Mr. Lancaster does not inform us why an English gentleman should choose a Sunday morning, of all days in the week, to cruise about at an early hour with ladies on board, nor does he supply the public with information as to the movements of the Deerhound during the hour and a half which elapsed between his exit from the harbour and the appearance of the Alabama. The preceding paragraph, however, supplies the omission. THE ENGAGEMENT. At length the Alabama made her appearance in company with the Couronne, the latter vessel conveying her outside the limit of French waters. Here let me pay a tribute to the careful neutrality of the French authorities. No sooner was the limit of jurisdiction reached, than the Couronne put down her helm, and without any delay, steamed back into port, not even lingering outside the breakwater to witness the fight. Curiosity, if not worse, anchored the English vessel in handy vicinity to the combatants. Her presence proved to be of much utility, for she picked up no less than fourteen of the Alabama's officers, and among them the redoubtable Semmes himself. So soon as the Alabama was made out, the Kearsarge immediately headed seaward and steamed off the coast, the object being to get a sufficient distance from the land so as to obviate any possible infringement of French jurisdiction; and, secondly, that in case of the battle going against the Alabama, the latter could not retreat into port. When this was accomplished, the Kearsarge was turned shortly round and steered immediately for the Alabama, Captain Winslow desiring to get within close range, as his guns were shotted with five-seconds shell. The interval between the two vessels being reduced to a mile, or thereabouts, the Alabama sheered and discharged a broadside, nearly a raking fire, at the Kearsarge. More speed was given to the latter to shorten the distance, and a slight sheer to prevent raking. The Alabama fired a second broadside and part of a third while her antagonist was closing; and at the expiration of ten or twelve minutes from the Alabama's opening shot, the Kearsarge discharged her first broadside. The action henceforward continued in a circle, the distance between the two vessels being about seven hundred yards; this, at all events, is the opinion of the Federal commander and his officers, for their guns were sighted at that range, and their shell burst in and over the privateer. The speed of the two vessels during the engagement did not exceed eight knots the hour.