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We present an analytical formulation of gravitational lensing using familiar triaxial power-law mass distributions, where the 3-dimensional mass density is given byThe deflection angle and magnification factor are obtained analytically as Fourier series. We give the exact expressions for the deflection angle and magnification factor. The formulae for the deflection angle and magnification factor given in this paper will be useful for numerical studies of observed lens systems. An application of our results to the Einstein Cross can be found in Chae, Turnshek, & Khersonsky (1998).Our series approach can be viewed as a user-friendly and efficient method to calculate lensing properties that is better than the more conventional approaches, e.g., numerical integrations, multipole expansions.

A generally accepted approach to the interpretation of the damped Lya absorption systems seen in QSO spectra is to attribute them to cases in which the observerÏs sight line passes through high column density gaseous disks of galaxies, i.e., spiral galaxies or the progenitors of spirals. Here we consider an alternative natural possibility, consistent with available observational data, namely that the absorption is simply associated with neutral gas in giant hydrogen clouds that could be associated with any type of gaseous galaxy or protogalaxy. At high redshift such galaxies could be the progenitors of di †erent types of galaxies (e.g., ellipticals, spirals, etc.) that are observed at the present epoch. We show that the observational data, which include low to moderate redshift data recently obtained with HST and moderate to high-redshift data compiled using large ground-based telescopes over the last decade, coupled with some reasonable assumptions about the properties of giant hydrogen clouds in galaxies, can be used to form the two-dimensional distribution for the number of damped Lya systems in redshift and column density, over the redshift interval 0.1 \ z \ 3.5. This can be further extended to redshift z \ 0 d2N/dz dN H I , using information provided by 21 cm observations of nearby galaxies. By combining this result with the assumption that damped Lya absorbers behave like the giant hydrogen clouds in our Galaxy and neighboring galaxies, we show that the mass spectrum of clouds responsible for the damped Lya absorption systems can be derived. This mass spectrum can be used to study the evolution of the giant hydrogen cloud population associated with galaxies over cosmic time. An interesting feature of the mass spectrum is that its steepness increases systematically as the redshift decreases over the interval 3 \ z \ 0.75. This e †ect is likely to be related to intensive star formation processes, which lead to the destruction of highmass clouds and the formation of numerous low-mass clouds.

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