. The London, Edinburgh and Dublin philosophical magazine and journal of science. above dimensions have to be deduced by meansof a connected system of equations, it becomes necessary tomake a suitable choice of axes of reference. Let X be theaxis of the electric displacement, and Y that of the magneticdisplacement at a point in the medium. For an isotropicmedium (the only case we have at present to consider) these Phil. Mag. S. 5. Vol. 34, No. 208. Sept. 1892. T 254 Mr. W. Williams on the Relation of Dimensions are mutually at right angles, and Z is at right angles to both,being the intersecti
. The London, Edinburgh and Dublin philosophical magazine and journal of science. above dimensions have to be deduced by meansof a connected system of equations, it becomes necessary tomake a suitable choice of axes of reference. Let X be theaxis of the electric displacement, and Y that of the magneticdisplacement at a point in the medium. For an isotropicmedium (the only case we have at present to consider) these Phil. Mag. S. 5. Vol. 34, No. 208. Sept. 1892. T 254 Mr. W. Williams on the Relation of Dimensions are mutually at right angles, and Z is at right angles to both,being the intersection of the electric and magnetic equi-potential surfaces. Let this relation between the directionsof the axes of reference and the displacements hold for everypoint of the medium, so that the axes constitute an instanta-neous system at every point. In passing therefore frompoint to point in the medium, and for different epochs at thesame point, the axes and the displacements preserve theirrelative directions, while their absolute direction in space ingeneral alters. Fi<r. Let AO be a closed electric circuit, and BO a correspond-ing closed magnetic circuit, both being circles in planes atright angles to each other. Taking instantaneous axes asabove, every element of the circuit AO is &x, and of thecircuit BO is dy, while an element of the intersection of theplanes of the circuits is ~dz. The length of the circuit AO is2d#, and of the circuit BO, 2d?/, while the surfaces of thecircuits are ultimately 2(B^B£)> and S(dydz). We havetherefore :— 1. Circulation H=2(Hd#) = C. 2. Circuitation E = 2(EBa?)=E. 3. Surface-integral of D = 2(D(ty32) = £. 4. Surface-integral of B = 2 (Hftadz) = m. To express these dimensionally, we have to neglect thesummation 2, and substitute for d#, ~fty, ~ftz respectively X,Y, Z. The relations then become : — 1. D =JcE. 2. C =eT~\ 3. e =D(YZ). 4. C =DT-X. 5. E =E(X)=Cw = mT-1 = B(XZ)T-1, 6. B =fiB. of Plitjskal Quantities to Directions in Sp
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