Originally the French who firmly supported dynastic succession of the house of Bourbon, now anyone who supports the hereditary monarchy.
As opposed to Orleanists who supported the Duke of Orleans, a.k.a. Louis Philippe - Louis XIV's brother.
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Uses:
only those Frenchmen were admitted who were deep-rooted legitimists, and patriotic views were expressed to the effect that one ought not to go to the French theater
Leo Tolstoy. War and Peace: 10 (Book Ten)
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the truest and bluest of Legitimists, would never have forgiven her,
Marcel Proust. In Search of Lost Time [volume 1] Swann’s Way