 Solution

Let $u$ and $v$ be the real object and image distances from the thin lens. The object-screen separation gives $u+v=L$, and the thin lens equation gives $uv=fL$. Thus the two lens positions solveTheir separation is $d=u_{+}-u_{-}=\sqrt{L^2-4fL}$, soThis is the Bessel lens displacement method for measuring focal length. It uses the screen-object separation and the displacement between two sharp-image settings, reducing the need to locate the lens's effective optical center. It assumes the thin-lens or corresponding principal-plane approximation; $L>4f$ is what provides two distinct real-image settings.