Eckenstein's Comparative Studies in Nursery Rhymes can now be seen at the Internet Archive in various formats:  http://www.archive.org/details/comparativestudi00eckerich.

I wouldn't place much trust in her commentaries, though they are fun to read. She belonged to the then-fashionable school of folklorists that we might call 'Romantic Frazerian', full of excited speculation on ancient origins and hidden meanings, Sacred Kings and Sun Heroes, euhemerised pagan deities and all manner of arcane fantasies that serious scholars have long ago dismissed as groundless, but which are still firmly believed by the credulous because they have read simplified re-hashes of those ideas in the Reader's Digest or the like. Don't fall into that trap!