Why are there circumflex accents in these words? -- κορῶναι, αἶγες, αἶγας

Question

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?

Answer 1

Per Wikipedia:

σωτῆρα (sōtêra) Law

If the accent comes on the penultimate syllable, it must be a circumflex if the last two vowels of the word are long–short. This applies even to words ending in -ξ -x or -ψ -ps [...] In most cases, a final -οι -oi or -αι -ai counts as a short vowel