Ebenezer Cunningham

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”