This question is about the Homeric dialect.
crow/gull = ἡ κορώνη, κορώνης, 1st decl., feminine
goat = αἴξ, αἰγός, 3rd decl., masculine or feminine
In the nominative plural, the copy of Homer I have has these with circuflex accents:
σκῶπές τ' ἴρηκές τε τανύγλωσσοί τε κορῶναι (Odyssey 5.66)
οἵ τέ σε πεφρίκασι λέονθ᾽ ὡς μηκάδες αἶγες (Iliad 11.383)
Αἴξ also has the circumflex in the accusative singular and plural:
αἶγας ὁμοῦ καὶ ὄϊς, τά οἱ ἄσπετα ποιμαίνοντο. (Iliad 11.245)
But κορώνη appears as κορώνην in the accusative, without a circumflex.
The grammar I have handy for Homeric Greek is Pharr (4th ed). He actually uses αἴξ as one of his examples for the third declension (p. 293, #694), but doesn't explain the accents. Re κορώνη, I haven't noticed this type of circumflex in other first-declension -η words.
Can anyone explain what is going on here?