Page:Aether and Matter, 1900.djvu/227

 while the vorticities at these points are connected by the relation

$$(\xi',\ \eta', \zeta')=\kappa (\xi,\ \eta,\ \zeta), $$

The formula of von Helmholtz for the velocity (u, v, w) of the fluid in terms of the vorticity, being of type

$$u=\int\left(\eta\frac{d}{dz}-\zeta\frac{d}{dy}\right)\frac{1}{r}d\tau,$$

gives

$$(u',\ v',\ w')=k\kappa (u,\ v,\ w).$$

But the systems will maintain their correspondence of configuration throughout succeeding time, only provided always

$$(u',\ v',\ w')=k(u,\ v,\ w); $$

hence $$\kappa=1$$ while k is arbitrary. Thus if any vortex-system is compared with another one expanded as regards linear scale k times, and the vorticity is at each point unaltered, so that the circulations of the vortices in the new system are all increased k² times, then their subsequent histories will correspond exactly.

The circulation of the vortex is however in the dynamical theory an unalterable constant, so that the one system cannot be changed by natural processes into the other. Let us try therefore to avoid this difference by a change of the time scale as well, so that $$t'=\lambda t$$; then for continued correspondence

$$(u',\ v',\ w')=k\lambda^{-1}(u,\ v,\ w): $$

hence $$k\kappa=k\lambda^{-1}$$ so that $$\kappa=\lambda^{-1}$$; and the strengths of the vortices are altered in the ratio $$k^{2}\lambda^{-1}$$, which must be a constant. Thus if the scale of time is increased &lambda; times, and that of linear magnitude $$\lambda^{-\frac{1}{2}}$$ times, and the corresponding vortex filaments are of the same strengths, the systems will continue permanently in correspondence. This is however on the assumption that the vorticity is around a vacuous core, or a fluid core so thin that its actual section does not affect the mutual actions of the vortices: for the change of linear scale will alter the volume of the core of each ring. There is under these conditions nothing in the hydrodynamical forces to fix the scale of magnitude of an isolated vortex-system with