We present the results of a systematic search for gravitationally-lensed arcs in clusters of galaxies located in the Hubble Space Telescope Wide Field and Planetary Camera 2 data archive. By carefully examining the images of 128 clusters we have located 12 candidate radial arcs and 104 tangential arcs, each of whose length to width ratio exceeds 7. In addition, 24 other radial arc candidates were identified with a length to width ratio less than 7. Keck spectroscopy of 17 candidate radial arcs suggests that contamination of the radial arc sample from non-lensed objects is ~30-50%. With our catalog, we explore the practicality of using the number ratio of radial to tangential arcs as a statistical measure of the slope beta of the dark matter distribution in cluster cores (where rho_{DM}propto r^{-beta} at small radii). Despite the heterogeneous nature of the cluster sample, we demonstrate that this abundance ratio is fairly constant across various cluster subsamples partitioned according to X-ray luminosity and optical survey depth. We develop the necessary formalism to interpret this ratio in the context of two-component mass models for cluster cores. Although the arc statistics in our survey are consistent with a range of density profiles -- beta<~1.6 depending on various assumptions, we show that one of the prime limiting factors is the distribution of stellar masses for the brightest cluster galaxies. We discuss the prospects for improving the observational constraints and thereby providing a reliable statistical constraint on cluster dark matter profiles on <~100 kpc scales.
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