Georg Autenrieth, A Homeric Dictionary

ἀνά , by apocope ἄν (ἀν), before labials ἄμ (ἀμ): up, opp. κατά.—I. adv., ἄνα (with anastrophe), hortative, up! quick! Il. 18.178, Od. 18.13; up there, thereon, μέλανες δ᾽ ἀνὰ βότρυες ἦσαν, Il. 18.562; back, ἀνά τ᾽ ἔδραὐ ὀπίσσω, Il. 5.599, ἀνὰ δ̓ ἴσχεο, ‘hold up,’ ‘refrain,’ Il. 7.110. The use with verbs ‘in tmesi’ is of course adverbial; likewise when a subst, occurs in a case that defines the adv. (thus showing the transition to a true preposition), ἂν δ᾽ ἄρα Τηλέμαχος νηὸς βαῖνε (νηός local or part. gen.), Od. 2.416.—II. prep., (1) w. gen., only ἀνὰ νηὸς ἔβην, Od. 9.177, see the remark on Od. 2.416 above.—(2) w. dat., up on, upon, Il. 1.15, Il. 15.152, ἀνά τ᾽ ἀλλήλῃσιν ἔχονται, hold on (close up) ‘to’ one another, Od. 24.8.—(3) w. acc., up to, up through, Il. 10.466, Od. 22.132, Il. 22.452; of motion, ἀνά generally denotes vague direction (up and down, ‘up through,’ ‘throughout’), ἐννῆμαρ μὲν ἀνὰ στρατὸν ὤχετο κῆλα θεοῖο, Il. 1.53, whereas κατά rather indicates motion toward a definite point or end (Il. 1.483, 484); with the idea of motion less prominent, Il. 13.117, 270; of time, ἀνὰ νύκτα, Il. 14.80; βασιλῆας ἀνὰ στόμ᾽ ἔχων, ‘bandying their names up and down,’ Il. 2.250 ; ἀνὰ θυ_μὸν φρονεῖν, ὁρμαίνειν, θαμβεῖν, ὀίεσθαι, Il. 2.36, Od. 2.156, Od. 4.638; ἀν᾽ ι?θυ?ν, ‘straight forward,’ Il. 21.303; following the governed word, νειὸν ἀν（ά), ‘up and down’ the field, Od. 13.32.