Franklin is the person responsible for putting together all of the narratives that make up the novel. In a sense, he's like the fictional editor of The Moonstone. He's also the closest this novel comes to a romantic hero. After all, he's the guy who eventually gets the girl. He is described by each of the narrators in some detail, and he contributes a narrative from his own point of view. And yet he's a very hard character to pin down. Why is that?
Well, Betteredge has a theory: he thinks that Franklin is hard to pin down because of his foreign education. Franklin was educated partly in England, partly in France, partly in Germany, and partly in Italy. Betteredge thinks that Franklin's complexity is a direct result of this:
He had come back with so many different sides to his character, all more or less jarring with each other, that he seemed to pass his life in a state of perpetual contradiction with himself. He could be a busy man, and a lazy man; cloudy in the head, and clear in the head; a model of determination, and a spectacle of helplessness, all together. (188.8.131.52)
Franklin's time in foreign countries has actually changed his personality. He "had his French side, and his German side, and his Italian side—the original English foundation showing through, every now and then" (184.108.40.206). Franklin is like the Moonstone itself: he has lots of different facets, and looks different depending on the light and the circumstances.
This description is one of the first times that changefulness is associated with foreignness. Notice how his "Englishness" is the only thing that's solid about him – it's the "foundation." Of course, this is all according to Betteredge's description, and we already know that we can't necessarily trust all of his opinions about foreignness. After all, Franklin claims that Betteredge's theory about the "foreign sides" to his character is totally silly. But it's true that Franklin does seem to change his mind with astonishing frequency, so maybe there is something to Betteredge's idea.