Analytical tools are developed that allow the 10Be measured in ice cores to be inverted to yield quantitative information about the long term time dependence of the cosmic ray flux outside the Earth's magnetic field. It is concluded that the historic changes in both the magnitude, and direction of the geomagnetic dipole, and the inter- latitudinal mixing in the atmosphere, produce effects that can be large compared to the standard deviation of 22 year average 10Be data, and appropriate procedures are developed to compensate for them. The 10Be response function is shown to have peaked near 1.8 GeV/nucleon since 1950, while during the Maunder minimum it approximated 0.8 GeV/nucleon. This shows that the 10Be data are approximately a factor of two more sensitive to cosmic ray modulation than are high latitude sea level neutron monitors. The Z ≥ 2 cosmic radiation contributes 30% to 49% of the observed 10Be, introducing a substantial rigidity dependence into the temporal variations. A mathematical model is developed that computes the 10Be flux arriving at points in the polar cap, taking into account polar wander, changes in the geomagnetic moment, and various degrees of atmospheric mixing. It is shown that polar wander introduces variations into the observed flux that may be approximately three times the standard deviation of 22 year average 10Be data, and that there may be significant differences between the northern and southern hemispheres. The mathematical model, together with the temporal changes in the nitrate concentration observed in ice cores, indicate that the secular changes in 10Be due to climate driven changes in atmospheric circulation were <4% during the little ice ages. Two methods are outlined that allow independent determinations of the characteristics of the region of the atmosphere that contributes to the 10Be observed in the polar caps. For the Antarctic, this has approximated the region south of 40¿S for the past millennium. Frequently, 10Be measurements are made using 4--8 year ice samples, and it is shown that the unresolved 11 year variations of the cosmic radiation are the dominant source of statistical variation in these data, and a procedure developed to largely eliminate them. Failure to eliminate these unresolved 11 year variation greatly diminishes the value of the 10Be data in the study of the long term changes in the galactic cosmic radiation.