_July 18th._--At last there appears a probability of this Turkish question being amicably settled. On Saturday I was told that despatches were just come from Sir Hamilton Seymour of a more favourable character, and representing the Imperial Government as much more disposed to treat, with a real disposition to bring the negotiations to a successful issue. My informant added that Palmerston predicted that none of the projects and proposals which have emanated from the different Courts would be accepted at Petersburg, which he thought they all would. Yesterday I saw Clarendon, and found matters even in a still more promising state. After the Cabinet Walewski went to him, and communicated to him very important news (of a later date than Seymour's letters) from St. Petersburg, which to my mind is decisive of the question of peace. It appears that both France and Austria have been concocting notes and projects of a pacific tendency to be offered to the Emperor. There have been several of these, some framed at Constantinople, others at Paris. A short time ago the French Government prepared one, which it submitted to ours; Clarendon thought it would not answer, and told them so. They asked whether he had any objection to their sending it off to St. Petersburg and Vienna and making the experiment. He replied, none whatever, and though he did not think it would succeed, he should rejoice if it did, as, provided the affair could be settled, it did not matter how. In the meantime he drew up his own project of a Convention, which went to Paris, and received the cordial approbation of the Emperor; and this document is now on its way to Petersburg. In the meantime the French project was sent there, Castelbajac took it to Nesselrode, who read it very attentively, and said that he liked it very much, but that he could give no positive answer till he had submitted it to the Emperor. The same afternoon he saw the French minister again, and told him that he had laid the project before the Emperor Nicholas, and that His Majesty was not only satisfied, but grateful for it, 'non seulement satisfait, mais reconnaissant,' and that the only reason he did not at once close with it was that his ally, the Emperor of Austria, had also submitted a proposal, and he did not like to take another from another Court exclusively without previous communication with him. Clarendon thinks that his proposal will be still more agreeable to the Emperor than the French one, and that he will probably end in taking it; nor will there be any difficulty in this, because our's is so fully concurred in by France as to be in fact her's as much as our's. SUCCESSION DUTIES BILL.