It is possible that the original manuscript read “the ball or the director” and that the second the was accidentally deleted by Oliver Cowdery while copying. Earlier in his copying from 𝓞 into 𝓟, Oliver deleted the repeated the in a similar conjunctive noun phrase:
On the other hand, Alma 37:38 reads “a ball or director” rather than “a ball or a director” (that is, without repetition of the article, here the indefinite a). And in that instance, the original manuscript is extant and reads without the a. Consequently, repetition of the article is probably not necessary for “the ball or director” in Mosiah 1:6. In each case, we let the earliest textual sources determine whether articles should be repeated for nominal conjuncts. For further discussion, see conjunctive repetition in volume 3.
Summary: Retain in Mosiah 1:16 the phraseology “the ball or director” (that is, without any repeated the), in accord with the reading of the earliest textual source (here the printer’s manuscript).