Hercules was charged for his third (or fourth) labor with the capture of the Arcadian Hind, a sacred deer belonging to the goddess Artemis, an uncatchable animal. Hercules chased it all over Greece until he got sick of the chase and shot it in the leg. He then carried it back, but who did he run into but Artemis? He talked his away out of that one and brought it in for full credit. (Presumably they ate the deer)
Poirot's Arcadian Deer is a beautiful young lady's maid being sought by a poor country mechanic who met her at a rich fellow's house when he was there to do repairs. Poirot discovers that she was the maid to a Russian ballet dancer. His chase leads him all over England and to the maid's native Italy... and a graveside. Does Poirot give up there? Mais, non! On a hunch, he goes to Switzerland to the hospital bed of the dancer herself. Guess who was playing lady's maid at a friend's house and met a cute mechanic? Aww.
Didn't enjoy this one as much, it seemed like a straight-forward chase with a predictable end. Christie flirted with a subplot involving blackmail, which had the effect of confounding the chase, but I felt it didn't amount to much.
What to learn from The Arcadian Deer? If I were writing a chase story, sure: the irrelevant subplot wasn't bad, and Christie gave the reader credit by not allowing the story to seriously consider the death theory for long. But it's not a mystery. Only the one twist (the mechanic's lady not being a maid) is fair in the sense that it was telegraphed, but I thought it was pretty weak.
Edit: I may have written off that subplot too soon. While at first I considered it weak, I thought it over and realized that in addition to confounding the chase, it did so in a clever way: it introduced the theme of mistaken identity into the plot. Everyone thought the maid being asked about was the more recent maid, not the dead one. Poirot got led down the garden path because he was not specific enough about his quarry. There are lots of potential ways to confuse a chase, but this confuses matters in a thematically appropriate way, and constitutes an additional clue.
Sorry, kid, they can't all be winners. But hey, this puts Poirot in Switzerland, just in time for the story of the live capture of the Erymanthean Boar!