Page:Aether and Matter, 1900.djvu/214

circuital quality of these vectors: the direction vector of the front of the latter train is proportional to $\left(l\epsilon ^{\frac {1}{2}}+{\frac {pv}{c^{2}}}\epsilon ^{\frac {1}{2}},\ m,\ n\right)$, and its velocity of propagation is

$p\epsilon ^{-{\frac {1}{2}}}/\left\{\left(l\epsilon ^{\frac {1}{2}}+{\frac {pv}{c^{2}}}\epsilon ^{\frac {1}{2}}\right)^{2}+m^{2}+n^{2}\right\}^{\frac {1}{2}}.$

Thus, when the wave-train is travelling with velocity V along the direction of translation of the material medium, that is along the axis of x so that m and n are null, the velocity of the train relative to the moving medium is

$V\epsilon ^{-1}/\left(1+{\frac {Vv}{c^{2}}}\right),$

which is, to the second order,

$V\left(1+{\frac {v^{2}}{c^{2}}}\right)/\left(1+{\frac {Vv}{c^{2}}}\right)$ or $V-{\frac {v}{\mu ^{2}}}-\left({\frac {1}{\mu }}-{\frac {1}{\mu ^{3}}}\right){\frac {v^{2}}{c}}.$

The second term in this expression is the Fresnel effect, and the remaining term is its second order correction on our hypothesis which includes Michelson's negative result.

In the general correlation, the wave-length in the train of radiation relative to the moving material system differs from that in the corresponding train in the same system at rest by the factor

$\left(1+2l{\frac {pv}{c^{2}}}\right)^{-{\frac {1}{2}}}$, or $1-lv/\mu c$,

where l is the cosine of the inclination of the ray to the direction of v; it is thus shorter by a quantity of the first order, which represents the Doppler effect on wave-length because the period is the same up to that order.

When the wave-fronts relative to the moving medium are travelling in a direction making an angle θ', in the plane xy so that n is null, with the direction of motion of the medium, the velocity V' of the wave-train (of wave-length thus altered) relative to the medium is given by

${\frac {\cos \theta '}{V'}}={\frac {l\epsilon }{p}}+{\frac {v\epsilon }{c^{2}}},\ {\frac {\sin \theta '}{V'}}={\frac {m\epsilon ^{\frac {1}{2}}}{p}},$

where $\left(l^{2}+m^{2}\right)p^{2}=V^{-2}$. Thus