Published online by Cambridge University Press: 24 October 2008
The methods of geometrical optics provide an approximate solution of the equation of wave propagation, if the following condition, given by De Broglie, is satisfied. If V is the phase velocity, λ the wave-length, l the direction of greatest increase of V, and θ the angle between l and V, then the condition is that
that is, the relative change of V over a distance of the order of θ must be small†.
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† In De Broglie's statement of the condition the direction l is left undefined. For a plane wave it can be proved that l most be taken as the direction of greatest increase of V. I am indebted to Mr L. H. Thomas for pointing oat that if the wave is not plane, the condition (l) will contain other terms involving the curvature of the wavefront. It seems unlikely that these terms will be of importance in the cases discussed in this paper.
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† It should be noticed that though the cnrvatnre has its greatest value at the apse, the quantity Z has its maximum some distance away from the apse. Since Z contains the factor cos θ it is zero at the apse. There can therefore be no general relation between Schrödinger's and De Broglie's conditions. For instance, for a particle in a circular orbit, Z = 0, whatever the size of the orbit.
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