Now the second term depends only on end points, and therefore has no effect on path. The first term contains only the second power of aberration magnitude; and hence it has much the same value as if everything were stationary. A ray that was straight, will remain straight in spite of motion. Whatever shape it had, that it will retain.
Only cos ε, and variations in α², can produce any effect on path; and effects so produced must be very small, since the value of cos ε is
√(1−α²sin²θ).
A second-order effect on direction may therefore be produced by irrotational motion, but not a first-order effect. A similar statement applies to the time of journey round any closed periphery.
Michelson's Experiment.
We conclude, therefore, that general etherial drift does not affect either the path of a ray, or the time of its journey to and fro, or round a complete contour, to any important extent. But that taking second-order quantities into account, the time of going to and fro in any direction inclined at angle θ to a constant drift is, from the above expression,
T1 + T2 = 2T cos ε / 1−α² = √(1−α²sin²θ) / (1 − α²) × 2T,
where 2T is the ordinary time of the double journey without any drift.
Hence some slight modification of interference effects by reason of drift would seem to be possible; since the time of a to-and-fro light-journey depends subordinately on the inclination of ray to drift.
The above expression applies to Michelson's remarkable experiment[10] of sending a split beam to and fro, half along and half across the line of the earth's motion; and is, in fact, a theory of it. There ought to be an effect due to the difference between θ = 0 and θ = 90°. But none can be detected. Hence, either something else happens, or the ether near the earth is dragged with it so as not to stream through our instruments.