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THE GIFT OF ..C,At**vsAj^4. niediiini surrounding the magnets and tlie fluid surround- ing tiie pulsating or oscillating bodies contain lieterogeneities is more delicate. In the hydrodynaniic case the heterogeneities should be fluid, and it is practically impossible, on account of the action of gravity, 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 their density, it is easily seen, by means of our registering device, that 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 particles would probably be the best method to employ. Experiments which we shall per- form later will give, however, indirect proofs that the fields have exactly the expected character. 16. 0/! I'osh-ible Extensioiis of the Analogy. — We have thus found, by elementary reasoning, a very complete analogy between the geometric properties of hydrodynamic fields and electric or magnetic fields for the case of statical phenomena. And, to some extent, we have verified these results by experiments. It is a natural question then, does the analogy extend to fields of greater generality, or to fields 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 fields considered with both elec- trostatic and magnetic fields. If there exists a perfect hydro- dynamic analogy to electromagnetic phenomena, the hydrodynamic fields considered will, presumably, turn out to be analogous either to electrostatic fields only, or to magnetic fields only, but not to both at the same time. The question therefore can now be raised, would our hydrodynamic fields in an eventually extended analogy correspond t« the electrostatic or the magnetic fields ? To this it must be answered, it is very probable that only the analogy to the INVESTIGATION OF GEOMETRIC PROPERTIES. 25 electrostatic fields will hold. As an obvious argument, it may be emphasized 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 of 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, for 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 an 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 flow which do not pass through the bodies produciug the motion. It is easily precon- ceived, that if the condition of the oscillatory nature of the fluid motion be insisted upon, the required motion cannot be pro- duced by fluid pressure in a perfect fluid. A cylinder, for instance, making rotary oscillations around its axis will produce no motion at all in a perfect fluid. Quite the contrary is true, if the fluid be viscous, or if it have a suitable transverse elasticity, 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 Hydrodynamic Analogy to the Fields of Stationary Eleetroinagnetisjn. — 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. Fig. 9. This analogy is that discovered by v. Helmholtz in his research on the vortex motion of perfect fluids. According to his celebrated results, a vort«x 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 vort€x 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 GEOMETRIC 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- 28 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 with 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 the fields whose geometric properties we have investigated. II. ELEMENTARY INVESTIGATION OF THE DYNAMI- CAL PROPERTIES OFHYDRODYNAMIC FIELDS. 1. The Dynamics of the Electne or the Mdgnetic 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 wliich these visible dynamic phenomena depend. Faeaday's idea, for instance, of a tension parallel to, and a pressure perpendicular to the lines of force, as well as Maxweli-.'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 have 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 OP 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- MELL 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. Resultant Force against a Pulsating Body in a Synchronously Oscillating Current. — 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 the 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 periodic, with a mean value zero in the period. It will then be brought only into oscillation, and no progressive motion will result. But 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 pulsations, synclironous 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 maximum in one direction, and a minimum when the acceleration has 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, AVe thus find the result : A pulsating body in a synchronoi^sly oscillating current is subject to the action of a resultant force, the direction of tchich is thai of the acceleration in the current at the time when the pulsating body has its maximum volume. 3. Mutual Attraction and Repulsion between Two Pidsaiing 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 proprtional to two param- ?,2 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 puhaling in the same phase there is an apparent attraction ; between bodies pulsating in the opposite phase there is an apparent repulsion, tlie force being proportional to the product of the two intensitieji of pulsation, and proportional to the inverse square of the distance. 4. DiscuJision. — 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 Cour/)MK'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 with 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 opposite j)liase 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. But we cannot discover the reason for it in the present limited state of 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 PROI'ERTIKW. 33 Thus, taking the facts 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 : Pulsaiing bodies act upon each other a.s if they were electrically charged particles or magnetic poles, but ir'ith the difference that charges or poles of the same sign attract, and charges or poles 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 has 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. Af 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-f/, 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 FIELDS OF FORCE. surface with the pivot as center, and the equilibrium will be neutral for a horizontal motion, and stable for a vertical motion. 6. Experimentfi with Pulsating Bodies. — Having one pulsator in the pulsation balance, take another in the hand, and arrange the •itiiiiiiiiiiitiiiiiii 'tHIJHttllltf t}}\ ^W,\\K\K\\\W.\\K, ik'.kk'.'.'.k^'.t.kt.kV k>^k^kl.^^M.k..l^^^.. ■.■.■■>.■.« ^^^s^^^^s^Y Fig. 11. generator for pulsations 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 repulsion appears (Fig. 1 2, 6). With the torsion balance it may INVESTIGATION OF DYNAMICAL rROPEETIES. 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 vibi-atory 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. Action 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 with the law of poles reversed. 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. 1 3, a) 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 FORCK. 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 i)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, c). If the oscillating body be turned around, the position of equilibrium will be on the other side. Exactly the same small o c Fio. 13. lateral displacement is observed when a short magnet is brought into the transverse position in the neighborhood of the pole of a long bar magnet which has the same freedom to move as the pulsating body. 8. Force a(/nin.st an Oacillatiing 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 equal and opposite to that exerted by the oscillating body upon the pulsating 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 apparent actions at a distance, where not only the two bodies but also a third one, the fluid, are engaged, no general conclusion can be 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 dififerent 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 Fig. 14. 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 osalUatliig body in tlie hydrodynamic field will be subject to the action of a force similar to that acting upon an elementary magnet in the magnetic field, the only difference being the difference in the signs of the forces which follows from the opposite pole-law. 9. Experimental Investigation of the Force exerted by a Palsat- 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 ill the hand, it will be seen that attraction takes place when the pulsating 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 witli magnetism can be followed further if the pulsating body 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 between Two OsciUnVnig Bodies. — 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 b, so that the axes of oscillation lie in the same line. The experiment will then correspond to tliat with magnets in longitudinal position. We get attraction in the case, (Fig. 15, a), when the oscillating bodies INVESTIGATION OP 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, 6), 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- e ee b e €>© 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. We 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- S3'mmetric positions. In all of them it is easily shown that the force inversely corresponds to that between two magnets in the corresponding positions. 11. notations of the Oscillating Body. — We have considered hitherto only the resultant force on the oscillating body. But in general the two forces acting at the two extreme positions also form a eouj)le, like the two forces acting on the two poles of a mag- net. The first effect of this couple is to rotate the axis of oscil- lation of the body. 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 pivot. 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 body may now be replaced by an oscillating body. INVESTIGATION OF DYNAMICAL PROI'KRTIKS. 41 Except for the direction of the force, we shall got rotations corres- ponding to those of a compass needle under the influence of a magnet. The position of equilibrium is always the position of greatest attraction (Fig. 15, a, 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, 6, and if it oscillates at right angles to this line, it will be the position indicated in Fig. 16, rf, 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- b 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 Aiudogous to Those of Teinjyorm-y Magnetimii. — We have already considered the forces between bodies which 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 u^wn 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 upon the bodies. ^Ye shall see how it is in this respect in the hydrodynamic 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 difiFerent masses of water, and if these have not exactly Fig. 17. the same motion, it will be subject in these two positions to kinetic buoyancies not exactly equal 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 upon the oscillation a progressive motion. That the average force which produces this progressive motion is strictly analogous to the force depending upon induced magnetism or electrification by influence, is easily seen. As we have already shown in the preceding 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 TROPERTIES. 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, 6 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 diamagnetic polarity. And, as is evident at once from the similarity of these figures, the heavy body in the hydrodynamic 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 : Tlie light body icill move in the direction of decreasing, the lieavy 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 with increasing distance. Therefore, the light body will be re- pelled, and the heavy body attracted by the pulsating or the oscil- lating body. To make this experiment we sus}:)end 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)elled. If one is sufficiently near, the small induced oscillations of the light body may also be observed. If the pul- sating body be brouglit near the heavy body, an attraction of simi- lar intensity is observed. In both cases it is seen that the force decreases 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. If for the pulsating body we substitute an oscillating body, the same attractions and repulsions are observed. Both poles of the oscillating bodj' exert exactly the same attraction on the heavy body, and exactly the same repulsion on the light body, and even INVESTIGATION OF DYNAMICAL PKOl'ERTIES. 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 tlie action of a mag- net on a piece of soft iron, or on a piece of bismuth. 14. SimiJtaneous 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 fi-eely 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 nejir 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 he made, all show- ing in the most striking N\-ay the analogy between the magnetic and the hydrodynamic forces. 15. OrieiUation of Cylindrical £odie-i. — 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 {wwerful 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 4G FIELDS OF FORCE. The light cylinder is attaciied from below and the heavy cylinder from alKjve, and on bringing near a pulsating 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 corresponds to the diamagnetic body, the longitudinal position. 16. Neutral Bodies Acted Upon by Two or More Pulsating or OarUldtinc) 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 tlie other were removed. For the direction of the great&st increa.se or decrease of the energy in the field due to both magnets is in general altogether different from the parallelogram- 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 contra.st to the principle of the parallelogram of forces, considered, it must be enii)hasized, as a physical principle, not merely as a mathematical principle ; i. 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 iiave great interest here, becati.se they are well suited to show how the analogy between hydrodynamic and magnetic phenomena goes even into tlie 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 surface, the ])iece of iron will be attracted to a point vertically above the pole. If a south pole be placed in the same vertical .symmetrically above the surface, nothing peculiar is observe = — I J"A ■ V<^(^T + fia.^- curl Gdr. 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 thin sheet, in which the scalars or the vectors of the field change 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 {(I) 4> = 1 JB 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 KO FIELDS OF FORCE. forces, the factor of the fi rst term being the mechanical force de- pending on the electric field, and the factor of the second term being tlie force depending upon the magnetic field, f, = (div A)a„ - ia^va + (curl aj x A, ^^' f „ = (div B)b„ - i-bf V^ + (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 motion 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 J(A-a„+B-bjV, which is simply the product of the energy per unit volume into the velocity. Finally, the last terra 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 - (a. V)A + 1 (A . aJV - (b„ • V)B -f 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- ponent of which against a plane whose orientation is given by the unit normal N is K(A N) - (1 A ■ ajN + \{B ■ N) - (i B • 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- DYNAJQC EQUATIONS OF E].ECTEOMAGNr:TIC 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. Tliis 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 diflFerences of opinion with regard to the details 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 principle of 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 of the electromagnetic field that we possess. 5. The forces in the Electromagnetic Field. — What particularly interests us is the expression for the mechanical forces in the field, (4, g). 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„ - ib^ V/3 + (curl bj X B. . This is a force per unit volume, and if our theory is correct, this expression should give the true distribution of tiie 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 tiiis 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, tiie 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 combined 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 + c , where c is the true electric current, and c^ the fictitious current, by DYNAMIC EQUATIONS OF ELECTROMAGNETIC 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 bj 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 term 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 cannot 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 this with perfect certainty ; if we integrate the elementary forces for the whole volume of a body, we shall arrive at the ti-ue 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 (a) F = /(div B)bjT - /ib^v/3t?T + /(curl bj x Bdr, 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(?T - / W^^d-r + /(curl b) x BcZt + J, where J = J(div B)b rfr - /(b • b ) VyS^T - Jib^ V/Srfr + /(curl bj X ^dr. To reduce the expression for J we consider the first term. Transforming according to well known formulae, we get /(div B)b(ZT = - jBvbc^T = - Jfib V(?t - /(curl bj x '^dt. Substituting, we get J reduced to three terms, (^") J = - /Bb Vrfr - /(b ■ bj V/3(^T - fibl^^dr. Introducing in the first of these integrals B = /8b + /3b^, we get - /Bb V^T = - //3(bb.v),) • • (DJ 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, £, while this velocity may exist in the fluid bodies ; just as in the free ether we have no 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 Hydrodynamic 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 (6) f = (div A)a„ - Ja^va + (curl aj x A, i. 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 (6). 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 (^,) f, = - (div A)a„ -f ialva - (curl aJ 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 specific momentum a and the ener- getic velocity A, according to the equation of connection (b) A = aa -|-A^. PEOPERTIES OF THE HYDKODYNAMIC FIELD. 99 Performing the differentiation and making use of the equation of continuity (2, a), we have ^ + (dxvA)a+-^ = f-Vi.. Introducing in the first left liand term the local time-derivation, ^ + Ava + (divA)a + -^^^ = f-Vi., or, transforming the second left hand member according to the vector formula, aa 1 dX (c) -^ + Aa V + (curl a) x A + (div A)a -\ -tt = f — Wp- Using the equation of connection (6) and performing simple trans- formations, we get for the second term in the left hand member Aav = aaav + A^av = ^ava' -f- A^av = V(^aa^ -f A^ a) — |aVa — aA^V- Introducing this in (c), 5a ., , . ^ . IdA, {d) -\- (curl a) X A — aA^V = f — Vp. -^ 4- v(i«a^ + A, • a) -f - -g -F (div A)a - laVa Now, we can split the equation in two, requiring that the vector a satisfy the equation da. (e) ^=-V(;)-f-^«a^ + A,a), and we find that the other vector A^ must satisfy the equation (y) i!^ = i- (div A)a + -i-aV« - (curl a) x A -|- aA^v. 100 FIELDS OF FORCE. Both equations are different from the corresponding equations (5, b) and (5, c). 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 (B). 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 (D,), (-Dj), (-Dj). 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 (DJ upon the fundamental fluid, since in a fluid having the properties (Z),) ■ • • {D^ 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 (i)J 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 individually, we have dA and, reasoning as before, we find that under this condition the fluid system will exert per unit volume the force {E^ i^= — (div A)a + JaVa — (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 equations of hydrodynamic 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 opposite sign of the forces exerted by the fields. PROPERTIES OF THE HYDKODYNAMIC FIELD. 101 In regard to the closeness of this analytical anology, we have to remark that we do not know with perfect certainty which of the expressions (^,) 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. We cannot, therefore, decide which of the two forms that we have found for the analogy is the most fundamental. But we know 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 (^3) *3 = — (•li*^ -A^)* — A^av — (curl a) x A, and (B,) f^ = - a„ (div a)a — a„ (curl a) x a. The fact, which we have just proved, that the laws of the elec- tric or magnetic fields and of the hydrodynamic fields can be rep- resented by the same set of formulae, 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 necessarily 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 imply an analogy of the things related. The subject 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 formulae to an analogy of perfectly concrete nature, such as that represented by our experiments. VI. FURTHER DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. 1. According to the systems of formulae which we have de- veloped, the hydrodynamic 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 witli 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 existunder 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 THK ANALOGY. 103 3. Steady State of Motion. — The first form of the analytical analogy, in which we supposed local conservation of the energetic specific momentum, 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 (a). 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. Irrotational Circulation Oxiiside 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 Fidd 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. 105 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 Ampeee'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. Restricted Generality of the Field for the Case of Vibratm-y 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 = E, with the conditions for the surrounding fluid, (6) a=a„ E=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, g). 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 /«\ 1 dA^ (c) a dt f — (div A)a + Ja^Va -f 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. Penodic Functions. — 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,/", of the period t ; thus («) f{t + t) =/(«). A DEVELOPMENTS AND DISCUSSIONS OF THE ANALOGY. 107 The values of the function / should 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 (6) ^£y^t)dt = o, (c) ^^p\f(t)Ydt^i. 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)=V'2sin27r{*- + 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. We may thus write (e) f;'nf{t)dt where the first is that of the " induced," the second that of the " energetic " motion. The first of these equations differs from equation (a) 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 dt ' = curl b, dB dt '' = — curl a, A = = Bx ' Bx "' , BB , BB^ , 8B ■' Bfi ^ " By ^ •- By "' , BB, , BB, , BB may be represented by one vector equation, (12) ABv=C'. Between the two vectors defined by (11) and (12) there is the relation (13) AvB = AB V +(curlB)x A. Special Fonnalre of Transformation. — The following formulae are easily verified by cartesian expansion : (14) div aA = a div A + A- V «, APPENDIX. 135 (lo) div(A X B)= — AcurlB+ B- curl A, (16) curl (a V /3) = V a X V /S. If the operation curl be used twice in succession, we get (17) curP A = V div A— v^A. Integral Formulce. — If dr he the element of a closed curve and ds the element of a surface bordered by this curve, we have (18) /Arfr = /curl Ads (Theorem of Stokes). If ds be the elemeut of a closed surface, whose normal is directed positively outwards, and dT an element of the volume limited by it, we have (19) JAc?s = JdivAcZr. T)-ansformation of Integrals Involving Products. — Integrating the formula (16) over a surface and using (18), we get (20) . /av;8-dr = /vax V/8-(/s. Integrating (14) and (15) throughout a volume and using (19), we get (21) Ja ■ V adr = — fa div Adr + facA ■ ds, (22) J' A curl Bdr = /b • curl Adr — Ja 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 FIELDS OF FORCE. (23) Ja V BdT = - Jb div Adr, (24) J'aB V rfr = -/BA V dT. Integrating equation (13) and making use of (23), we get (25) Jb div A dr = -/AB S7 d- t /(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.