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We propose a new technique, which we call the lens parallax method, todetermine simultaneously the redshift distribution of the faint blue galaxiesand the mass distributions of foreground clusters of galaxies. The method isbased on gravitational lensing and makes use of the following: (1) theamplitude of lensing-induced distortions of background galaxies increases withredshift; (2) the surface brightnesses of galaxies decrease steeply withredshift. The distortions of galaxy images due to lensing are thus expected tobe inversely correlated with surface brightness, allowing us to obtain relativedistances to galaxies as a function of surface brightness. If the redshifts ofthe brightest galaxies are measured, then the relative distance scale can beconverted to mean galaxy redshifts as a function of surface brightness.Further, by comparing the angular sizes of lensed galaxies with those ofsimilar galaxies in empty control fields, it is possible to break the so-calledmass sheet degeneracy inherent to cluster mass reconstruction techniques whichare based purely on image ellipticities. This allows an unambiguousdetermination of the surface density of a lensing cluster. We describe aniterative algorithm based on these ideas and present numerical simulationswhich show that the proposed techniques are feasible with a sample of ~ 10 richclusters at moderate redshifts ~ 0.3-0.4 and an equal number of control fields.The numerical tests show that the method can be used to determine the redshiftsof galaxies with an accuracy of dz ~ 0.1-0.2 at z ~ 1-1.7, and to measure themasses of lensing clusters to about 5% accuracy.Comment: 31 pages, uuencoded compressed postscript file containing 10 figures,  to be published in the Sep. 20 issue of Ap

The Lens Parallax Method: Determining Redshifts of Faint Blue Galaxies through Gravitational Lensing