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Mixtures with anilin as liquid 1.

O,H,NO2
I. •32 (1.081)
CH CO,C,H........ I. .39 (977)

8.6 4.64
7.2 3.52

1.0 1.0
.9 .9

The densities enclosed in brackets were not observed, but calculated from the densities of other mixtures or of the component liquids on the supposition of no shrinkage in mixing.

Of the 28 pairs of liquids examined, 24 show a value of the ratio 1A2/(1A12A2) lying between 9 and 1.1, while of the remaining 4, 2 (ethyl oxide and anilin and ethyl oxide and dimethylanilin) show what appears to be a constant value of the ratio having in one case the value ·7 and in the other 8; in the other two cases, namely, those of ethyl oxide and dimethyl anilin and carbon disulphide and methyl iodide, the theoretical equation (6) does not apply as the value of 1A2/(1A12A2) is not constant for different values of pi Some mixtures of ethyl oxide and CS, gave a precipitate disappearing only with shaking; so that perhaps they ought not to rank as genuine mixtures of two liquids. We will not enquire at present more closely into these exceptional cases, nor discuss the great class of exceptions formed by watery solutions and mixtures. In the "Laws of Molecular Force" the method of treating watery solutions was pointed out, and the method will be improved and developed in another paper devoted to the surface-tensions of watery solutions alone. As to watery mixtures it will suffice to instance as the most extreme case of exceptional capillary behaviour the well-known one of mixtures of water and amyl alcohol, water with a surface-tension of 74 when mixed with only 2.5 per cent. of amyl alcohol having a surface-tension of 3.7 has its surface-tension reduced to 2.8, which is even lower than that of the small amount of added alcohol. It is clear that cases of this sort are complicated with quite another class of phenomena from those we are discussing in connexion with normal liquid mixtures, and that we have a right to set them apart for separate study.

The result A=(A12A2), which is the outcome of this investigation on the attraction of unlike molecules, has an important bearing on the interpretation of the data as to the attraction of like molecules contained in the "Laws of Molecular Force;" for evidently the expression Am2 for the attraction of two like molecules must be regarded as the product of two parameters Am characteristic of each molecule. The investigation of the attraction of like molecules from this point of view will be taken up in my next paper, "Further Studies on Molecular Force."

Melbourne, January 1894.

XXI. On Electromagnetic Induction in Plane, Cylindrical, and Spherical Current-Sheets, and its Representation by Moving Trails of Images. By G. H. BRYAN, M.A.*

1.

PART I.-GENERAL EQUATIONS-PLANE SHEETS.

IT

Introduction.

T is well known that if a very thin, indefinitely extended plane sheet of metal of finite conductivity is placed in a varying magnetic field due to the presence of moving magnetic poles in its neighbourhood, induction-currents are set up in the sheet and the field of force due to these currents may be represented by a moving trail of images.

In the present paper the surface-conditions which hold at the surface of a plane, cylindrical, spherical, or other conducting sheet of uniform small thickness are deduced directly from the fundamental laws of electromagnetic induction. By working directly with the scalar magnetic potential, and avoiding the introduction of the vector potential and the quantity which Maxwell denotes by P, the investigations are much simplified. Moreover, in at least one comparatively recently published paper the boundary conditions satisfied by the vector potential at the surface of separation of two different media have been erroneously stated, and for this reason it is advantageous to employ a method which obviates the difficulty.

The results will be employed to show how the field due to the presence of a magnetic pole of varying intensity in the neighbourhood of a plane, cylindrical, or spherical currentsheet may be represented by means of a moving trail of images. By the principle of superposition the effect of any number of poles of varying intensity can be deduced, and the corresponding expressions for the field can thus be obtained when the variable inducing system of magnets is of the most general possible character, as, for example, one (or more) magnetic poles moving about in any manner whatever.

In this way a synthetic solution of the problem of induction in current sheets is obtained. The phenomena of induction in spherical and other current sheets have been treated at considerable length from an analytical point of view by Larmor, Lamb, and Niven, and the last-named writer has made some attempt, in the case of a sphere, to interpret the results by means of images; but the present investigation

* Communicated by the Physical Society: read May 11, 1894.

seemed desirable for many reasons, and it is hoped that it will overcome some of the difficulties, and elucidate some of the obscurities which present themselves in most treatments of this interesting application of the principles of electromagnetism.

Fundamental Assumptions.

2. The laws of electromagnetic induction assert that in bodies at rest

I. The total current across any enclosed portion of a surface which always contains the same particles is equal to 1/4π of the line-integral of magnetic force round the curve bounding the surface.

II. The rate of decrease of the surface-integral of magnetic induction across any enclosed surface which always contains the same particles is equal to the line-integral of electromotive force round the curve bounding the surface.

In applying these laws to an infinite dielectric separated into two portions by a thin conducting sheet, it is usually assumed that the disturbance produced by the inducing system is not a very rapidly alternating one, so that displacement currents in the dielectric have no appreciable magnetic effect*. With this assumption, the magnetic force in the dielectric will always be derivable from a potential which will only depend on the inducing system and the currents in the sheet. In other words, the state of the dielectric will be given by an "equilibrium theory."

It is also assumed that the induction-currents at any point distribute themselves uniformly throughout the thickness of the sheet. This requires that the disturbance shall not be a very rapidly alternating one, and also that the thickness of the sheet shall be very small compared with the other linear dimensions of the system (such as the distances of the moving poles, the radius of the sheet if spherical, &c.).

Surface Conditions at a Plane-Current Sheet.

3. Let the plane of the sheet be taken as the plane of x, y, let the thickness of the sheet be c, and specific conductivity C. Let 1, 2 be the magnetic potentials on the negative and positive side of the sheet respectively, the current function in the sheet at any point. Apply Law I. to the circuit formed by going along the positive side of the sheet from the origin to any point and returning along the negative side from that

* Watson and Burbury, 'Mathematical Theory of Electricity and Magnetism,' ii, § 405.

point to the origin. Since the thickness of the sheet is small, the terms contributed to the line-integral of the magnetic force by the passage of the circuit from one side of the sheet to the other may be neglected and we obtain

N2−N1=4π&+constant.

*

(1)

Now apply Law II. to any circuit s drawn in the plane of the sheet. Let S,, S2 be two surfaces drawn in the dielectric. infinitely near to the positive and negative faces of the sheet respectively and both bounded by the curve s, and let P, Q be the components of electromotive force at any point. Then, assuming the magnetic permeability of the dielectric to be unity, we have

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S2

and since corresponding elements of the near surfaces S,, S2 are equal, it is evident that

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at the surface, so that the normal component of magnetic induction is continuous, as it should be †.

Again, the equations of conduction give

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where dn is the element of the outward-drawn normal to s and the surface-integral taken over the area S of the sheet bounded by the curve s.

Maxwell, 'Electricity and Magnetism,' ii. § 653.

The magnetic permeability of the sheet itself will not affect the conditions of the problem unless it is required to proceed to a higher order of approximation by taking into account first powers of the thickness (or unless the sheet is formed of soft iron whose magnetic permeability may be large).

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