contaminating M31 stars as follows. For a star at POS2 of magnitude [*I*]{} and color , a number of stars

$$N = 1.2 \times C_1(I,V - I)/C_2(I,V - I)$$

of similar magnitude and color were subtracted from the POS1 sample, where $C_1$ and $C_2$ denote completeness fractions computed at POS1 and POS2, respectively, and the factor 1.2 corrects for the misplacement of POS2. In most cases $N$ is not an integer, and stars were subtracted once, twice, or not at all as determined by whether a randomly generated number fell above or below the fractional portion of [*N*]{}. In total, about 3600 stars were subtracted from the POS1 sample to account for contamination by M31.

The completeness tests also showed that the level of crowding leads to relatively large photometric uncertainties, ranging from $\pm0.19$ mag RMS at $I = 22.0$ to $\pm0.46$ mag RMS at $I = 25.0.$ However, the uncertainties in [*V*]{} and [*I*]{} are correlated, and the color uncertainties range from $\pm0.08$ mag RMS at $I = 22.0$ to $\pm0.32$ mag RMS at $I = 25.0.$ This is illustrated in Figure \[fig:returned\], where we have plotted the colors and magnitudes of artificial stars as returned by the DAOPHOT/ALLFRAME 