Exploring Expedition. The United States Corvette Vincennes, Captain Charles Wilkes, the flag ship of the Exploring Expedition, arrived at New York on Friday, June 10th, from a cruise of nearly four years. The Brigs Porpoise and Oregon may shortly be expected. The Expedition has executed every part of the duties confided to it by the Government. A long list of ports, harbors, islands, reefs, and shoals, named in the list, have been visited and examined or surveyed. The positions assigned on the charts to several vigias, reefs, shoals, and islands, have been carefully looked for, run over, and found to have no existence in or near the places assigned them. Several of the principal groups and islands in the Pacific Ocean have been visited, examined, and surveyed; and friendly intercourse, and protective commercial regulations, established with the chiefs and natives. The discoveries in the Antarctic Ocean (Antarctic continent, -- Observations for fixing the Southern Magnetic pole, &c.) _preceded_ those of the French and English expeditions. The Expedition, during its absence, has also examined and surveyed a large portion of the Oregon Territory, a part of Upper California, including the Columbia and Sacramento Rivers, with their various tributaries. Several exploring parties from the Squadron have explored, examined, and fixed those portions of the Oregon Territory least known. A map of the Territory, embracing its Rivers, Sounds, Harbors, Coasts, Forts, &c., has been prepared, which will furnish the information relative to our possessions on the Northwest Coast, and the whole of Oregon. Experiments have been made with the pendulum, magnetic apparatus, and various other instruments, on all occasions, -- the temperature of the ocean, at various depths ascertained in the different seas traversed, and full meteorological and other observations kept up during the cruise. Charts of all the surveys have been made, with views and sketches of headlands, towns or villages, &c., with descriptions of all that appertains to the localities, productions, language, customs, and manners. At some of the islands, this duty has been attended with much labor, exposure, and risk of life, -- the treacherous character of the natives rendering it absolutely necessary that the officers and men should be armed, while on duty, and at all times prepared against their murderous attacks. On several occasions, boats have been absent from the different vessels of the Squadron on surveying duty, (the greater part of which has been performed in boats,) among islands, reefs, &c., for a period of ten, twenty, and thirty days at one time. On one of these occasions, two of the officers were killed at the Fiji group, while defending their boat's crew from an attack by the Natives.