Thus the path of the ray when the aether is at rest is the curve which makes **fds**/V least; but when it is in motion it is the curve which makes **fds**/(V+lug-m y -I-nw) least, where (l,m,n) is the direction vector of Ss.

The latter integral becomes, on expanding in a series, **fds**/V f(udx+vdy+wdz)/V2-1-f(udx+vdy+wdz)2/V3ds+..., since ids = dx.

Moreover, this is precisely the condition for the absence of interference between the component of a split beam; because, the time of passage being to the first order **fds**/V f(udx+vdy+wdz)V2, the second term will then be independent of the path (43 being a single valued function) and therefore the same for the paths of both the interfering beams. If therefore the aether can be put into motion, we conclude (with Stokes) that such motion, in free space, must be of strictly irrotational type.