By using the trajectory-tracing technique, cutoff rigidities for Palestine, Dallas, and Midland, Texas, have been calculated as a function of various zenith and azimuth angles. Extensive analysis of the trajectory calculations shows that there is a systematic uncertainty involved in computing the lowest allowed rigidity, and this uncertainty may be a significant fraction of the penumbral width. Continuation of the trajectory-tracing process below the Stormer cutoff allows an evaluation of the reentrant albedo, showing that the average invariant latitude of the guiding center of the trajectory at the albedo origin is the same as the average invariant latitude of the guiding center of the particle trajectory at the detection point. No significant difference in the cutoff rigidities for these locations is found when the external magnetic fields present in the magnetosphere are added to the geomagnetic field of internal origin.