Page:Aether and Matter, 1900.djvu/226

 Hence, given any one existing system of electrons with pointnuclei, another system is possible in the same aether having all distances and times reduced in any the same ratio, and electric displacement and magnetic flux independently reduced in any other the same ratio. But if the electrons of this correlated system are to be of the same strengths as the original ones $$\phi/k$$ must be unity; hence the scale must be altered in the same ratio throughout, as regards length, time, and the inductions. Thus, given any existing steady system of electrons, the same system altered to any other scale of linear magnitude is possible if there are none but electric actions. This is on the hypothesis which is here generally adopted, that the dimensions of the nucleus of an electron are so small, compared with the mutual distances of electrons, that these dimensions are not sensibly involved in the forces between them. If this condition is left out the constancy of volume of the nucleus will have to be taken into consideration in the dimensional transformation, so that k must be unity; and this indefiniteness of linear scale in a material body cannot exist. The size of a molecule would also be rendered determinate if residual non-linear terms in the aethereal equations became sensible at intermolecular distances. Thus, these saving hypotheses being excluded, if the atoms of matter were constituted electrically, and the forces between them were wholly of electric origin, there would be nothing to determine the scale of an isolated system as regards time and space: and different systems need not be always of the same scale of magnitude as regards their atomic structure.

. A similar deficiency of definite scale would also be expected to exist in any hydrodynamical theory or illustration which would construct an atom out of vortex rings. Thus let us consider a system of vortex rings, (&xi;, &eta;, &zeta;) being the vorticity at the point (x, y, z), and compare with another system in another space (x', y', z') such that the coordinates of corresponding points are connected by the relation

$$(x',\ y',\ z')=k(x,\ y,\ z), $$