It is to be observed that this effective current satisfies the condition of incompressible flow, which by definition (or rather by the æthereal constitution) is necessarily satisfied by the total current (u, v, w) of the previous memoirs; for the additional terms which represent the magnetism clearly satisfy the stream relation.+ The remainder of the scheme of electrodynamic relations is established as in the previous memoirs. Thus (F, G, H) now representing simply f(u1, v1, w1)r−1dt, which satisfies the stream relation dF/dx+dG/dy+ dH/dz = 0 because (u1, vi, w1) is a stream vector, we deduce an electric force (P, Q, R) acting on the electrons, where Pcq-br-dF/dt-dvdx, also an æthereal force (P', Q', R') straining the æther, where the function being determined in each problem so as to avoid æthereal compression. Across an abrupt transition, F, G, H and the normal component of (u, v1, w1) must be continuous, thus making up the four necessary and sufficient interfacial conditions. The gradients of F, G, H are, however, not continuous when there is magnetisation or dielectric convection, on account of the effective interfacial current sheets before mentioned. The exact value of the mechanical force (X, Y, Z) per unit volume, comes out as * It is proposed to call a flow-vector which obeys this condition a stream, the more general term flow or flux including cases like the variable stage of the flow of heat in which the condition of absence of convergence is not satisfied. The two main classes of physical vectors may be called fluxes and gradients, the latter name including such entities as forces and being especially appropriate when the force is the gradient of a potential. Lord Kelvin's term circuital flux has previously been used to denote a stream vector; but it is perhaps better to extend it to a general vector which is directed along a system of complete circuits. † The (u, v, w) of § 13, however, included a part arising from convection of electric polarisation. Notice that when this is transferred to the magnetism, as here, we have u = u'+df/dt+df' dt + pp: thus when there is no conduction and p is therefore wholly convected so that 8p/dt is null, the stream character of the total current simply requires d(f+f'), dx+d(g+g') /dy + d(h+h')/ dz = p, so that the formulation is now easier and more natural. In these formula, with the exception of the one for (u1, v1, w1) above, (A, B, C) includes the quasi-magnetism arising from electric convection, while (u, v, w) is the total electric current that remains after all magnetic effect of whatever type has been omitted. It is to be noted that the final terms in X involve in strictness the æthereal force, instead of the electric force as in § 39. It follows from the formula for (P, Q, R) that hence Faraday's circuital relation holds good provided the velocity (p, q, r) of the matter is uniform in direction and magnitude. Again, since (F, G, H) is a stream vector, where (u, v, w) represents the total current of Maxwell, and (A, B, C) the whole of the magnetism and the quasi-magnetism of convection: hence dy dẞ = 4TU, so that Ampère's circuital relation holds, with the above definition of (a, B, 7), under all circumstances. But in circumstances of electric convection these two circuital relations would not usually by themselves form the basis of a complete scheme of equations, as they do when the material medium is at rest. To complete the scheme, the above dynamical equations must be supplemented by the observational relations connecting the conduction current with the electric force, the electric polarisation with the electric force, and the magnetism with the magnetic force. In the simplest case of isotropy these relations are of types u' P, f' (K-1)/4c2P, A = x+(rg'-qh'). = = It is to be observed that the physical constants which enter into the expression of these relations will presumably be altered by motion through the æther of the material system to which they belong: but because there is nothing unilateral in the system, a reversal of this motion should not change the constants, therefore their alteration must depend on the square of the ratio of the velocity of the system to that of radiation, and would only enter in a second approximation. The various problems relating to electric convection and optical aberration worked out in §§ 14-16, pp. 225-229, will be found to fit into this scheme. I take the opportunity of correcting an erratum in p. 226, lines 16, 17, which should read ¥1 = }(1+K−1)wcr2 + Ar2(cos 20—3) + A′ 2 Br(cos 0-4)+B'r1, with of course different values of the constants. 2. In a material dielectric the bodily mechanical forcive is derived from a potential (K-1)F2/87, and there is also a normal inward traction KF2/87 where it abuts on conductors. For the thin dielectric shell of a condenser this forcive could be balanced by a hydrostatic pressure (K-1)F/8′′ together with a Maxwell stress consisting of a pressure F2/87 along the lines of force and an equal tension at right angles to them: in fact this reacting system gives the correct traction over the faces of the sheet and the correct forcive throughout its substance. If the sheet has an open edge the tractions on that edge are however not here attended to; when the sheet is thin these are of small amount, and their effect is usually local, as otherwise the nature of the edge would be an important element. Moreover, in the most important applications of the formula the edge is of small extent, so that they form a local statically balanced system. The stress above specified will thus represent the material elastic reaction, provided the strains in the different elements of volume, which correspond to it, can fit together without breach of continuity of the solid material. This condition will be secured if the shell is of uniform thickness so that F is constant all over it in that case, therefore, the elastic reaction in the material will make up a pressure KF2/87 along the lines of force and a pressure (K-2) F2/87 in all directions at right angles to them, which is the result obtained for solids in § 76. If, however, the coatings of the condenser are not supported by the dielectric shell, the elastic reaction in the shell will be simply a pressure (K-1)F/87 uniform in all directions. This is what actually occurs in the case of a fluid dielectric, where such support is not mechanically possible. * It appeared from § 79 that in glass there is actually an increase of volume under electric excitation, while the mechanical forces would produce a diminution: and the same is true for most dielectric liquids, the fatty oils being exceptions, though by a confusion. between action and reaction the result was there stated as the opposite. It thus appears that in general an intrinsic expansion, in addition to the effects of the mechanical force, accompanies electric * In the cognate case of magnetisation of ferrous sulphate solution, Hurmuzescu finds a contraction of volume. VOL. XIII. 2E excitation of material dielectrics. This circumstance will perhaps recall to mind Osborne Reynolds' theory of the dilatancy of granular media, which explains that the discrete elements of such media tend to settle down under the mutual influences of their neighbours so as to occupy the smallest volume, and therefore any disturbing cause has a tendency to increase the volume. In § 80, on the influence of electric polarisation on ripple velocity, the result stated for dielectrics should be doubled. It is to be remarked that a horizontal dielectric liquid surface becomes unstable in a uniform vertical electric field when the square of the total continuous vertical electric displacement exceeds the moderate value K1K2(K2 + K1) '{(p:—p;)gT}' electrostatic units, T being the capillary tension. For a conducting liquid instability ensues when the square of the surface electric density exceeds {p—p1)gT}* elec 2 2 27(K2-K1) 2π trostatic units. In exciting a dielectric liquid by the approach of an electrified rod it must often have been noticed that when the rod is brought too near, the liquid spurts out vigorously in extremely fine filaments or jets: the fineness of the filaments may be explained, in part at any rate, after Lord Rayleigh ('Phil. Mag.,' 1882, "Theory of Sound," § 364), without assuming an escape of electricity into the liquid, as arising from the circumstance that it is only narrow crispations of the surface, and not extensive deformations, that become unstable. The opportunity is taken to correct other errata in the memoir, 'Phil. Trans.,' A, 1897, as follows: page 252, line 15, read 8′′ for 4. : June 9, 1898. The Annual Meeting for the Election of Fellows was held this day. The LORD LISTER, F.R.C.S., D.C.L.., President, in the Chair. The Statutes relating to the election of Fellows having been read, Professor Bonney and Mr. R. H. Scott were, with the consent of the Society, nominated Scrutators to assist the Secretaries in the examination of the balloting lists. The votes of the Fellows present were collected, and the following Candidates were declared duly elected into the Society : : Preston, Professor Thomas, M.A. Scott, Alexander, M.A. F.I.C. Taylor, Henry Martyn. Thanks were given to the Scrutators. June 9, 1898. The LORD LISTER, F.R.C.S., D.C.L., President, in the Chair. Professor W. Haswell (elected, 1897) and Professor Amagat (elected a Foreign Member, 1897) were admitted into the Society. The following Papers were read : : I. "On a New Constituent of Atmospheric Air." By Professor II. "On the Position of Argon, Helium, and Krypton in the IV. "Experiments on Aneroid Barometers at Kew Observatory and their Discussion." By Dr. C. CHREE, F.R.S. VOL. LXIII. 2 F |