 
 research. The reason is obvious. We are all kinetic machines. 
 Instinctive kinetic knowledge is laid down in our muscles and
 
 126 FIELDS OF FORCE. 
 
 nerves as an inheritance from the accumulated dynamic exper- 
 ience of our ancestors, and has been further developed without 
 interruption from the time of our first motions in the cradle. And 
 furthermore, while we have this invaluable instinctive knowledge 
 of the fundamental principles of dynamics, we have at the same 
 time an objective view of dynamic phenomena as of no other 
 physical phenomena, from the fact that we have the power of 
 following and controlling the phenomena of motion by several of 
 our senses at the same time, while for other phenomena, such as 
 sound, light, or heat, we have only one special sense, and for still 
 others such as electricity, magnetism, or radioactivity, we have 
 no special senses at all. 
 
 No wonder, therefore, that at the time when science grew np 
 dynamics soon developed into the model science, from the formal 
 point of view the most perfect of physical sciences, and in this 
 respect second only to pure mathematics. This also explains why 
 the dynamic side of physical phenomena has always offered the 
 best point of attack for research, while, on the other hand, it gives 
 the obvious reason why we may be tempted to overestimate the 
 value of our dynamic constructions. 
 
 But if a reaction against exaggeration has been necessary, noth- 
 ing can be gained by giving up advantages which, for subjec- 
 tive reasons at least, are combined with the kinetic direction of 
 research, whatever be the final objective result of these researches. 
 The reaction has taught us that problems should be stated in a 
 perfectly unprejudiced way. 
 
 10. The. Coiiipnrative Method. — It is such a way of conducting 
 the investigation of the relations betw^een physics and kinetics, 
 which we have tried to realize in these researches on fields of force. 
 The essence of the method is, that kinetic systems are made the 
 subject of pure phenomenological research. Their laws and pro- 
 perties are made the subject of impartial investigation, but w'ith 
 constant attention to the analogies and the contrasts between the 
 laws found for the dynamic system and the laws of physical 
 phenomena.
 
 GENERAL CONCLUSIONS. 127 
 
 And this comparative method is applicable far outside the 
 limits of our special problem of fields of force. Indeed, it is the 
 method used by such authors as Boltzmaxx, Helmholtz, 
 Hertz, and Wir.LARD Gibbs, in their profound researches in the 
 dynamical illustration of physical laws and phenomena, especially 
 those of heat and thermodynamics. These researches are un- 
 completed, just as are our researches on fields of force, and will 
 probably remain so for a long time. But the more they have 
 advanced, the stronger has been the demand for rigorousness of 
 methods ; the more have the methods of construction been forced 
 back and the impartial comparative method advanced. And no 
 one has emphasized conservative and safe methods more strongly 
 than WiLLARD Gibbs. In the preface to the last work which he 
 has left us he expresses this in the following plain words : 
 
 "Difficulties of this kind have deterred the author from at- 
 tempting to explain the mysteries of nature, and have forced him 
 to be contented with the more modest aim of deducing some of 
 the more obvious propositions relating to the statistical branch of 
 mechanics. Here there can be no mistake in regard to the agree- 
 ment of the hypotheses with the facts of nature, for nothing is 
 assumed in that respect. The only error into which one can fall, 
 is the want of agreement between the premises and the conclu- 
 sions, and this, with care, one may hope, in the main, to avoid." 
 His method is exactly the same as that which we have tried to 
 ■ employ, namely the impartial research of each branch of physics 
 by itself, but with comparison of the resulting laws, and with the 
 greatest possible caution with respect to the conclusions to be 
 drawn from the analogies and the contrasts presenting themselves. 
 The method is that of comparative anatomy. Is it too sanguine 
 a hope, that this method will, sooner or later, unveil for us the 
 relations of the different physical phenomena, just as the methods 
 of comparative anatomy successively give us an insight into the 
 relation between the different kinds of living beings? 
 
 11. On the Value of the Comparative Method for Instruction 
 in Theoretical Physics. — I cannot leave the discussion of this com-
 
 128 FIELDS OF FORCE. 
 
 parative method without seizing the occasion to emphasize its vahie 
 also in instruction in theoretical physics. The results obtained 
 by this method and the discovery of similar laws in apparently 
 perfectly different branches of physics makes an unexpected con- 
 centration of instruction possible. And if the principle be carried 
 out, and similar facts presented in similar ways, the analogies will 
 facilitate, to a degree not to be overestimated, the power of the 
 student to comprehend and assimilate the matter. Especially will 
 this be the case when the analogies give us the opportunity to 
 throw light upon obscure theories, such as those of the electromag- 
 netic field, by means of perfectly plain and comprehensible theories 
 such as those of the hydrodynamic field, in which every step can 
 be made by rigorous mathematical conclusions, by elementary in- 
 ductive reasoning, or by experiment. 
 
 And yet, this saving of labor, so imperatively demanded in our 
 days whenever possible, is perhaps less essential in comparison 
 with the independence relative to the methods and the results, 
 which the student will gain when he observes how similar methods 
 can be used, and similar laws obtained, in apparently widely dif- 
 ferent branches of physics. This will teach him to judge better the 
 value of the methods, and give him independence of view for his 
 future work as an investigator. 
 
 The arrangement of instruction according to principles by which 
 the analogies at our disposal are used as much as possible for the 
 benefit of the student, is a problem which has its own charm, in- 
 voluntarily attracting the attention of the investigator engaged in 
 research on these analogies. Time does not allow me to enter 
 upon the details of my experiments in this direction. But before 
 concluding these lectures, I wish to answer an objection, which 
 seems to lie near at hand, against the use to a greater extent of 
 these analog-ies in instruction. 
 
 12. Theory and Practice. — It seems to be an obvious reflec- 
 tion, that instruction conducted according to the plan thus indi- 
 cated will be of an exceedingly abstract nature, tending to develop 
 in a purely theoretical direction, and to draw attention away from
 
 GENERAL CONCLUSIONS. 
 
 129 
 
 practically useful points. To take the nearest example: hydro- 
 dynamics is useful if it teaches us to understand and calculate 
 water motions occurring practically. Now water is practically 
 homogeneous and incompressible, and hydrodynamics of practical 
 use will have to direct the attention to the investigation of the 
 motions of this simple medium, and not to the abstract fluid sys- 
 tems considered by us, with density and compressibility varying 
 according to laws never occurring practically. 
 
 I was of this opinion myself when I commenced my study of 
 these extraordinary fluid systems. Nothing was further from my 
 thoughts than to expect practical results from investigations of 
 this abstract nature. But as the result of conversations with sci- 
 entific friends who were interested in the dynamics of the ocean 
 and the atmosphere, I happened to see that certain theorems 
 which I had developed to investigate the motions of my abstract 
 fluid system had immediate bearing upon the motion of these two 
 media." And the reason why these theorems had not been discov- 
 ered a long time before was obvious. To work out the science 
 of the motion of fluids in a practical form investigators had 
 always considered the fluids as homogeneous and incompressible, 
 or, in the most general case, as compressible according to an 
 idealized law, so that the density depended upon the pressure 
 only. But these very suppositions precluded from consideration 
 the primary causes of the motions in the atmosphere and the 
 sea. For these primary causes are just the difi^erences of density 
 which do iwt depend upon the pressure, but on other causes, such 
 as diff-erences of temperature and salinity in the sea, and diifer- 
 ences of temperature and humidity in the atmosphere. While 
 the old theorems of the practical hydrodynamics did not allow 
 us to take up from the beginning the discussions of the circu- 
 lations of the atmosphere and the sea, the thoerems which I had 
 developed for my impractical fluid systems gave at once a very 
 simple view of the atmospheric and oceanic circulations. If, there- 
 fore, it be considered a question of practical importance to mas- 
 ter the dynamics of these two universal media on which we 
 17
 
 130 FJEl.DS OF FORCE. 
 
 human beings are in such a state of dependence, then the methods 
 of this theoretical hydrodynamics are not impractical. And I 
 do not think that this is an isolated fact. For the more we ad- 
 vance in theoretical and practical research, the more we shall dis- 
 cover, I think, that there is really no opposition between theory 
 and practice. 
 
 I hope that you will allow me to exemplify this in the addi- 
 tional lecture to-morrow, in which I shall consider the hydrody- 
 namic fields of force in the atmosphere and the sea.
 
 APPENDIX. 
 
 Vector Notation and Vector Formulae. 
 
 A vector with the rectangular components A^, A^, A^ is desig- 
 nated by A. 
 
 A vector with the rectangular components B^, B^, B^ is desig- 
 nated by B. 
 
 A vector with the rectangular components C^, C , C, is desig- 
 nated by C. 
 
 Vector Sum. — The three scalar equations, 
 
 A + B =C, 
 
 X ' X X' 
 
 A +B =C, 
 
 are represented by one vector equation, 
 
 (1) A + B = C. 
 
 C is called the vector sum of the two vectors A and B, 
 
 Scidat- Product. — The scalar quantity A^B^ -j- A^B^ + AB, is 
 designated by AB and called the scalar or dot-product of the 
 vectors A and B, 
 
 (2) kB = AB,+AB^ + AB,, 
 
 Vector Product. — The three scalar equations, 
 
 AB,^-AB^^=C^, 
 AB^-A^B = C,^, 
 AB^-AB^=C,^, 
 
 are represented by one vector equation, 
 
 (3) A X B = C. 
 
 The vector C is called the vector- or cross-product of the two 
 vectors A and B. The definition states that the vector product 
 
 131
 
 132 FIELDS OF FOllCE. 
 
 C is normal to each of the vector-factors A and B, and is directed 
 so that the positive rotation according to the positive screw rule 
 around the vector C rotates the first vector- factor, A, towards the 
 second, B. Change of the order of the factors, therefore, changes 
 the sign of the vector-product. 
 
 Triple Products. — In a scalar product one vector-factor can be 
 a vector-product. For this triple product it is easily proved that 
 dot and cross can be interclianged, and that circular permutation 
 of the factors is allowable, thus 
 
 .^. ABxC = CAxB = BCxA 
 
 ^ ^ =AxBC = CxAB = BxCA. 
 
 In a vector-product one factor itself may be a vector-product. 
 Cartesian development easily gives the formula 
 
 (5) A X (B X C) = -(AB)C + (AC)B. 
 
 Linear Derivdtion of a Scalar Qaanfiti/. — The three scalar 
 equations, 
 
 da 
 A = ;^j 
 
 da 
 
 A=^, 
 
 oz 
 
 are represented by one vector equation, 
 
 (6) A = \70L. 
 
 The differentiating symbol V or " del " represents a vector opera- 
 tion with the three component-operations d jdx, <-> jdy, d jdz. The 
 vector A or v^ sliows the direction of greatest increase of the 
 values of the scalar function a, and represents numerically the 
 i-ate of this increase. The vector — vx is called the gradient of 
 the scalar quantity a (compare the classical expressions pressure- 
 gradient, temperature gradient, etc.).
 
 APPENDIX. 133 
 
 Sjiher'wal Derivation of a Scalar Quantity. — The sum of the 
 secoud derivations of a scalar quantity may be called the spheri- 
 cal derivative of this quantity, and the operation of spherical de- 
 rivation may be designated by V^, thus 
 
 d'^a d^'a d^-a. ., 
 
 Divergence. — The scalar quantity dAjdx + dAJdy -\- dAJdz 
 
 is called the divergence of the vector A, and designated by div 
 
 A, thus 
 
 dA dA^ dA^ ,. ^ 
 
 Curl. — The three scalar equations, 
 
 dy dz ~ '' 
 
 dA^ dA^ _ 
 
 '^dz ~'dx ~ y' 
 
 dx dy * ' 
 
 define a vector C, which is called the curl of the vector A, and 
 the three scalar equations are represented by the one vector equa- 
 tion, 
 (9) curl A = C. 
 
 Spherical Derivation of a Vector. — The three scalar equations, 
 
 ^'A d^-A d^A^ ^^, 
 
 d'A^ d'A^ d^A 
 'M' "^ "a/" ^ ^dz 
 
 a- A 0"A, ^
 
 134 FIELDS OF FORCE. 
 
 define the vector C, which is called the spherical derivative of A, 
 and the three scalar equations are represented by one vector 
 equation, 
 
 (10) V'A = C. 
 
 Linear Operations. — The three equations, 
 
 , dB, dB , dB, ^ 
 
 ox •' ^y ^^ 
 
 , dB , dB , dB, 
 
 ^' dx + " dy + ~ dz 
 
 , dB , dB , dB ^ 
 
 ^•^^ + ^^aF + ^^^='^^' 
 
 may be represented by one vector equation, 
 
 (11) AvB = C. 
 The three scalar equations, 
 
 . dB\. , dB^ , dB ^ , 
 
 A — - + A - -^ + A ~--^= CJ, 
 " dx ^ ' dx ' dx " ' 
 
 dB , dB^ , dB 
 
 dy ■' dy dy 
 
 , dB, , dB^ . dB, ^, 
 
 ^ dz '' dz ~ dz '^ 
 
 may be represented by one vector equation, 
 
 (12) ABv=C'. 
 
 Between the two vectors defined by (11) and (12) there is the 
 
 relation 
 
 (l;i) A vB = AB V + (curl B) X A. 
 
 t^pecial Fornvihc of Tran.fonnidion. — Tlie following formulie 
 are easily verified by cartesian expansion : 
 
 (14) div aA = a div A + A- V '^j
 
 APPENDIX. 135 
 
 (15) div (A X B) = — A • curl B + B • curl A, 
 
 (IB) curl (a V /3) = V ^ X V /3. 
 
 If the openition curl be used twice in succession, Me get 
 
 (17) curP A = V div A - v' A. 
 
 Integral Fonnuke. — If dr be the element of a closed curve 
 and (Is the element of a surface bordered by this curve, we have 
 
 (18) jAr^r = /curl A-r/s 
 
 (Theorem of Stokes). If ds be the element of a closed surface, 
 whose normal is directed positively outwards, and dr an element 
 of the volume limited by it, we have 
 
 (19) f^'^^^ = Jdiv Af/r. 
 
 Transformation of Integrak Involving Products. — Integrating 
 the formula (16) over a surface and using (18), we get 
 
 (20) faS7l3-dr = / V a x V /S • ds. 
 
 Integrating (14) and (15) throughout a volume and using (19), 
 we get 
 
 (21) J A • V Mh = — fa div Adr + JaA ■ ds, 
 
 (22) J A • curl Bdr = Jb ■ curl Ar/r - /a x B ■ ds. 
 
 If in the first of these integrals either a or A, in the second either 
 A or B, is zero at the limiting surface, the surface integrals will 
 disappear. When the volume integrals are extended over the 
 whole space, it is always supposed that the vectors converge towards 
 zero at infinity at a rate rapidly enough to make the integral over 
 the surface at infinity disappear. 
 
 Performing an integration by parts within a certain volume of 
 each cartesian component of the expressions (11) and (12) and 
 supposing that one of the vectors, and therefore also the surface- 
 integral containing it, disappears at the bounding surface of the 
 volume, we find, in vector notation.
 
 136 FIEI>DS OF FORCE. 
 
 (23) Ja V Bdr = - /B cliv Adr, 
 
 (24) Jab V (h = -/BA V dr. ■ 
 Integrating equation (13) and making use of (23), we get 
 
 (25) JB div A dr = -J'AB V d- r /(curl B) x A dr. 
 
 For further details concerning vector analysis, see : Gibbs-Wil- 
 son, Vector Analysis, New York, 1902, and Oliver Heaviside, 
 Electromagnetic Theory, London, 1893.
 
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