∫AB ds / V = minimum
If the medium, instead of being stationary, is drifting with the velocity v, at angle θ to the ray, we must substitute for V the modified velocity V cos ε + v cos θ; and so the function that has to be a minimum, in order to give the path of a ray in a moving medium, is
Time of journey = ∫AB ds / V(cos ε + α cos θ)
= ∫AB (V cos ε − v cos θ) / V²(1 − α²) ds = minimum
where α is the ratio v/V.
Path of Ray, and Time of Journey, through an
Irrotationally Moving Medium.
Writing a velocity-potential φ in the above equation to a ray, that is putting
v cos θ = δφ / δs,
and ignoring possible variations in the minute correction factor 1−α² between the points A and B, it becomes
Time of journey = ∫AB cos ε / (1 − α²) · ds / V − (φB − φA) / V²( 1−α²) = minimum.