Voöbus, Arthur. "Part 1: On the Rite of Baptism." "Chapter Three: The Ointment Prayer." Liturgical Traditions in the Didache. Stockholm: ETSE, 1968, 41-50.
Voöbus observes that, in conjunction with the baptismal ritual, an anointing with oil appears in the Apostolic Constitutions (Voöbus 1968, 41). This is also reflected in a Coptic papyrus fragment of the Didache, where a prayer for anointing appears appended to chapters 9-10. In both instances the prayer is associated with allowing prophets to pray as they wish.
The prayer is associated with an aromatic ointment or a perfume (Voöbus 1968, 42). The interpretation is not clear, according to Voöbus, expect that the text of the Apostolic Constitutions identifies it as a prayer associated with the aroma. The question of what specific use of ointment is meant is more difficult (Voöbus 1968, 43). Voöbus discusses the common use of anointing and praying for the sick. This would not intuitively be connected with the Eucharist, where the passage appears. However, ointment was also associated with baptism. Voöbus finds such an association in Jewish baptismal practice, as well as in Jewish ceremonial meal practices (Voöbus 1968, 44). Unfortunately, this fails to create clarity in the view.
A search for the function of the anointing is not immediately rewarding. Voöbus finds that the Apostolic Constitutions makes mention of "the immortal eon" (Voöbus 1968, 45). This does suggest the dichotomy between the present, fallen age and that of immortality. From this foundation, Voöbus is able to move to the concept of baptism, where one moves from the former to the latter (Voöbus 1968, 46). When asking about a reason for the choice of a word related to fragrance, Voöbus finds more associations. A Syriac version of Acts describes a fragrance along with the coming of the Holy Spirit at Pentecost (Voöbus 1968, 46). Paradise is regularly described in terms of aromatic plants. Voöbus thus finds good smells as related to moving into the heavenly realms as one does in baptism (Voöbus 1968, 47). Fragrance therefore can be well seen as a concomitant with baptism. Voöbus notes that the ointment prayer was grouped with various prayers of thanksgiving, the meaning of εὐχαριστία (Voöbus 1968, 49). When used in later documents, "oil of the eucharistia" is descriptive of anointing at baptism.