On my return on board, the Admiral had retired to his cot; and I had no means of communicating with him until next morning. I could then, however, perceive that under all the severity of disappointment he experienced from our failure, and the loss of the Hannibal, he felt that no honour had been lost; that every effort had been made to obtain success; and that he derived comfort from the sanguine hopes he entertained that an opportunity might present itself in which he should be able to retrieve the loss. He ordered me to take a flag of truce and wait upon the French Admiral, and propose an exchange of prisoners; which M. Linois refused, alleging that it was not in his power to establish a cartel for the purpose until he obtained the sanction of the Minister of the Marine at Paris, to whom he had sent off an express as soon as the firing had ceased on the preceding day; but he consented to send over the officers on parole. The object of the French Admiral was very obvious. He concluded we wanted the men to replace those that had been killed and wounded; but he thought, justly, that sending over officers on parole would be a harmless act of courtesy, from which we could reap no immediate benefit. It will be seen that, subsequently to the second action, (only six days after the first,) when his despatches from Algeziras had scarcely reached Paris, he did not wait for an answer from the Minister of the Marine: but circumstances had altered. We had taken a line-of-battle ship, and burned two first-rates; and he now wished to get as many of the crew back as possible. We shall now transcribe the Rear-admiral's official account of the battle: this affords additional particulars, and proves that every step had been taken to insure success; which, but from circumstances that often blight the fairest prospects, would have had a very different result. We shall venture to add, that, had the gallant Admiral hesitated to make the attempt, he would have rendered himself obnoxious to animadversions, not only from all the squadron under his command, but from every one on the Rock of Gibraltar who witnessed the enemy's squadron of inferior force setting, as it were, that of Sir James at defiance; while it would have afforded the French and Spaniards a just, or, at least a plausible subject of exultation. But Sir James, with that decision of character and coolness, when in the presence of an enemy, for which he was always remarkable, did not hesitate one moment on the attack, which was made with consummate skill, and maintained with a perseverance never surpassed. Caesar, Gibraltar, 6th July 1801. SIR,