For consider a slab of matter moving flatways with velocity _u_; let its internal etherial density be μ², and let the external ether of density 1 be stationary. Let the forward speed of the internal ether through space be _xu_, so that a beam of light therein would be hurried forward with this velocity. Then consider two imaginary parallel planes moving with the slab, one in advance of it and the other inside it, and express the fact that the amount of ether between those two planes must continue constant. The amount streaming relatively backwards through the first plane as it moves will be measured by _u_ times the external density, while the amount similarly streaming backwards through the second plane will be (u-xu) times the internal density. But this latter amount must equal the former amount. In other words, u×1 must equal (u-xu) × μ². Consequently _x_ comes out x = (μ²-1)/μ²; which is Fresnel's incontrovertible law for the convective effect of moving transparent matter on light inside it. The whole subject, however, may be treated more generally, and for every direction of the ray, on the lines of Chapter X, thus:-- Inside a transparent body light travels at a speed V/μ; and the ether, which outside drifts at velocity _v_, making an angle θ with the ray, inside may be drifting with velocity _v´_ and angle θ´. Hence the equation to a ray inside such matter is 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.