In this, all the arguments pro and con ought to be examined, before we come to a judgment. Probability wanting that intuitive evidence which infallibly determines the understanding and produces certain knowledge, the mind, if it will proceed rationally, ought to examine all the grounds of probability, and see how they make more or less for or against any proposition, before it assents to or dissents from it; and, upon a due balancing the whole, reject or receive it, with a more or less firm assent, proportionably to the preponderancy of the greater grounds of probability on one side or the other. For example:-
If I myself see a man walk on the ice, it is past probability; it is knowledge. But if another tells me he saw a man in England, in the midst of a sharp winter, walk upon water hardened with cold, this has so great conformity with what is usually observed to happen that I am disposed by the nature of the thing itself to assent to it; unless some manifest suspicion attend the relation of that matter of fact. But if the same thing be told to one born between the tropics, who never saw nor heard of any such thing before, there the whole probability relies on testimony: and as the relators are more in number, and of more credit, and have no interest to speak contrary to the truth, so that matter of fact is like to find more or less belief. Though to a man whose experience has always been quite contrary, and who has never heard of anything like it, the most untainted credit of a witness will scarce be able to find belief.
The king of Siam. As it happened to a Dutch ambassador, who entertaining the king of Siam with the particularities of Holland, which he was inquisitive after, amongst other things told him that the water in his country would sometimes, in cold weather, be so hard that men walked upon it, and that it would bear an elephant, if he were there. To which the king replied, Hitherto I have believed the strange things you have told me, because I look upon you as a sober fair man, but now I am sure you lie. (Essay Concerning Human Understanding, 4.15.5)
The first paragraph of this selection makes it clear why Hume found it so convenient to appeal to the King of Siam as an example in Enquiry, Section X, Part I: Hume wants to generalize this sort of weight-balancing to all reasoning, and especially to that concerning matters of fact (e.g., reasoning about marvels and miracles, the topic of Section X). Both Campbell and Shepherd, although they approach the question from different angles, try to use this (with interesting results) to show that Hume's account, which he uses to rule out belief about miracles on the basis of testimony, can't make the sorts of distinctions it needs to make if it is to succeed (the quotation in the previous post gives an example of this for Campbell).
Prior to the whole miracles debate, there are two other cases I have found in which it is used. Berkeley considers it briefly in his Alciphron (he is unimpressed); and Butler appeals to it in the Introduction to The Analogy of Religion.