An archaic or dialectal variant of 'again' or possibly a phrase meaning 'by again,' used in old English or Scottish variants.
This appears to be either a very rare variant or specialized spelling combining 'bye' (by) with 'gaein' or 'again,' possibly representing Middle English or Scottish English pronunciation patterns where words were spelled more phonetically.
Some words in the historical record appear so rarely that scholars aren't entirely sure if they're typos, regional dialects, or deliberate spelling choices—byegaein might be one of those ghost words that tells us how much linguistic variation existed before modern standardized spelling.
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