  Air fluorescence detectors traditionally determine the dominant chemical
composit ion of the ultrahigh energy cosmic ray flux by comparing the averaged
slant depth of the shower maximum, $X_{max}$, as a function of energy to the
slant depths expect ed for various hypothesized primaries. In this paper, we
present a method to make a direct measurement of the expected mean number of
protons and iron by comparing the shap es of the expected $X_{max}$
distributions to the distribution for data. The advantages of this method
includes the use of information of the full distribution and its ability to
calculate a flux for various cosmic ray compositi ons. The same method can be
expanded to marginalize uncertainties due to choice of spectra, hadronic models
and atmospheric parameters. We demonstrate the technique with independent
simulated data samples from a parent sample of protons and iron. We accurately
predict the number of protons and iron in the parent sample and show that the
uncertainties are meaningful.
