T´ = ∫ ds / ((V/μ) cos ε´ + v´ cos θ´) = min.,
where sin ε´ / sin θ´ = v´ / (V/μ) = α´.
This may be written
T´ = ∫ cos ε´ ds / V/μ (1 − α´²) − ∫ v´ cos θ´ ds / V²/μ² (1 − α´²);
the second term alone involves the first power of the motion, and assuming that μ²v´ cos θ´ = dφ´/ds, and treating α´ as a quantity too small for its possible variations to need attention, the expression becomes
T´ = μT cos ε´ / (1 − α´²) − (φ´B − φ´A) / V²(1 − α´²),
T being the time of travel through the same space when empty. Now, if the time of journey and course of ray, however they be affected by the dense body, are not to be more affected by reason of etherial drift through it than if it were so much empty space, it is necessary that the difference of potential between two points A and B should be the same whether the space between is filled with dense matter or not (or, say, whether the ray-path is taken through or outside a portion of dense medium). In other words (calling φ the outside and φ´ the inside potential function), in order to secure that T´ shall not differ from μT by anything depending on the first power of motion, it is necessary that φ´B−φ´A shall equal φB−φA: i.e. that the potential inside and outside matter shall be the same up to a constant, or that μ²v´ cos θ´ = v cos θ; which for the case of drift along a ray is precisely Fresnel's hypothesis.
Another way of putting the matter is to say that to the first power of drift velocity
T´ = μ T − ∫ (μ² v´ cos θ´ − v cos θ) ds / V²,
and that the second or disturbing term must vanish.