Georg Autenrieth, A Homeric Dictionary

ἄν (1): modal adv., indicating a condition; essentially equivalent to κέν, and of less frequent occurrence. The use of ἄν is less exactly defined in Homer than in Attic Greek; besides the regular usages in Attic (viz. in conclusions expressed by the secondary tenses of the ind., and by the opt., or by the inf. representing these, and joined to εἰ or relative words, ἐάν, ὅταν, etc., in conditional clauses that take the subjunctive), Homer employs ἄν with the subj. in independent sentences, and κέ (rarely ἄν) with the fut. indicative. In final clauses the use of ἄν or κέ prevails, and is not uncommon even with the opt. in conditions. On the other hand the potential opt. occurs without ἄν (κέ) oftener than in Attic. The following examples will illustrate the most important of these peculiarities of usage:—(1) ἄν w. subj. in independent sentence, οὐκ ἄν τοι χραίσμῃ κίθαρις, ‘perchance the harp may avail thee not,’ Il. 3.54, cf. Il. 1.205.— (2) ἄν w. fut. ind., αὐτὸν δ᾽ ἂν πύματόν με κύνες.. ἐρύουσι, ἐπεί κε<*> τις κτλ., ‘me like enough last of all will dogs drag about, after I am slain,” etc., Il. 22.66.—(3) ἄν w. opt. in final clause, σὺ δέ με προΐεις.. ὄφρ᾽ ἂν ἑλοίμην δῶρα, Od. 24.334.—(4) ἄν w. opt. in condition, στεῦτο γὰρ εὐχόμενος νι_κήσεμεν, εἴπερ ἂν αὐταὶ | Μοῦσαι ἀείδοιεν, Il. 2.597.