He does give a reference, but I have a feeling it'll be hard to track down:
PÖSCHL, T., 'Der Stross', Handbuch der Physik, 6, Chapter 7. Berlin, J.. Springer, 1926.
A local technical library has Handbuch der Physik, 1958, in English, but there were no articles by Poschl. I managed to get Handbuch der Phsyik, 1928 (though I had requested 1926) via inter-library loan. Unfortunately, it is in German. Judging by the diagrams, it seems that only the results are graphed, also. I.e., unless I'm missing something, there is no development of the underlying theory, and, indeed, mathmatical expressions of the solutions are not usually given. (The statement of the problems are, however, made mathematically explicit.) There are very few footnotes; it seems the most relevant one is:
1) S. Kap. 4, Ziff. 17 ds. Bd. des Handb.
I will backburner looking for this particular analytical solution. I can always compare the numerical models I'm developing against analytical formulae that I do have. Maybe the next time I'm in Princeton, I can ask a librarian for help with Poschl's sources.