This paper test both Clark's negative expondential model and Newling’s quadratic exponential model against the population densities in districts of Kaohsiung Metropolis. It proposes four theoretical amendments and supplements to the population density function and the related theories. First, Newling’s conclusion, derived from the first derivative of the quadratic exponential model, that “the highest point of the urban population density lies at the place which is b/2c away from the center of the city,” is true only when the b parameter (density gradient) is positive, and when there is a density crater at the center of the city. Otherwise, the highest point lies at the center when the city is undergoing "centralization", and does not have a density crater. Meanwhile, the place which is b/2c away has the lowest point of the population density. Second, after examining Clark’s negative exponential model, Newling concludes that “in the same period, the farther the region is away from the center of the city, the higher the rate of population growth will be” This point is not true in the following two aspects: (1)it is not appropriate to assume that the density gradient will “decline exponentially as time marches on”; on the contrary, it will increase accordingly when the city is experiencing its period of “centralizatio n”, (2) instead of the previous interpretation, the implication of this formula should be revised as:“the rate of population growth is higher everywhere outside than in the center of the city.” Third, Berry’s arguments about the “Western cities” and“Non-Western cities” cannot be sustained since he was misled by the temporal phenomenon. There should have been no differences in the distributive pattern of the population density between these two kinds of cities, since, if investigated both backward and forward in longitude, they both have the“centralization" and "decentralization" processes. Fourth, the quadratic exponential model explicates the distributive phenomenon much better than the negative one. Moreover, theτe are significant divergences in their interpretative power between the two models, especial1y at the stage of “centralization” when the city is newly established, and at the last stage of declination after“decentralization.”