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 From the PiiaosopiitCAL Magazine for June 1872. 
 
 S 
 
 vu^ 
 
 ^ OcJC^ 
 
 
 
 (f) 
 
 NEW DISCUSSION 
 
 OF THE 
 
 HYDRODYNAMICAL THEORY OF MAGNETISM. 
 
 SY 
 
 Phofessoh CHALLIS, M.A., LL.D.. E.E.S. 
 
 A THEORY of Magnetism founded on propositions in Hy- 
 drodynamics was originally proposed by me in the Num- 
 bers of the Philosophical Magazine for January and February 
 1861. The same theory, considerably modified, is given in my 
 work ^ On the Principles of Mathematics and Physics F and the 
 Number of the Philosophical Magazine for July 1869 contains 
 an article the purpose of which is to make the mathematical part 
 of the theory more complete. In the present communication I 
 propose to discuss anew the principles on which magnetic phe- 
 nomena are explained according to this theory, with the view of 
 correcting or extending the results previously obtained. The 
 discussion will necessarily involve to a considerable extent the 
 hydrodynamical theory of galvanic currents, 
 
 I. I assume, as I have already done, (1) that all the active 
 forces in nature are different inodes of pressure, under different 
 circumstances, of a universal elastic aether, which may be mathe- 
 matically treated as a continuous substance pressing always pro- 
 portionally to its density ; (2) that all visible and tangible bodies 
 consist of inert spherical atoms of constant magnitude, held, 
 when undisturbed, in positions of equilibrium by attractive and 
 repulsive forces, the laws of which result both from the active 
 pressure of the aether and the passive resistance of the atoms due 
 to the constancy of their form and magnitude. The aether at 
 rest is accordingly assumed to be everywhere of the same den- 
 
 A 
 
2 
 
 Prof. Challis on the Hydro dynamical 
 
 sity ; and it is, further^ supposed that the atoms are so small 
 that even in dense bodies the space which they occupy is very 
 small compared with the intervening spaces. 
 
 2. In general, when the atoms of any substance are in posi- 
 tions of stable equilibrium, the attractive forces acting on any 
 atom in the direction of any straight line drawn through its 
 centre will just counteract each other ; and the same will be the 
 case with respect to the repulsive forces. But it is also con- 
 ceivable that the equilibrium of the atom may be maintained by 
 the mutual counteraction of attractive and repulsive forces. For 
 instance, let the substance be a long rectangular bar of steel, 
 slightly increasing in density by regular gradations from one 
 end to the other, and of uniform density in any transverse sec- 
 tion. Then the direction of the resultant of the molecular 
 attractive forces will be parallel to the length of the bar, as well 
 as that of the resultant of the atomic repulsive forces ; and it is 
 therefore supposable that by the former any atom may be as 
 much attracted towards the denser end as by the latter it is re- 
 pelled towards the rarer end. This, in fact, is assumed in the 
 present theory to be the case in a permanently magnetized bar 
 pf steel, the gradation of density being conceived to be originally 
 generated by the usual processes of magnetization, and to be 
 maintained exclusively by the proper atomic and molecular 
 forces of the steel. In soft iron a like gradation of density, 
 produced by the action of an extraneous magnet, or that of a 
 galvanic current, continues only so long as the action lasts. In 
 diamagnetic substances, such as bismuth, gradation of density 
 is similarly produced and maintained by the action of a magnet, 
 but the direction in which the density increases is opposite to 
 that in soft iron under analogous circumstances. Also the ap- 
 nlication of heat under certain conditions gives rise to magnetic 
 or galvanic phenomena, which, according to this theory, are re- 
 ferable to the production of gradations of atomic density by the 
 dynamical action of the heat. 
 
 3. In all cases of the existence of such gradation of density, 
 it can be shown on the above-stated hypotheses, by reasoning 
 which will be presently indicated, that motions either of the 
 sether within the body, or of the body relative to the sether, have 
 the effect of producing accelerations of the sether. The secondary 
 currents thus generated are considered to be the immediate cause 
 of magnetic or galvanic attractions and repulsions. Also, in the 
 Theory of Frictional Electricity which I proposed in the Philo- 
 sophical Magazine for October 1860, electrical attractions and 
 repulsions are accounted for in the same manner. Hence as the 
 generation of secondary streams under the above-stated condi- 
 tions is a fundamental proposition in the hydrodynamical theories 
 
Theory of Magnetism » 
 
 of these three physical forces^ I shall now endeavour to give as 
 exact a proof of it as may be possible. The mathematical rea- 
 soning relating thereto under the head of the Theory of Electric 
 Force in pages 545-547 of my work ^ On the Principles of 
 Physics ^ is incomplete and inaccurate. 
 
 4. Suppose a current of the sether to traverse a substance con- 
 sisting of atoms so arranged that their number in a given space 
 increases regularly^ but by very small gradations, in a given di- 
 rection, and conceive the whole of the space occupied by atoms 
 to be very small compared with the intervening space. Also, the 
 motion being (at first) assumed to be steady, let V be the ave- 
 rage velocity, and p the average density of the sether at a certain 
 position A, and let B be another position distant by from A 
 in the direction in v/hich the atomic density increases. Then if 
 D be the proportion of the space occupied by atoms to the whole 
 of the given space, or, as it may be called, the contraction of 
 channel, the quantity of fluid which passes a unit of area at A 
 in the unit of time is Vp(l— D). But it has been proved (see 
 Phil. Mag. for June 1864, p. 458) that when a stream is inci- 
 dent on a small sphere, the mean flow in the original direction 
 is not altered by the disturbance of the lines of motion caused 
 by the reaction of the sphere; and the same is the case if there 
 be many such spheres, provided the contraction of channel they 
 produce is very small. Hefice, supposing the initial generation 
 of the stream to have been such that the same quantity of fluid 
 was made to pass each element of a given transverse section in a 
 given small interval, according to the above reasoning this con- 
 dition will be fulfilled at any subsequent epoch at each transverse 
 section, notwithstanding the presence of the atoms ; so that the 
 quantity of fluid which passes through an elementary space at 
 B will be the same in the same interval as that which passes 
 through an equal space at A. Consequently 
 
 
 Hence, passing from small differences to differentials, the rea- 
 soning being independent of the magnitude of Zz, we have 
 
 dp JD 
 pdz {\ — J))dz yaz 
 
 {a) 
 
 Now, the motion being steady, the fluid unlimited in extent, and 
 no extraneous force acting, if pQ be the value of p where V = 0, 
 
 the equation p=PqC> ‘ 2 a 2 is applicable at all points, even when the 
 effect of the reaction of the atoms is taken into account. Con- 
 
 A 3 
 
4 Prof. Challis on ike Hydrodynamical 
 
 dY 
 
 sequently, by employing this equation to eliminate from (a), 
 it will be found that 
 
 a^p _ a^Y^ 
 
 dJ) 
 
 Y^dJ) 
 
 nearly. 
 
 
 pdz~d^-Y^ {l-\))dz' 
 
 The force on the left-hand side of this equation, which is entirely 
 
 due to the gradation of atomic density expressed by takes 
 
 effect in producing the streams I have called secondary ] and the 
 reasoning shows that the intensity of such streams is indepen- 
 dent of the direction of the primary current. (See the Theory 
 of Electricity in the Philosophical Magazine for October 1860, 
 art. 18). 
 
 5. In the applications of the above views respecting the gene- 
 ration of secondary streams to the Theories of Electricity, Gal- 
 vanism, and Magnetism, I have assumed (Principles of Physics, 
 pp. 547 & 548) that the 'primary stream, the velocity of which 
 is V, might be one which relatively passes through all terrestrial 
 substances in consequence of the earth^s motion about the sun, 
 and that two other primary streams would similarly be due to 
 the eartVs rotation about its axis, and the motion of the solar 
 system in space. It was also argued that the resulting secondary 
 stream would be the sum of those which the three primaries, 
 supposed to be steady motions, would produce separately, and 
 that it would consequently be quam proximo steady. 
 
 But it has since appeared to me that this argument cannot be 
 maintained, and that the generation of the secondary stream 
 must be ascribed to the resultant at each instant of the three 
 primary velocities. Since there is reason to conclude that the 
 motion of the solar system is comparable with the earth’s orbital 
 motion, that resultant would be subject to large variations, to 
 which there is nothing corresponding in the observed intensity 
 of magnetism. Hence the movements of the earth fail to ac- 
 count for the primary velocity of the theoiy. 
 
 6. The formula (6) shows that the force which generates the 
 secondary streams varies as the square of the primary velocity V, 
 
 since the factor- - may be considered to be constant. Now, 
 
 since no observed magnetic variations are attributable to varia- 
 tions of the primary velocity such as those which the composi- 
 tion of the earth^s velocity with that of the solar system might 
 be supposed to produce, a fortiori the much smaller variations 
 of the earth^s velocity in its orbit can have no perceptible effect. 
 This inference agrees with a conclusion drawn by the Astronomer 
 Royal from discussions of the Greenwich Magnetical Observa- 
 
Theory of Magnetism. 5 
 
 tions made in 1848-1857_, and in 1858-1863. In the volume for 
 1867^ p. ccxiv^ he says, “We are justified in stating that there 
 is no certain evidence for Annual Inequality.^^ The same result 
 is arrived at with respect to the Horizontal Force. The annual 
 inequalities deduced from the theory in ^ The Principles of Phy- 
 sics^ (pp. 657-660) were obtained by employing the ^argumcnt 
 which is shown above to be untenable. 
 
 It is true, however, that General Sir E. Sabine has deduced 
 very small annual inequalities of Dip, Total Force, and Declina- 
 tion by discussions of observations taken at Kew, Toronto, Ho- 
 barton, St. Helena, and the Cape of Good Hope (see Phil. Trans, 
 for 1863, p. 307). But it is possible that these may be due to 
 a cause, distinct from the earth^s motions, which is adverted to 
 in art. 36 of the communication on magnetic force in the Philo- 
 sophical Magazine for February 1861, and will be more fully 
 treated of in the sequel of the present communication. 
 
 7. It remains, therefore, to determine by what means the 
 streams to which the theory ascribes the magnetism of a steel 
 magnet are generated, the magnet being either a straight bar or 
 in the form of a horseshoe. This question I shall endeavour to 
 answer by taking account of the theories of atomic repulsion and 
 molecular attraction proposed in the Numbers of the Philoso- 
 phical Magazine for March and 'November 1859 and February 
 1860, and in ^The Principles of Physics^ (pp. 459-464). For 
 this purpose the following general hydrodynamical theorem, 
 which, as far as I am aware, has not hitherto been recognized, 
 will be made use of : — Whenever the lines of motion in a given 
 fluid element are normals to a continuous surface, so that the 
 element is changing form by reason of the motion, the function 
 udx-^vdy -\-wdz is an exact differential. In proof of this theo- 
 rem it seems sufficient to say that the change of form of a given 
 element in consequence of convergency or divergency of the 
 lines of motion is a distinctive property of a fluid, whereby its 
 motion is separated from that of a solid, and that the^ integra- 
 bility of that differential function is the sole and necessary ana- 
 lytical expression of this property. 
 
 8. This being understood, we have, as is known, for a fluid, 
 defined by the relation between the pressure and density ex- 
 pressed hy p^a^p, the general equation 
 
 a'^Nap.logp+^ + ^'+/W=:0, 
 
 where a' is put for ku [k being a known numerical factor, the 
 theoretical determination of which I have discussed in previous 
 communications), and {d^) =-udw-\-vdy-{-wdz. No extraneous 
 force being supposed to act, this equation is applicable to all 
 
6 Prof. Challis on the Hydrodynamical 
 
 points of the fluid at all times ; and if the fluid be disturbed 
 within a limited space and be of unlimited extent, there must 
 
 be distant points at which or ^ 
 
 vanishes together with V, and the density has a constant value 
 pQ, Hence 
 
 V2 
 
 p=:p^e a'‘^dt^2a^ (^c) 
 
 This equation applies generally to unsteady motion. Now it is 
 to be observed that, whether the motion be steady or unsteady, 
 the investigation of the equation [a) is the same, because in both 
 cases the mean quantity of fluid which passes a given transverse 
 area in a given time is not sensibly altered by the reaction of one 
 
 dY 
 
 or more small spheres. Hence, eliminating from [a) by 
 means of (c), the result is 
 
 a!Hp_ WD 
 
 pdz {I -J))dz 
 
 
 (d) 
 
 which equation difi’ers from that for steady motion by having an 
 additional term on the right-hand side. If this equation be ap- 
 plied in a case in which V represents the velocity in vibratory 
 motion, the additional term will have as much positive as nega- 
 tive value, so that the mean effect of the impulses it indicates 
 will be zero. The other term is indicative of impulses towards 
 the denser part of the substance, whether V be positive or nega- 
 tive, just as when the motion is steady. 
 
 9. Recurring now to the before-mentioned theories of atomic 
 repulsion and molecular attraction, according to which the repul- 
 sion which keeps the atoms asunder is due to vibrations emana- 
 ting from individual atoms, while the counteracting attractions 
 result from composite vibrations emanating from a congeries of 
 atoms Constituting a molecule, it will appear that the maximum 
 velocity and breadth must be supposed to be much greater in the 
 latter vibrations than in the others. Also it is presumable that 
 the maximum velocity may very much exceed the velocity of the 
 earth in its orbit, or that of the solar system in space, and yet be 
 small compared with the value of a', which is about 190,000 
 miles per second. In the supposed case of the atoms being con- 
 strained to take positions such as to produce a regular gradation 
 of atomic density from one end to the other of a steel bar, at- 
 traction-vibrations, propagated in the direction from the denser 
 to the rarer end, will continually counteract the atomic repulsions 
 urging the atoms towards the rarer end, without being neutra- 
 lized by attraction-vibrations of the same order propagated in 
 
7 
 
 Theory of Magnetma , 
 
 the opposite direction. To the velocity (V) in these outstanding 
 vibrations it is reasonable to attribute the generation, in the 
 manner explained above, of the magnetic streams of the theory. 
 In fact the generation of such streams may be regarded as a re- 
 action arising from the state of constraint into which the sub- 
 stance is put by the abnormal relative positions of its atoms. 
 According to these views the magnetism of a steel bar is in no 
 sensible degree due to the earth^s motions relative to the sether, 
 but results from vibratory motions of the aether of the order of 
 those by which, in previous researches, I have endeavoured to 
 account theoretically for intrinsic molecular forces. This is an 
 important correction of the principles I have hitherto adopted 
 in the hydrodynamical theory of magnetism. 
 
 10. It is a general law of magnetic streams that they are re- 
 entering. The streams, for instance, which issue from that which 
 is assumed to be the denser end of a magnet are turned back, 
 and, after flowing in the direction of the magnetos length, enter 
 it at the other end. This general law admits of being accounted 
 for theoretically as follows. It is evident that the acceleration 
 of a mass of unlimited dimensions by the action of a finite pres- 
 sure on a finite surface is an infinitesimal quantity of the third 
 order, and that, consequently, if the mass be a fluid as nearly 
 incompressible as the sether is assumed to be, such pressure pro- 
 duces absolutely no movement of the whole mass in either a finite 
 or an infinite interval of time. Hence the displacement, by the 
 pressure, of any portion of the fluid in the direction of its action 
 must immediately give rise to the displacement of an equal por- 
 tion in the contrary direction. In other words, there can be no 
 permanent flow of the fluid across any plane perpendicular to 
 the direction of the impulses. To satisfy this condition the mo- 
 tion must take place in reentering courses or circuits.* Thus 
 the existence of complete circuits as a necessary condition of mag- 
 netic, as also of galvanic phenomena, is accounted for by the 
 hydrodynamical theory. I am not aware that any other a priori 
 explanation of this very general and prominent characteristic of 
 physical currents has been given. 
 
 11. Another general law relating to galvanic and magnetic 
 circuits may be referred to the hydrodynamical fact that the 
 currents of a fluid always take the easiest course — that is, the 
 course in which the least resistance from the inertia of the fluid 
 is to be overcome. In a bar magnet there is very little magne- 
 tism about its middle part in directions transverse to its length, 
 the magnetic action taking place chiefly about the two ends, as 
 is shown by immersing the magnet in iron-filings. The hydro- 
 dynamical explanation of these facts is as follows. About the 
 middle of the magnet there is no transverse impulse capable of 
 
8 
 
 Prof. Cliallis on the Hydrodijnmnical 
 
 overcoming in any sensible degree the inertia of the circumja- 
 cent fluid; while the impulses in the direction of the increase of 
 density have the effect of causing a stream to flow out of the parts 
 near the denser end in courses which^ by reason of the inertia of 
 the fluid beyond^ are at first made divergent^ and eventually are 
 turned completely backward : these return-currents are then op- 
 posed by the inertia of the fluid beyond the other end of the 
 magnet^ and are made to converge towards that end in such 
 manner as to fulfil by the easiest courses the necessary condition 
 of motion in circuits. According to this view the courses are 
 determined by a law of least action. (See a mathematical theory 
 of this kind of motion in the Philosophical Magazine for July 
 1869.) 
 
 12. Like considerations are applicable if the magnet has a 
 form different from that of a straight bar ; for instance^ if it has 
 the form of a horseshoe. In this case, as the two ends are 
 brought near each other, the mean course of least resistance is 
 along the axis of the magnet, and the stream passes out of one 
 end immediately into the other. Also the tendency of the 
 stream to escape from the curved part of the magnet by reason 
 of centrifugal force is still opposed by the inertia of the external 
 fluid. In order to make the insulation of the current more 
 complete, that part is usually covered with sealing-wax, this 
 substance not having the property of easily conducting setherial 
 streams. 
 
 The same hydrodynamical principles account for the flow^ of a 
 galvanic current through conducting substances of very irregular 
 forms. In consequence of the resistance to emergence arising 
 from the inertia of the surrounding sether, the currents are con- 
 fined within the boundaries of the conductors just as a stream 
 of water is confined within channels, such as rigid pipes, or ves- 
 sels of any form through which it is compelled to flow. Only^ 
 in the case of the galvanic current, the condition of a complete 
 circuit is required to be fulfilled in order that the flowing may 
 take place. 
 
 13. It might be urged as an objection to the foregoing rea- 
 soning, that if an unlimited mass of elastic fluid, such as the 
 aether is conceived to be, were to receive, within certain limits of 
 distance from a centre, impulses directed from the centre and 
 equal in all directions, the effect of these impulses would not be 
 neutralized by reaction from the inertia of the surrounding fluid. 
 This objection admits of being answered by an appeal to hydro- 
 dynamical principles which I proposed long since, and have re- 
 peatedly insisted upon, although they have not hitherto received 
 general recognition. I have pointed out that when an clastic 
 fluid receives impulses equally in all directions from a centre, 
 
9 
 
 Theory of Magnetism, 
 
 either at a given distance during a limited interval of time; or at 
 a given instant through a limited space^ the subsequent motion 
 cannot be a solitary wave of condensation or of rarefaction ; for 
 in such case the condensation would vary inversely as the square 
 of the distance from the centre; whereas the mathematical solu- 
 tion of the problem shows that it varies simply as the inverse of 
 the distance. To meet this difficulty I now adhere to the argu- 
 ments I adduced in an article in the Philosophical Magazine for 
 January 1859; and in paragraph 10 of an article in the Philoso- 
 phical Magazine for June 1862; although subsequently I adopted 
 a different view. According to those arguments the law of the 
 simple inverse of the distance holds good only in case the dis- 
 turbance gives rise both to condensation and rarefaction and the 
 resulting motion is consequently vibratory. It must therefore 
 be admitted that, whether the original impulses are vibratory or 
 not; alternations of condensation and rarefaction are actually 
 produced; and it seems evident that this effect must be attri- 
 buted to the obstacle opposed to the impulsive action by the 
 inertia of the surrounding mass of fluid. This explanation is; 
 I think; complete when it is supplemented by. the consideration 
 that; according to the hydrodynamical principles above referred 
 tO; vibratory motion; accompanied by alternate condensation and 
 rarefaction; may be shown to be proper to an elastic fluid ante- 
 cedently to any suppositions respecting particular modes of dis- 
 turbance. (See the demonstration of Prop. X. in the Philoso- 
 })hical Magazine for December 1854; and that of Prop. XL in 
 the ^Principles of Mathematics;^ pp. 201-205.) 
 
 14. Again; when the motion of an elastic fluid is supposed to 
 be in directions perpendicular to a given plane; the usual pro- 
 eess for determining the velocity and condensation at any point 
 conducts; as I long since remarked; to a contradictory result; in- 
 dicative of faulty reasoning. (See ^ Principles of Mathematics;^ 
 pp. 193-195.) As in the foregoing case of central motion; 
 the contradiction is significant of an effect of the inertia of the 
 fluid not taken into account by that process. By first proving; 
 antecedently to the consideration of arbitrary modes of disturb- 
 ance; that the fluid is susceptible; by reason of its inertia; of 
 spontaneous vibratory motions partly parallel and partly trans- 
 verse to an axiS; and thence arguing that arbitrarily impressed 
 motions must be regarded as actually composed of such primary 
 motions; I have shown that the above-mentioned contradiction 
 disappears. (See Prop. XI. above cited.) The conclusions 
 arrived at in this and the preceding paragraph respecting the 
 generation of vibratory motions by impulses that are not vibra- 
 tory; are of essential importance in accounting for a large class 
 of phenomena of light on the hypothesis of undulations, Also 
 
10 
 
 Prof. Challis on the Hydrodynamical 
 
 certain effects of motions of the air — for instance,, the sounds of 
 recognizable pitch resulting from the mutual collision of a&ial 
 streams^ or from the diversion given to such streams by encoun- 
 tering solid obstacles^ are explainable on the same principles. 
 
 15. I take occasion here to remark that the generation and 
 propagation at the surface of water of a series of circular waves 
 in forms which appear to be independent of the mode of dis- 
 turbance, or shape of the disturbing body, are, I think, referable 
 to dynamical reasons analogous to those adduced above. Also 
 the series of small waves which are seen to precede a cylindrical 
 rod when it is held vertically and moved horizontally through 
 water in which it is partly dipped, may be similarly accounted 
 for. (These ripples,^”^ together with the broader waves which 
 follow the rod under the same circumstances, are described and 
 discussed by Professor W. Thomson in an article in the Philo- 
 sophical Magazine for November 1871.) Supposing the fluid 
 to be one of perfect fluidity, the foregoing argument, which is 
 based on that supposition, leads to the conclusion that the gene- 
 ration of the ripples may be ascribed to the obstacle opposed to 
 the motion of the rod by the inert mass of fluid in front, and 
 that the waves behind are broader than those before by reason 
 of the reluctance with which the mass behind, on account of its 
 inertia, follows the rod. 
 
 16. The motions which have been thus far considered are all 
 such that each element of the fluid is at each instant changing 
 its form, and the lines of motion are normals to continuous sur- 
 faces, so that udx-\-vdy + wdz is always and everywhere an exact 
 differential. This may be true even supposing the motion to be 
 in directions perpendicular to a given plane, because, as I have 
 indicated above, the rectilinear motion may be composite, in which 
 case the change of form of the fluid elements takes place with 
 respect to each of the. component motions. There are, however, 
 cases of the motion of a fluid in which each element maintains 
 always the same form, either because the whole mass moves or 
 rotates as if it were solid, or consists of an unlimited number of 
 parts which individually so move. Such motions are distinguished 
 by the analytical circumstance that for them udx + vdy -\-wdz is 
 integrable by factor. To prove this is the object of the fol- 
 lowing argument, in which, for the sake of brevity, the fluid is 
 supposed to be incompressible. 
 
 17. For proving a proposition of this kind it is necessary to 
 employ the general equations of hydrodynamics, in order that 
 the reasoning may depend on the fundamental principles which 
 these equations express. 1 shall therefore begin by drawing 
 an inference from that which I call the equation of continuity, 
 namely : — 
 
Theory of Magnetism, 
 
 dy^r /dylr^ dyfr^ d\lr^\ 
 — " +^( ' = 
 
 dt 
 
 \dx^ dy 
 
 + 
 
 dz^ J ' 
 
 0. 
 
 11 
 
 • (<^) 
 Since 
 
 which takes account only of space^ time^ and motion. 
 
 it follows that the left-hand side 
 
 ^ dylr ^ dyjr ^ d^lr 
 
 dy 
 
 dx dy ^ dz 
 
 of this equation is the complete differential coefficient of ‘yfr with 
 
 l'^)=0, md 
 
 respect to the coordinates and the time ; so that 
 
 \dtj' 
 
 by integration = an arbitrary quantity not containing t, 
 Hence^ since yjr does not change with the time, the equation 
 — C = 0 shows that each surface of displacement maintains an 
 invariable position. Now' there are only two ways in which this 
 condition can be fulfilled when the forms of the elements are also 
 invariable ; either the motion is in straight lines perpendicular 
 to a fixed plane, or in circles about a fixed axis. 
 
 18. First, let the motion be in directions perpendicular to a 
 plane, which w'e will suppose to be the plane of xy, and let the 
 velocity along any line the coordinates of which are x and y be 
 f{Xj y) . Then we have ^^ = 0, ?; = 0, w =f{oc, y) ; so that 
 udx + vdy + wdz becomes f{xy y)dz, which is not integrable per se, 
 
 but plainly may be made integrable by the factor 777^ - Then 
 
 {d^y. 
 
 W 
 
 -d%—dz‘y and by integrating, 'v/r=^ + ^(/). 
 
 ■j\x, y) ^ - — ■ 
 
 it is showm above that is independent of if ; so that ^(^)=0. 
 Hence, since y^ is equal to a constant C, C = 0 is the general 
 equation of the surfaces of displacement, which, accordingly, 
 are planes perpendicular to the axis of Also the motion will 
 be the same at all points of a given filament of the fluid parallel 
 to the axis of z\ but, since w=:f{x, y), it may be supposed to 
 vary from one filament to another. The proposition is thus 
 proved for this case, the result having been obtained by means 
 of a factor. 
 
 19. Next let the motion be in circles about the axis of z. 
 Then, V being the velocity at the distance r from the axis, w'e 
 have at the point xyz 
 
 
 \y 
 
 z;= — , = 
 
 and udx-\-vdy + wdz= — — {ydx~-xdij)y which is not an exact 
 
 differential. It is evident that the factor ^ will make it such, 
 
 \r 
 
12 Prof. Cluillis on the Hydrodynamical 
 
 and VvC shall thus have 
 
 m-- 
 
 ocdy-^ydx 
 
 = f/. tau""^ -• 
 
 X 
 
 y . 
 
 Hence by integrating^ i|r=tan“^-^ no arbitrary function of t 
 
 X 
 
 being added^ because it has already been shown that '\/r is equal 
 to a constant C which is independent of the time. Consequently 
 
 C^tan""^-, or y^xid^nC, C being an arbitrary arc. This 
 
 general equation of the surfaces of displacement indicates that 
 the motion is in circles about the axis of z. This result having 
 been arrived at by means of a factor^ the proposition that 
 udx + vdy -{-v:dz is integrable by a factor for this kind of motion 
 is thereby demonstrated. 
 
 20. It is now to be observed that although the general equa- 
 tion {e) is satisfied by the two supposed kinds of motion, the 
 possibility of such motions is not pj'oved till the other general 
 equations have been taken into account. Yet, according to the 
 essential principles of applied calculation, the circumstance that 
 that general equation has been satisfied cannot be without sig- 
 nificance j and it is on this account necessary to inquire whether 
 and under what conditions the other general equations are satis- 
 fied by the same motions. 
 
 21. Taking, first, the motion in parallel straight lines, since 
 u — Q, = and w^f{xyy)y it is evident that the general equa- 
 tion of constancy of mass, 
 
 du dv dw __ ^ 
 dx dy \dz ^ 
 
 if) 
 
 is at once satisfied, and it only remains to take account of the 
 dynamical equations 
 
 !+(§)='>- 
 
 equations 
 
 By substituting 
 there will result 
 
 the values of u, v, and lo in these 
 dp 
 
 dz dy 
 
 :=03 
 
 whence it follows that [dp)=0j and p is constant, 
 last of the three equations is equivalent to 
 
 Since the 
 
 - + 
 dz^ 
 
 diu dw dw dw ^ 
 
 — -f - =0, 
 
 dt dx dy dz 
 
 it may be remarked that the lorcgoin 
 
 g reasoning, since 
 
 2 ^ — 0 
 
13 
 
 Theory of Magnetism. 
 
 and 2 ; = 0^ does not exclude finite values of ^ and and con- 
 sequently it is possible that w may vary from one line of motion 
 to a contiguous one. Thus it has been shown that the supposed 
 motion in parallel lines satisfies all the general equations {e), 
 (/), and [g). 
 
 22. Proceeding, now, to the case of rotatory motion about 
 the axis of it will be found, on substituting in the equation (/) 
 
 ^^ 2 / 07 
 
 for 2 /, Vj w the respective values 0, that the result is 
 
 y dY ^ cc 
 r dx r 
 
 r 
 
 dy 
 
 = 0 . 
 
 This is a partial difi’erential equation, the solution of which by 
 the usual process is V = P(r). It is thus proved that the circular 
 motion is a function of the distance from the axis and of arbi- 
 trary value. It remains to ascertain under what dynamical con- 
 ditions this kind of motion is possible. 
 
 23. Since = the equations to be used for this purpose are 
 
 dp 
 
 dx 
 
 da da du ^ . 
 
 + s+“s+'5;='''l 
 
 dp dv dv dv ^ 
 
 drj + di+"di+’’T)=°- 
 
 V‘) 
 
 By substituting in these equations the foregoing values of ii and 
 Vj it will be found that 
 
 [dp) = — dr — r—rr d . tan ^ -• 
 
 ^ ^ ^ r dt X 
 
 In order that the right-hand side of this equation may be an 
 
 dY 
 
 exact differential, we must have -^ = 0; so that V is a function 
 
 of r without containing t, and the motion is thus sliown to be 
 steady. Hence also 
 
 [dp) __ V2 ^ 
 
 dr r ^ 
 
 that is, the centrifugal force is counteracted by variation of pres- 
 sure with the distance. Since the right-hand side of this equa- 
 tion is necessarily positive, the pressure p continually increases 
 with the distance. 
 
 24. Suppose, in consequence of what has now been proved, 
 that for the case of motion in parallel straight lines we have 
 = and for the circular motion — and that the sum 
 of the two equations gives y = C. Since this composite equation 
 is of the same form as the components, it follows that, so far as 
 
14 
 
 Prof. Challis on the Hydrodynamieal 
 
 the principle of continuity is concerned^ the two motions might 
 coexist. It may also be readily shown that this coexistence is 
 compatible with the equation of constancy of mass. For if 
 
 
 and 
 
 ^2, ^2, 
 
 represent respectively the components 
 
 of the velocity for the two motions, and if + 
 
 we have, as already proved. 
 
 du^ dv^ ^ dn.\ 
 dx dy dz 
 
 : 0 , 
 
 du ^ ^ ^^<2 
 dx dy dz 
 
 = 0, 
 
 ± 0 . 
 
 and, by adding the two equations, 
 
 du ^dv ^ dw 
 dx dy dz 
 
 Having by this result proved that the component motion fulfils 
 the condition of constancy of mass, we have, further, to inquire 
 whether udx '^-vdy + ivdz can be made integrable by a factor; 
 and if so, by what factor. For this purpose we have 
 
 
 \y 
 
 
 Yx 
 
 2 /; = y), 
 
 which values of u, v, and iv evidently show that udx + vdy + wdz 
 is notan exact differential. In order that it may be made inte- 
 grable by a factor, the following equation of condition, as is 
 known, must be satisfied : 
 
 /dv dw\ /d 
 
 dw du\ 
 dx dz) 
 
 - tw [ 
 
 f du 
 \dy 
 
 dv 
 
 dx 
 
 ;)= 0 . 
 
 Putting, for shortness, /for /(.r, y), and for V its value F(r),the 
 result of substituting the values ofw, w will be found to be 
 
 ydf xdf rF{r) 
 
 fdy'^fdx-^^i\r)' 
 
 This equation makes it evident that the equation of condition 
 cannot be satisfied unless / be a function of r. Putting there- 
 fore /(r) for /, we have 
 
 = 1 + 
 
 rF-{r) 
 
 W 
 
 (i) 
 
 and by integration, 
 
 /w _ ^ 
 
 F(r) b’ 
 
 h being an arbitrary constant. This result establishes a relation 
 between /(r) and F(r), by taking account of which it will be seen 
 that 
 
 udx + vdy + wdz 
 
 =rF(r)( 
 
 xdy-^ydx ^ dz 
 
 “I 17 
 
Theory of Magnetism, 
 Hence the required factor is , and 
 
 15 
 
 
 X 
 
 Thus the differential equation of any surface of displacement is 
 
 , , xdy—ydx 
 
 ds—~b . — — > 
 
 and by integration the equation of the surface becomes 
 
 ?=c — itan' 
 
 -xl 
 
 [k) 
 
 These results give the means of defining exactly the character 
 of the motion. According to the principle of easy divisibility, 
 we may conceive the fluid to be divided into an unlimited num- 
 ber of infinitely thin cylindrical shells having the axis of 5* for 
 their common axis; and if Y(r) be the rotatory velocity of a 
 given shell at the distance r from the axis, then will /(r), or 
 rJF(r) 
 
 — ^ , be the velocity of the same shell parallel to the axis. Con- 
 sequently the motion of any given point of the shell will be in a 
 spiral. The form and position of the spiral may . be inferred 
 from the above equation of the surfaces of displacement when 
 the values of the constants b and c are given. The spiral is 
 left-handed or right-handed according to the sign of b, 
 
 25. We have still to determine the dynamical conditions of 
 this motion. From what has been proved, the components of 
 the velocity have the following values : 
 
 v = ^F(r), M;=/(r)=^^F(r); 
 
 and for determining the pressure, since v, w are independent 
 of < 2 *, we have 
 
 dp du du r. dp ^ dv ^ dv r. dp ^ dw dw ^ 
 
 *+“*+•’5=“’ 5 ;+“*+'%='’' £+”s+V,='’' 
 
 The last of these equations gives 
 
 J = ” f W + f mf'{r)=0. 
 
 Hence the equations for determining p are the same as for simple 
 
 (dT)) 
 
 rotation about the axis, and, just as for that case, = — ^ 
 
 or the centrifugal force is counteracted by the increment of the 
 pressure with the distance from the axis. 
 
 (The foregoing investigation takes account of all the cases of 
 
16 Prof. Challis^s on the Hydro dynamical 
 
 motion for which iidx -Vvdy ■\-wdz is iiitegrable by a factor. 
 When that differential is exacts the general equation (e) conducts 
 to a unique result belonging to a different class of motions^ as I 
 have elsewhere shown : Principles of Mathematics^ pp. 186-188. 
 The equation is in that case satisfied by motion^ which, 
 
 by reason of the other general equations, is restricted to motion 
 along an axis of longitudinal and transverse vibrations. Mo- 
 tions of the vibratory class are inapplicable to galvanic and mag- 
 netic phenomena.) 
 
 26. The mathematical investigation concluded above gives 
 the means of explaining in what manner a galvanic current flows 
 to any distance along a fine cylindrical wire of copper. Together 
 with the movement of a stream along the wire, both within it 
 and outside, and symmetrical with respect to its axis, there must 
 be transverse circular motion about the axis ; for otherwise, since 
 by the contraction of channel the velocity is greater, and the 
 pressure less, within the wire than in the circumjacent space, 
 the fluid would flow from all sides towards the axis, and thus a 
 stop would be put to the current. The spiral motion which re- 
 sults from the composition of the longitudinal and transverse 
 motions, being accompanied by centrifugal force due to the cir- 
 cular motion, has the effect of maintaining the current. When 
 the continuity of the wire, or of any other substance by which a 
 current is conducted, is abruptly broken, at the first instant the 
 stream issues from the terminal, and impinges on the surround- 
 ing fluid j but since it ceases at the same instant to be main- 
 tained by means of circular motion about an axis, the compound 
 motion is immediately converted into the kind for which 
 udx vdy wdz is an exact differential, and is consequently 
 turned back by having to encounter the inertia of an unlimited 
 mass of the fluid. The return course will be along the original 
 conductor if this be the only, or the readiest, channel. If, how^- 
 ever, another wire conductor should be in the neighbourhood of 
 the first, there would seem to be no reason wdiy the revulsion 
 due to the fluid^s inertia should not cause a partial return of the 
 fluid along this wire also. In fact it has been proved experi- 
 mentally by Faraday that such motion actually takes place. I 
 venture here to express the opinion that no explanation of this 
 induction of a galvanic current, other than one resting on hydro- 
 dynamical principles, is likely ever to be discovered. 
 
 27. The inquiry as to the origination of the rectilinear and 
 circular motions which combine to produce a galvanic current is 
 distinct from the preceding investigation, inasmuch as it involves, 
 together with deductions from hydrodynamical principles, con- 
 siderations respecting the chemical action between dissimilar 
 bodies, as also respecting the effect of the particular arrange- 
 
Theory of Magnetism. 17 
 
 merit of the atoms or molecules of the conducting substance. 
 On these points it will now be proper to make some remarks. 
 
 Galvanic currents may be conceived to be generated in the 
 same manner as magnetic, so far as regards the condition of a 
 gradation of atomic density, the gradation being in their case 
 produced and maintained by chemical action in such manner as 
 to exist permanently in the neighbourhood of the surfaces of 
 contact of the substances between which the action takes place. 
 The currents thus generated must, from what has already been 
 argued, fulfil the condition of flowing in a complete circuit in 
 order that they may be permanent ; for which reason it is neces- 
 sary to connect the poles of a galvanic battery by some material 
 (as copper wire) capable, in respect both to form and quality, of 
 conducting galvanic currents. But from the foregoing mathe- 
 matical argument it may be inferred that currents so conducted 
 can proceed only in spiral courses. Now, supposing the course 
 to be of this kind, it appears from experiment that the turn of 
 the spiral is always in the same direction relative to the course 
 of the current along the conductor. Hence it would seem that 
 such courses are impressed by the wire itself or other conducting 
 substance, because, as far as regards hydrodynamical conditions, 
 the course might either be dextrorsum or sinistrorsum. Suppo- 
 sing the mean direction of the current to be in the positive di- 
 rection of the ordinates Zy if the arbitrary constant h in the fore- 
 going equation (k) be positive, the value of z is greater as the 
 
 arc tan”^- is less, and therefore the spiral course is dextrorsum ; 
 
 X ^ 
 
 that is, to a person looking along the axis in the positive direc- 
 tion the turn of the spiral above the axis is from the left hand 
 to the right. But hydrodynamically it is equally possible that 
 
 y 
 
 b may be negative, in which case z is greater as tan~^ - is greater, 
 
 X 
 
 and the turn of the spiral is sinistrorsum^ 
 
 28. Assuming, for the reasons above given, that the direction 
 of the spiral course is determined by the atomic or molecular 
 constitution of the conductor, it is conceivable that this effect 
 may be attributable to a particular arrangement of the constituent 
 atoms, causing the path of least resistance, instead of being rec- 
 tilinear, to be continuously maintained in a spiral form. For 
 instance, such a modification of the path might be produced if 
 the mean retardation, due both to the reaction and the arrange- 
 ment of the atoms, operated in a direction not exactly opposed 
 to that of the stream. 
 
 29. The quantity of fluid which in a unit of time passes a 
 plane perpendicular to the axis of the wire is proportional to the 
 integral of f{r)dr between certain limits, and is therefore greater 
 
 B 
 
18 Prof. Challis on the Hydrodynamical 
 
 as f[r) is greater; and since &/(r)=rF(r), for a given value of r 
 f[r) is greater as the circular motion F{r) is greater. The latter 
 function may be considered to be the exponent of the capacity 
 of the substance for generating spiral motion, and thereby con- 
 ducting galvanic currents. This property, which exists in very 
 diflFerent degrees in diflPerent substances, seems to be possessed 
 in an eminent degree by copper, wires of this metal being gene- 
 rally used for conducting galvanic currents. I shall have occa- 
 sion to advert again to the origination of spiral motion by the 
 agency of copper. 
 
 30. The mathematical theory being incapable of determining 
 by itself whether the spiral motion is dextrorsum or sinistrorsurn, 
 it remains to inquire how far this question may be decided by 
 combining experiment with the theory. Experiment, in the 
 first place, may be taken to indicate that the direction of the 
 theoretical circular motion about a rheophore is related in a 
 constant manner to the direction of the current along the rheo- 
 phore, so that when the latter is given the other should admit 
 of being inferred. In order to determine that relation I make 
 the provisional hypothesis that currents flow, or tend to flow, 
 out 0 / the poles which experimenters positive, and into those 
 which they call negative. That pole of a horizontal magnetic 
 needle which is southward is called the positive pole. Hence, 
 by the hypothesis, magnetic streams flow out of it, and enter into 
 the pole which is northward ; whence it follows, by the hydro- 
 dynamical theory of the magnetic needle (see infra, art. 33), that 
 the horizontal components of the terrestrial streams flow south- 
 ward both in the northern and southern magnetic hemispheres, 
 and by consequence, as the dipping-needle shows, that the ver- 
 tical components flow upwards. Thus the total terrestrial streams 
 issue from the earth at and about the north magnetic pole, and 
 enter into it at and about the south magnetic pole. 
 
 31. Again, that pole of a voltaic pile or galvanic battery which 
 has the zinc terminal plate is called positive, and that which has 
 the copper terminal plate negative. Hence, by the hypothesis, 
 the galvanic current along the connecting rheophore outside the 
 nile or battery flows from the zinc to the copper. 
 
 [I have previously taken the current to be in the contrary di- 
 rection, having had difficulty in deciding between the discordant 
 indications of experimenters on this point. The direction shown 
 by arrow^s and the signs + and — in the figure in art. 682 of 
 Atkinson’s edition of ‘ Ganot ’ does not accord with the present 
 hypothesis ; neither, in fact, is it consistent with directions simi- 
 larly indicated in other cases in the same work.] 
 
 32. By referring to two experiments described in arts. 732 
 and 734 of Atkinson’s ‘ Ganot,’ it will be seen that the coexist- 
 
19 
 
 Theory of Magnetism, 
 
 ence of the horizontal terrestrial current with a vertical galvanic 
 current produces an action on the rheophore tending westward 
 or eastward according as the vertical current is ascending or de^ 
 scending. Now on the hypotheses that the earth^s current is 
 southward, and the galvanic current in the direction from the 
 zinc terminal to the copper terminal, these facts are explainable 
 only on the supposition that the spiral motion is dextrorsum, that 
 term being taken in the sense defined in art. 27. For under 
 these circumstances the magnetic current and the circular part 
 of an ascending current are opposed to each other on the east 
 side of the wire and concur on the west side, and consequently, 
 by the hydrodynamics of steady motion, the result is an excess 
 of pressure westward-, and similarly, the magnetic current and 
 the circular part of a descending galvanic current are opposed on 
 the west side of the wire and concur on the east side, and the 
 consequent action is eastward. These inferences agree with the 
 experimental facts ; and accordingly this argument is decisive as 
 to the direction of the spiral motion on the above hypotheses 
 respecting the directions of the currents. 
 
 33. Supposing, in accordance with this conclusion, that the 
 spiral motion is always dextrorsum, we may proceed next to ac- 
 count for the results of Oersted^s experiment on the same hypo- 
 theses. The horizontal component of the earth^s current flowing 
 southward, it will follow that the end into which the proper 
 streams of a magnet flow in converging courses is that which, 
 when the magnet is suspended horizontally, points northward-, 
 for it is under these conditions that the terrestrial current gives 
 that end a northward direction, as may be thus shown. Con- 
 ceive the north end to deviate from the magnetic meridian 
 through a certain angle westward, and let the earth^s current 
 and the above-mentioned converging streams be both resolved 
 perpendicularly to the axis of the needle. Then the resolved 
 parts will be in opposite directions on the west side of the north 
 end, and in the same direction on the east side, and thus the 
 north end will be driven eastward. Similarly, since the needless 
 streams are divergent from the south end, this end will be driven 
 westward, J ust the opposite effects would take place if the de- 
 viation of the north end were eastward. Accordingly the needle 
 is in a position of stable equilibrium when it points northward 
 and the marked end is in the magnetic meridian. This theory 
 accounts for the directive action of the earth^s magnetism. 
 
 34. Now let a galvanic current be caused to pass/rom south 
 to north along a conducting-wire placed over the needle in the 
 magnetic meridian, as in one case of Oersted^s experiment. 
 (See the figure in art. 627 of ‘ Ganot,^ which is the same as that 
 in art. 698 of Atkinson^s edition.) Then supposing the con- 
 
20 Prof. Challis on the Hydrodynamical 
 
 verging streams of the magnet at the north end and the circular 
 part of the galvanic current to be resolved in a horizontal plane 
 passing through the axis of the needle and perpendicularly to 
 that axis^ the two portions will concur on the east side of the 
 needle and be opposed to each other on the west side, so that 
 the needle will be urged eastward. Similarly at the south end 
 the circular part of the galvanic stream and the resolved parts of 
 the divergent magnetic streams concur in direction on the west 
 side and are in opposite directions on the east side, and that end 
 will thus be urged westward. Hence on both accounts the north 
 end deviates towards the east. But according to Ganot and 
 Atkinson the deviation in this case of the experiment is towards 
 the west. What, then, is the explanation of this disagreement ? 
 Simply that the experimental result, as given by Ganot, is incon- 
 sistent with the course of the current from the zinc to the copper, 
 as indicated by the signs -f and — and by arrows. The result 
 for the same case of the experiment, as stated by M. de la Rive 
 {Traite de Eiectricite, vol. i. p. 207), and by Mr. Airy (Trea- 
 tise on Magnetism, top of p. 210), accords with the above deduc- 
 tion from the theory in giving an eastward deviation. It is, 
 further, to be remarked that Ganot^s statements (in art. 627) of 
 the results of the experiment in the four cases, together with the 
 indicated directions of the current, are in agreement with Am- 
 pere^s well-known rule for determining the direction of the de- 
 viation of the north end of the needle, and would also agree with 
 the hypothesis of spiral motion, if that motion might be assumed 
 to be sinistrorsum when the direction of the current is from the 
 zinc to the copper. But this supposition is inadmissible, be- 
 cause experiment taken in conjunction with the theory shows 
 conclusively that either the eurrent is from the zinc to the cop- 
 per and the spiral motion dextrorsum, or from the copper to the 
 zinc and the spiral motion sinistrorsum; but neither theory 
 nor experiment appears at present to be capable of deciding 
 which of these laws is the true one. In any case the inconsistent 
 statements of experimental results I have referred to, which have 
 caused me a great deal of perplexity, require to be rectified. 
 
 All the other cases of Oersted^s experiment may be similarly 
 explained by the hydrodynamical theory on the same hypotheses. 
 
 35. One of the most remarkable phenomena relating to mag- 
 netism is the eflPect which a mass of copper in the neighbourhood 
 of a magnetic needle has upon the number and extent of its vi- 
 brations. The hydrodynamical theory, combined with a simple 
 magneto-galvanic law established experimentally by Faraday, 
 offers the following explanation of this fact and of others of 
 the same class. Faraday found that when a plate of copper 
 (I| inch wide, ^ of an inch thick, and 12 inches long) was 
 
21 
 
 Theory of Magnetism. 
 
 placed with its faces at right angles to the line of junction of the 
 poles of a powerful horseshoe magnet, and the terminals of a 
 galvanometer were put in contact with the long edges, a galvanic 
 current was developed as soon as the plate was caused to move 
 transversely to the magnetic current (Phil. Trans. 1832, p. 151). 
 Now, according to hydrodynamics, the displacement of the 
 aether by the finite spherical atoms of the copper in motion 
 would give rise to a stream of aether in the opposite direction, due 
 to the reaction of the unlimited fluid mass. Hence, conceiving 
 the plate to be held with its faces horizontal and the long edges 
 parallel to the meridian, on moving it northward or southward a 
 southward or northward current would be produced, which would 
 coexist with the original magnetic current. The galvanometer, 
 in fact, indicates that under these circumstances a galvanic cur- 
 rent is generated, which flows from one of the long edges of the 
 plate and completes the circuit by entering at the opposite edge. 
 It follows from this fact, taken in conjunction with the theory, 
 that the motions in rectangular directions of the two above- 
 mentioned currents are partially converted into dextrorsum cir- 
 cular motion parallel to the meridian, and that this effect is 
 attributable to the molecular constitution or arrangement of the 
 atoms of the copper, although theory is at present incapable of as- 
 certaining the exact modus operandi. 
 
 36. It is also found by experiment that if the magnet be 
 moved and the copper be stationary, a galvanic current is equally 
 produced. This fact appears to admit of the following explana- 
 tion. The magnetic current and its lines of motion neces- 
 sarily partake of the motion of the magnet. Hence relatively 
 to the stationary atoms of the copper the current is a composite 
 one, the horizontal component of which flows in the direction of 
 the motion of the magnet and with the same velocity. Now, by 
 hydrodynamics, this horizontal stream generates, by reason of 
 the contraction of channel by the atoms and the inertia of the 
 fluid mass, a secondary stream flowing in the same direction as 
 the stream itself. Thus, besides the vertical current, there is a 
 horizontal current flowing in the direction of the magnetos actual 
 motion, and, therefore, in the direction contrary to that of the 
 virtual motion of the copper relative to the magnet. Conse- 
 quently the conditions for generating galvanic currents are ex- 
 actly the same in this case as when the copper was moved and 
 the magnet was at rest. 
 
 37. On the principles thus established we may proceed to 
 explain the whole of the class of phenomena which depend on 
 the relative motion between a magnet and a mass of copper. 
 But for this purpose it will be necessary to ascertain previously 
 the direction of the flowing of the current in Faraday^s elemeii- 
 
22 
 
 Prof. Challis on the Hydro dynamical 
 
 tary experiment above described. This point is left in ambi- 
 guity by the experiments, as not admitting probably of being 
 decided except by the combination of experiment with a true 
 theory. The present theory furnishes for deciding it the fol- 
 lowing considerations. 
 
 38. In the description of the fundamental experiment Faraday 
 states (referring to fig. 16 in plate iii. p. 131, of thfe Phil. 
 Trans, for 1832) that when the galvanometer-needle was de- 
 flected, its north or marked end passed eastward, indicating that 
 the wire A received negative and the wire B positive electri- 
 city.^^ As the wire A belongs to the terminal applied to the 
 eastern edge of the plate and the wire B to that applied to the 
 western edge (the two wires being in fact a single wire consti- 
 tuting the circuit), I was led by the above statement to suppose 
 that the current proceeded out of the plate on the west side and 
 entered it on the east side (Principles of Physics, p. 636). 
 But on the assumption that the spiral motion of the galvanic 
 current has been correctly determined by the foregoing argu- 
 ment to be dextrorsum, the truth of that supposition can be 
 tested as follows by the observed direction of the displacement 
 of the galvanometer-needle. The wire proceeding from the 
 western edge was made to pass beneath the needle in the direc- 
 tion from south to north. Hence the circular motion about the 
 rheophore would conspire on the west side of the north end of 
 the needle with the entering magnetic streams resolved perpen- 
 dicularly to the axis of the needle, and be opposed to them on 
 the east side. Thus there would be an excess of pressure on the 
 east side, and the deflection would be westward. But Faraday 
 says that the marked end passed eastward.^^ It must there- 
 fore be concluded, in order to reconcile the theory with experi- 
 ment, that the direction of the current is the reverse of that as- 
 sumed, proceeding out of the east side and entering at the west 
 side. It is true that the fact and theory would agree if the 
 spiral motion might be assumed to be sinistrorsum ; but the 
 previous argument is opposed to this supposition. There re- 
 mains, therefore, only the inference that the current actually 
 flows along the rheophore from the east to the west side of the 
 plate. Moreover, as will presently be shown, the phenomena 
 of the mutual action between a magnet and a mass of copper 
 relatively in motion are explainable by the theory only in case 
 the current has this direction, which will at least give evidence 
 of the consistency of the theory with itself. 
 
 39. Let us take the case of a circular plate of copper placed 
 under a magnetic needle and caused to rotate about an axis co- 
 incident with that of the magnet, and let the direction of rota- 
 tion be from west through north to east, or that of the move- 
 
23 
 
 Theory of Magnetism. 
 
 merit of the hands of a watch. Then conceiving at first the 
 copper to be at rest, let the magnetic streams which pass through 
 it, as well those of the magnet as the terrestrial streams, be re- 
 solved into components parallel and perpendicular to the faces 
 of the plate. It is clear that the vertical component of the 
 terrestrial magnetism can have no rotational effect ; and the 
 same is the case with respect to the horizontal components of 
 both kinds, inasmuch as these only give rise to vertical galvanic 
 currents, the effects of which, as regards rotation, neutralize 
 each other. We have, therefore, only to consider the vertical 
 components of the magnetos streams. Now as these streams 
 enter the needle at the north end and the needle is above the 
 copper, the vertical components will flow upwards at that end. 
 Again, the copper being now supposed to rotate in the direction 
 above stated, since the motion of the parts under the north end 
 is eastward, the secondary or induced stream will flow westward. 
 Comparing, therefore, these circumstances with the case of the 
 elementary experiment in which, when the magnetic current 
 flowed upwards and the induced current southward, the resulting 
 galvanic current flowed eastward (see art. 38), it will appear, by 
 imagining the horizontal directions to be turned through 90°, that 
 in the actual case the galvanic current flows southward. With re- 
 pect to what takes place at the south end, we have to consider that 
 the vertical components of the magnetos issuing streams pass 
 through the copper downwards, and that the direction of the in- 
 duced streams is changed from westward to eastward. Each of 
 these changes (as was shown by Faraday ^s experiments) causes 
 a reversion of the direction of the galvanic current, and conse- 
 quently the direction is southward at the south end as well as at 
 the north end. 
 
 After this determination we have only to inquire what effect 
 a southward galvanic current has on a magnetic needle above it. 
 The spiral motion being dextrorsum, the circular motion will 
 conspire with the entering streams at the north end on the east 
 side, and be opposed to them on the west side. Hence the north 
 end will be driven eastward, that is, in the direction of the motion 
 of the disk. So the circular motion will be opposed to the issu- 
 ing streams at the south end on the east side and conspire with 
 them on the west side, and the south end will consequently be 
 urged westward — that is, again in the direction of the motion of 
 the disk. These results agree with the known facts of the ex- 
 periment ; and this agreement, it is to be noticed, depends upon 
 the foregoing inference respecting the direction of the induced 
 galvanic current. 
 
 40. All the other instances of the mutual action between 
 a magnet and copper relativelj in motion may be referred to the 
 
24 
 
 Prof. Chaliis on the Hydrodynamical 
 
 same principles as those adopted and exemplified in the foregoing 
 case ; and their phenomena may be similarly accounted for. 
 
 41. To complete this review of the Hydrodynamical Theory of 
 Magnetism, it remains to discuss and correct the theoretical ex- 
 planations I have proposed respeeting certain phenomenaof Terres- 
 trial Magnetism. At the endof the article*^ On Atmospheric Tides^^ 
 in the Number of the Philosophical Magazine for January 1872, 
 I have intimated my abandonment of the hypothesis that the 
 lunar* diurnal variation of terrestrial magnetism is attributable 
 to gradations of density of the atmosphere produced by the moon^s 
 gravitational attraction. Having in that article succeeded in sol- 
 ving with sufficient generality the problem of the disturbance of 
 the atmosphere by the moon, 1 found that neither the law nor the 
 amount of the gradation of density due to the moon^s attraction 
 could account for the facts of the lunar-diurnal variation. Being 
 thus compelled to seek for another explanation, 1 reconsidered the 
 views advanced in my first essay towards a theory of magnetism 
 published in the Numbers of the Philosophical Magazine for 
 January and Pebruary 1861, and have come to the conclusion that 
 the gyratory motions of the sether there attributed to the rotations 
 of the bodies of the sun, moon, and planets about axes are strictly 
 deduced from the physical principles on which the theory is 
 founded, and must accordingly be regarded as necessary conse- 
 quences of the hypotheses of the theory. In fact 1 do not 
 think that I could say on this part of the subject any thing dif- 
 ferent from what is said in arts. 27-30 contained in the February 
 Number, which, after giving this reference, 1 consider it unneces- 
 cessary to reproduce here. 
 
 42. With respect to all magnetism which has a cosmical oxx^m, 
 the view that 1 now take is that it is due to gyrations of the 
 sether produced by the impulses it receives from the motions of 
 the constituent atoms of the bodies of the solar system. The 
 gyrations may either be immediately generated by the rotations 
 of the bodies about their axes, or indirectly result from disturb- 
 ances of the sether caused by their motions of translation. Ac- 
 cording to hydrodynamics, the motion of translation of a mass 
 constituted atomically (as stated in art. 1) will continually im- 
 press on the sether motions whose mean direction is at the first 
 instant directly contrary to that of the motion of the mass. 
 This impressed motion will be subsequently converted into cir- 
 culating or gyratory motion, because, according to the argument 
 in art. 10, there can be no permanent transfer of any portion of 
 the setherial fluid across any fixed unlimited plane. Such cir- 
 culating motions will necessarily partake of the motions of trans- 
 lation of the bodies which generate them, so as to have always 
 the same geometrical relations to these bodies, provided their 
 motions be uniform. 
 
25 
 
 Theory of Magnetism. 
 
 43. Accordingly the moon, since it completes relatively to the 
 sether at rest a revolution about its axis in a month, will gene- 
 rate gyrations, the effect of which might possibly extend to the 
 earth, and be perceived as a very small variation of terrestrial 
 magnetism. But as these gyrations are in the direction of the 
 moon^s revolution about her axis, it will be found on trial that 
 the disturbances of the magnetic needle which they produce are 
 not in accordance with the observed law of the lunar-diurnal 
 variation of magnetism. When, however, the effect of the dis- 
 turbance of the sether by the moon^s orbital motion is considered, 
 it will be seen that as the impressed motion is tangential to the 
 orbit and contrary in direction to that of the moon^s motion, the 
 generated circulating motion that reaches the earth will be/rowe 
 the 7'ight hand to the left of a spectator on the earth looking 
 towards the moon, and therefore in the direction opposite to that 
 of gyrations resulting from the moon^s rotation about her axis. 
 Also their effect, it may be presumed, would be much greater 
 than that of these gyrations. Now the circulating streams pro- 
 duced as above stated by the moon^s orbital motion would dis- 
 turb the magnetic needle in a manner conformable with the law 
 of the kmar-diurnal variation ascertained by observation. For 
 when the magnet is on the meridian and under the moon, these 
 streams, flowing eastward^ would oppose the needle’s entering 
 streams at its north end on t\i^east side, and the issuing streams 
 at its south end on the west side, so that by both actions the 
 western declination would be increased. The variation of the 
 declination in any other position of the needle relative to the 
 moon might on these principles be readily investigated ; and it 
 is easy to see that according to this theory there would be two 
 maxima and two minima of declination in the course of twenty- 
 four lunar hours. These deductions agree with the results of 
 observation given in p. (ccxliii) of the Greenwich Observations of 
 1867. 
 
 44. The solar-diumal variations of magnetism follow a differ- 
 ent law, there being but one principal maximum in the course of 
 twenty-four hours. This fact is accounted for by the circum- 
 stance that the sun’s heat produces, crjeteris paribus, a gradual 
 diminution of the density of the atmosphere in all directions 
 converging to the position of greatest heat, it being assumed, 
 conformably with the argument in article 4, that such grada- 
 tion of density, in consequence of its extending over a large 
 space, is capable of generating magnetic streams of sensible mag- 
 nitude. So far as this cause operates, we might expect that 
 there would be magnetic effects due to the rapid changes of tem- 
 perature and of atmospheric density in the day-time of a more 
 decided character than any due to the slower changes in the 
 
 c 
 
26 
 
 Prof. Challis on the Hydrodynamical 
 
 night-time ; and this anticipation is confirmed by observation. 
 Bat besides the magnetic variations due to changes of the tem- 
 perature and density of the atmosphere^ we may suppose that 
 effects are produced by solar gyrations analogous to those attri- 
 buted above to lunar gyrations. This cause might modify the 
 epoch of day-maximum (which, as observation shows, does not 
 occur at the hottest time of the day), and might also account for 
 what has been called the nocturnal episode. As to the reality of 
 the solar gyrations, I consider that we have ocular evidence in 
 the phenomenon of the zodiacal light ; for, according to the ar- 
 gument maintained in arts. 13 and 14, the rays which render 
 the zodiacal light visible might originate in a collision between 
 the gyratory motion and vibrations of different orders propagated 
 from the sun, without its being necessary to suppose that matter 
 other than the aether exists at points from which the rays pro- 
 ceed. It has been established by observation that the zodiacal 
 light extends beyond the radius of the eartlPs orbit (see an 
 article in the Phil. Mag. for February 1863). 
 
 45. The foregoing considerations enable me now to state the 
 cause alluded to in art. 6, to which I conceive that minute an- 
 nual inequalities of Dip, Total Force, and Declination might be 
 attributed. Let it be admitted that the solar gyrations produce 
 a sensible magnetic effect at the earth^s distance from the sun. 
 Then on the supposition that the gyrations are in circles sym- 
 metrically disposed relatively to the plane of the ecliptic (which 
 must approximately be the case), it will follow that the magnetic 
 effect varies with the earth^s varying distance from the sun, being 
 less as the distance is greater. Inequalities thus produced wmuld 
 be the same for the northern as for the southern hemisphere, 
 agreeing in that respect wdth the small annual inequalities which, 
 as stated in art. 6, General Sir E. Sabine has deduced from ob- 
 servation. 
 
 46. On the hydrodynamical principles which have been ap- 
 plied in the foregoing explanations, 1 make the following sug- 
 gestion relative to the cause of the secular variations of ter- 
 restrial magnetism. From what has been already argued (art. 
 42), the motions immediately impressed on the aether by the 
 eartlPs constituent atoms in consequence of its rotatory and or- 
 bital motions, result in circulating motions which, if the earth 
 moved uniformly in a rectilinear course, would be steady motions 
 having always the same geometrical relations to the position of the 
 earth^s centre. But the earth moves with varying velocity in a 
 curved path. Assuming, therefore, that the system of circulating 
 motions always tends to be steady and to maintain a rectilinear 
 course, it is conceivable that tlure may be continual minute shift- 
 ings of the axis of the system in directioiis transverse to the 
 
37 
 
 Theory of Magnetism, 
 
 orbital motion and towards the tangential direction_, analogous 
 in some degree to the shifting of the earth^s axis by precession 
 and nutation. Such displacements of the axis of the setherial 
 movements would cause it to take positions relative to the earth 
 continually more westerly, 
 
 47. I take this opportunity for rectifying an opinion respect- 
 ing ear th‘ currents which I have expressed in a note in p. 653 of 
 my work on the ^Princi[)les of Physics.^ Since the note was 
 written T have learnt from the Greenwich Observations that 
 magnets and the galvanometers employed as indicators of earth- 
 currents are simultaneously affected only by large magnetic dis- 
 turbanceSj and that generally no correspondences exist between 
 small magnetic variations and the indications of the galvanome- 
 ters. It does not appear, therefore, that earth-currents and 
 terrestrial magnetism have any special relation to each other, 
 the effect of the large magnetic disturbances on the earth-current 
 being only an instance of the ordinary mutual action between a 
 galvanic current and a magnetic current. 
 
 Having completed this review of the hydrodynamical theory 
 of magnetism, I think I may say that it is now supported by 
 arguments which should command the attention of physicists. 
 Considering the number and variety of the explanations of phe- 
 nomena it gives, which in fact might have been much ex- 
 tended, I do not see bow the inference that the facts of mag- 
 netism are. referable to hydrodynamical laws can be resisted. 
 
 Cambridge, April 19, 1872. 
 

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