From reading his Letters I got the impression that Kirith with a K was the way he spelled it while writing the book, and he changed it to C in one of his post-completion revisions whilst preparing it for publishing. I recall that Christopher didn't want him to do it because he felt that everyone would pronounce it "Sirith" as a result. I don't remember what Tolkien's exact reason was and why he didn't follow Chris's advice.
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All shall be rather fond of me and suffer from mild depression.
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