In a quick, fast, or speedy manner; with expedition or dispatch (archaic form of 'expeditiously').
From 'expedite' plus the adverb suffix -ly. This form represents an older or variant spelling of what we now more commonly write as 'expeditiously.'
Before 'expeditiously' became standard, people said 'expeditely'—it's a reminder that even common words have variant forms that sometimes lose out in the evolution of English.
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