The continuous limit of the multiple lens effect and the optical scalar equation

We study the continuous limit of the multiple gravitational lensing theory based on the thin lens approximation. We define a new, light-path dependent angular diameter distance d and show that it satisfies the optical scalar equation. The distance provides relations between quantities used in gravitational lensing theory (the convergence, the shear and the twist terms) and those used in scalar optics theory (the rates of expansion, shear and rotation).