An archaic or dialect word for errand; a task or journey undertaken to accomplish something specific.
From Old English 'ærende', related to Old Norse 'erindi'. The spelling 'arrand' represents a variant or dialectal form that eventually was displaced by the more common spelling 'errand.'
Arrand is a fascinating example of how older spelling variants survive in place names and dialects long after the standard spelling changes—you might find 'Arrand' as a surname or place name in regions where the old form became fossilized.
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