Now let us define the optical length of a given path (optical path length, OPL) as the distance traversed by a ray in a homogeneous isotropic reference medium (e.g., a vacuum) in the same time that it takes to traverse the given path at the local ray velocity. Then, if c denotes the propagation speed in the reference medium (e.g., the speed of light in vacuum), the optical length of a path traversed in time dt is dS = c dt, and the optical length of a path traversed in time T is S = cT. So, multiplying equation (1) through by c, we obtain S = ∫ ∫ A B d S = ∫ ∫ A B c v r d s = ∫ ∫ A B n r d s , where n r = c / v r is the ray index – that is, the refractive index calculated on the ray velocity instead of the usual phase velocity (wave-normal velocity). For an infinitesimal path, we have d S = n r d s , indicating that the optical length is the physical length multiplied by the ray index: the OPL is a notional geometric quantity, from which time has been factored out. In terms of OPL, the condition for Γ to be a ray path (Fermat's principle) becomes