The Second American Arctic Expedition under the command of Dr. E. K.Kane,U. S.N. sailed from this port on Tuesday last week. The expedition consists of one vessel, the Advance. The object of the expedition, next to a search for Sir John Franklin, is scientific discovery in that region. The ship is furnished with supplies for three years ; which can be made to last five or six years, by hunting and fishing. The cruise is expected to continue from 18 months to two years. The vessel is lined throughout with felt and cork to keep her warm : and furs and buffalo robes are on board for the use of the men in sleeping, and for wear when exposed to the cold. There are guns, rifles, and shot guns, pistols, and other weapons. Goods are taken out to make iriends of the natives, who must oe depended upon for sledge-men, sledges, dogs, &c. An Order of Council has been furnished by the Danish Government, instructing their representatives at the different settlements to furnish aid to the expedition. Daguerreotype apparatus and stock is taken out, with which to take views of places and scenes; and pictures of some of the persons in authority, and natives. It is expected that the advance will touch at St. John's, Newfoundland, for fresh provisions, and Dr. K. expects to reach Greenland in three week's time. Thence the earliest opportunity will be improved to proceed upon her errand of noble philanthropy, and scientific research. Dr. Kane was out on the former expedition with Lieut. De Haven, and the " Advance " is one of the vessels employed on the Grinnel Expedition. A competent naturalist and astronomer are aboard, and the corps selected are all men of true worth and decided courage. The " Advance " was accompanied down our Bay by many friends and well-wishers ; Judge Kane, of Philadelphia, was among them, to take an affectionate farewell of his noble son, than whom there is not a nobler spirit in our land. We have no hope of the main object of this expedition being successful, but we trust the " Advance " with its commander and associates will return safe again to our shores, to gladden the hearts of many Iriends.