From Hardee's "Rifle and Light Infantry Tactics" : "A regiment is composed of ten companies, which will habitually be posted from right to left, in the following order: first, sixth, fourth, ninth, third, eighth, fifth, tenth, seventh, second, according to the rank of captains."
I suppose that in practice, alpha designations were substituted for the numerical as a matter of practicality. Shifting positions as the seniority of company commanders necessarily changed over time would seem to cause a good deal of confusion.