The Admiral, with the body of the fleet and store-ships, is in the vicinity of Pigeon Island, at the entrance of Fort Royal Bay. Our operations to windward have been vigorous and effectual in point of time; and the privations of the troops have been considerable, and borne in a manner worthy of the character of British soldiers. From what has passed, I am of opinion the inhabitants of the country manifest a friendly disposition; and after the heights of Surirey shall be carried, which I expect will be strongly contested, the campaign will be reduced to the operations of a siege, and the defence of the fortress.--The services rendered by the captains and officers of the navy to windward have been great and essential, and the exertions of Captain Withers of the navy, principal agent for transports, peculiarly meritorious. "GEO. BECKWITH, Com. Forces." * * * * * "_Martinique, Heights of Surirey, Feb. 3._ "My Lord, "In my letter of the 1st inst. I had the honour to report, for His Majesty's information, the progress then made in our operations against the enemy. My expectation that Lieutenant-General Sir G. Prevost would meet them upon Morne Bruno, and that the heights of Surirey would be warmly contested, was realized in the course of the same day; and both were carried under the direction of the Lieutenant-General, with that decision and judgment which belong to this respectable officer, and much to the honour of Brigadier-General Hoghton, the officers and men of the Fusileer brigade and light battalion, engaged on that service. On the 2nd, it appeared to me to be desirable to extend to the right of our position; which was effected in a spirited manner by the King's infantry. An exertion was then made to carry the advanced redoubt; but, having soon reason to believe that it would have been acquired with a loss beyond the value of the acquisition, the troops were withdrawn; and the enemy abandoned it during the night, with another redoubt contiguous to it, with evident marks of disorder: both will be occupied and included in our position this night. Pigeon Island surrendered at discretion yesterday, which enables the shipping to enter Fort Royal Bay; all the batteries on the Case Naviere side have been destroyed and abandoned, a frigate and some other merchant-vessels burned, the lower fort abandoned, and all their troops withdrawn from Fort Royal to the principal fortress. I consider the investiture to be nearly completed, and we must now look for the operations of a siege. Time does not admit of details; but your Lordship will perceive that these operations have been effected in eight days from our quitting Barbadoes, notwithstanding heavy rains and most unfavourable weather, in which the troops have borne every species of privation in a manner worthy their character as British soldiers.