Page:CunninghamExtension.djvu/3

The second part deals with a new transformation or, rather, group of transformations, which shews that a uniform translation through space is not the only type of motion which it will be impossible to detect so long as our measures of time and space are of electrodynamic origin. It has been pointed out by Minkowski that in a space of four dimensions in which the coordinates are $\left(x,y,z,ct{\sqrt {1-1}}\right)$, the geometrical transformation employed by Einstein, is simply a finite rotational displacement of the whole space about y = 0, z = 0. The equation $\nabla ^{2}V=0$, i.e.,

$\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {p}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)V=0,$

is known to be invariant for such a transformation.

But this equation is invariant for a larger group of transformations than that of rotations, viz., for the group of conformal transformations in the four dimensional space, which, as is known, is built up out of inversions with respect to the hyperspheres of the space.^[1] 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. In that case, a fixed configuration transforms into one every point of which has the same velocity of translation. But, in the present case, under the simplest operation of the group a fixed system becomes one in which the whole is expanding or contracting radially about a point in a certain way, which, though analytically simple, is difficult to describe geometrically. But the important property of it is, that any sphere which is expanding with velocity equal to that of light transforms into a sphere expanding (or contracting) with equal velocity.

It may be remarked here that it appears to be impossible for a uniform velocity of rotation of an electromagnetic system to be obscured in the way in which the types of motion above mentioned are. For, without considering the electromagnetic equations at all, if a disturbance propagated with velocity c equally in all directions is to transform into the like, it follows that the space (x, y, z, ict) of the one system must be conformal

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1. This was pointed out to me by Mr. Bateman, a conversation with whom suggested the present investigation.