Ebenezer Cunningham

“ Thus if the single electron at rest has a spherical configuration, and there are no other than electromagnetic forces, we should expect it in motion to have a spherical configuration when measured by the variables x' y' z' which means that as measured by the variables x, y, z it will have the spheroidal shape as suggested by Lorentz.
The electron as conceived by Abraham, on the other hand, is spherical always as regards the variables x, y, z, and a prolate spheroid as regards x' y' z' the ratio of the axes being . ”

“ Larmor's scheme of the æther does not apply if that scheme is stated accurately as follows:— "The propagation of electromagnetic effects through space, relative to a given frame of reference, may be illustrated by the propagation of disturbances in a rotationally elastic medium, it being possible for a given frame of reference to construct such a medium, in which the rotational displacement at any point is proportional to the electric force at that point, and the velocity relative to the frame of a point of the medium is proportional to the magnetic force." ”

“ The question arises whether the theorem of relativity also holds for the types of motion of an electromagnetic system derived from one another by such a transformation. The following investigation shews that this is so, and develops a scheme of correlation between the physical quantities in the system and its transformation. The constitutive equations take exactly the same form as in the first part of the paper.
The motions that arise in these transformations are naturally a good deal more complicated than in that of Einstein. ”

“ Suppose axes of ξ η ζ to be B's system of coordinates moving with him with velocity v relative to A's axes of x y z, and coinciding with them at t=0.
Associated with a given point at a given time as marked by the values (x, y, z, t) will be unique values of (ξ, η, ζ, τ) , τ being B's measure of the interval elapsed from the time of his coincidence with A, and conversely. There must therefore be a linear transformation from the variables (x, y, z, t) to (ξ, η, ζ, τ) . ”

“ We may, however, prove that no conformal transformation in this space exists which will transform every point at rest in the space (x, y, z) into a point moving with angular velocity about a fixed line in the (XYZ) space. Thus, as in material dynamics, the equations of electrodynamics will not be preserved in the same form for a set of axes rotating uniformly relatively to the space in which they are satisfied by actual phenomena.
I.
Some considerations concerning the Lorentz-Einstein Transformation. ”

“ II.
4. The group of conformal transformations in four dimensions referred to in the introduction can be built up of transformations by inversion in the hyperspheres of the space.
To establish the new theorem of relativity it is sufficient to consider a single member of the group, i.e., a single inversion. Let the centre of inversion be taken as origin.
The analysis is best carried out in spherical polar coordinates, so that the geometrical correlation of the two systems is
the angular coordinates remaining unaltered. ”

“ The discussion was suggested by the fact that it has been proved that Maxwell's equations represent equally well the sequence of electromagnetic phenomena relative to a set of axes moving relative to the aether, as relative to a set of axes fixed in the aether. ”

“ On the other hand, since from what has been said above it appears that the electron will naturally retain the spherical shape as measured by the variables associated with the moving axes, it appears that some extraneous forces would be required to cause it to retain the spherical shape to an observer remaining at rest. ”

“ If a denote a volume which is moving with velocity , the corresponding volume A is given by In applying the transformation to the consideration of the number of free electrons per unit volume, it must be remembered that they are moving with different velocities, and that this velocity affects the relative volumes which they occupy in the two systems of coordinates. ”

“ The transformation of space and time variables mentioned above shows a means of explaining this dependence of the electromagnetic phenomena on relative motion only; and conversely it is a comparatively simple matter to show that it is the only means. For it is required, among other things, to explain how a light-wave travelling outwards in all directions with velocity C relative to an observer A, may at the same time be travelling outwards in all directions with the same velocity relative to an observer B moving relative to A with velocity v. ”

“ If, therefore, we confine our attention to a portion of matter contained within a volume which is small, but large enough to allow of the process of averaging commonly employed in molecular physics, we shall obtain results similar to those of the last section. ”

“ The longest-living author of this work died in 1977, so this work is in the public domain in countries and areas where the copyright term is the author's life plus 45 years or less. This work may be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works. ”

“ Inasmuch as energy and pressure are independent of thermodynamics, it seems appropriate to shew here that Planck's results for those quantities are an immediate result of the foregoing transformation. ”

“ Taking the expression for the force per unit area on a moving surface, [12]
this being a vector equation, we adapt it to the case where the surface is moving in the direction of the axis of x with velocity v.
The equations of the transformation are
Let the element of area upon which p' acts be ds, the normal having direction cosines (l, m, n) , and let the transformed area be dS, with normal (L, M, N) . ”

“ But if we take the Lorentz equation which is obtained from that above by the substitution of the vector for
the conditions are that
shall be continuous, provided there is no true electricity on the surface of separation. There is no difficulty in proving that these conditions are conserved under Mirimanoff's extended transformation.
It will be proved now that they are also conserved under the transformation by inversion, and therefore by the most general transformation of the whole group which leaves the fundamental equations invariant.
From the equations (18') , we obtain if we write ”

“ In the cavity at rest when the radiation is in equilibrium, the mean values of are each equal to , and that of is zero, the average being taken over any interval of time very small, but large compared with the periods of the constituent radiation. ”

“ If, therefore, we confine our attention to a group of electrons all having the same component of velocity in the one system and in the other, the volumes occupied, δa, δA, are connected by the relation (5) .
Thus this group contributes to the density of the free electricity as measured in the two systems different amounts, , . Using (5) , we have
Let be the mass velocity of the moving medium as distinguished from the velocity of the individual electron. ”