_Newmarket, October 12th._--This morning I met the Duke of Bedford on the heath, who told me he wanted very much to speak to me about certain communications he had received which made him extremely uneasy, and full of apprehension of coming difficulties, threatening the very existence of the Government. It seems that a short time ago Lord Aberdeen imparted to John Russell his wish to resign, and to place the Government in his hands. He said that he had only taken his present post because his doing so was indispensable to the formation of the Government, and had always contemplated Lord John's eventually succeeding him, and he thought the time was now come when he might very properly do so. He did not anticipate any insurmountable opposition in any quarter, and he should himself speak to Gladstone about it, who was the most important person to be consulted, and he was in fact only prevented doing so, as he had intended, by not being able to go to Scotland, where he had expected to meet Gladstone. Whether Aberdeen had spoken to Gladstone since his return to London, the Duke of Bedford did not know. No steps appear to have been taken with regard to Palmerston, nor does it appear that any progress was made in accomplishing this change. The Queen had been apprised of Aberdeen's intentions. Such was the state of things when a short time ago the Duke received a letter from Lord John, in which he said that matters could not go on as they were, and that there must be some changes; and that very soon he could no longer act without being primarily responsible for the policy of the Government--in other words, without resuming his post of Prime Minister. This is all the Duke knows, as Lord John entered into no explanations or details, and he is in total ignorance of the grounds of his brusque determination, and of what can have occurred to produce it. He sees, however, all the difficulties and embarrassments that in consequence of it are looming in the distance, and how very possible it is that the Government may be broken up. All this we very fully discussed, but without either of us being able to guess what it all means, or what the result will be of Lord John's putting his intentions into execution. LORD JOHN RUSSELL'S PRETENSIONS.