c.1510, from L. alacritatem (nom. alacritas) "liveliness," from alacer (gen. alacris) "cheerful, brisk, lively;" cognate with Goth. aljan "zeal," O.E. ellen, O.H.G. ellian.
Obviously related to the Italian musical term allegro... But I thought from the looks of the O.E. relative ellen that it would also be related to French élan -- but no!
1877, from Fr. élan, from élancer "to rush, dart," from O.Fr. elancer, from e- "out" + lancer "to throw a lance," from L.L. lanceare, from L. lancea "lance."
So no relation to alacrity. Oh well.
But alacrity is still an interesting word.