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The dispersion of light.

§ 50. There are two ways of attempting to explain the dispersion of colors, either by (like Cauchy) considering from location to location the variation of the equilibrium disturbance, or by considering as relevant the variation with respect to time. In one case it is the wave length, in the other one the oscillation period, that directly determines the propagation velocity, although at the end both have the same result.

If we would take the first path and also reproduce the explanation given by Cauchy — in its mathematical form — in our theory, then we would have to assume, that the equations summarized in (59) likely contain derivatives with respect to x, y, z, but not such with respect to t, and that namely, due to the smallness of m, the first term in (57) would vanishes. It is clear, that the propagation velocity must change with wave length, as soon as, for example, ${\mathfrak {M}}_{x}$, and ${\frac {\partial ^{2}{\mathfrak {M}}_{x}}{\partial y^{2}}}$ are standing next to one another. Namely, the latter magnitude gains with respect to the first a greater influence, the smaller the wavelength.

The straight opposite assumption would be, that only derivatives with respect to t, but none with respect to x, y, z occur in formula (59). Now, in so far, that the only magnitude of the first kind (whose introduction has proven to be necessary) is the term

$m{\frac {\partial ^{2}{\mathfrak {M}}_{x}}{\partial t^{2}}}$

in equation (57), we can say that the second mentioned view reduces the phenomenon to the mass of the co-oscillating ion.

That this explanation can really be achieved now, was already proven by v. Helmholtz and earlier also by me. The new form that I now give to the theory, makes no difference in this respect.