We understand that the objects of the expedition have been successfully accomplished. The party has traversed 4,000 miles of this frontier since the last of May. Their route was from this place to Michilimackinac, and to the Sault of St. Mary's, where a treaty was concluded with the Chippewas for the cession of a tract of land, with a view to the establishment of a military post. They thence coasted the southern shore of Lake Superior to the Fond du Lac; ascended the St. Louis River to one of its sources, and descended a small tributary stream of Sandy Lake to the Mississippi. They then ascended this latter river to the Upper Red Cedar Lake, which may be considered as the principal source of the Mississippi, and which is the reservoir where the small streams forming that river unite. From this lake they descended between thirteen and fourteen hundred miles to Prairie du Chien, passing by the post of St. Peter's on the route. They then navigated the Ouisconsin to the portage, entered the Fox River, and descended it to Green Bay. Then the party separated, in order to obtain a topographical sketch of Lake Michigan. Some of them coasted the northern shore to Michilimackinac, and the others took the route by Chicago. From this point they will traverse the eastern shore of the lake to Michilimackinac, and may be expected here in the course of a week. Governor Cass returned from Chicago by land. A correct topographical delineation of this extensive frontier may now be expected from the accurate observations of Captain Douglass, who is fully competent to perform the task. We have heretofore remained in ignorance upon this subject, and very little has been added to the stock of geographical knowledge since the French possessed the country. We understand that all the existing maps are found to be very erroneous. The character, numbers, situation, and feelings of the Indians in those remote regions have been fully explored, and we trust that much valuable information upon these subjects will be communicated to the Government and to the public. We learn that the Indians are peaceable, but that the effect of the immense distribution of presents to them by the British authorities, at Malden and at Drummond's Island, has been evident upon their wishes and feelings through the whole route. Upon the establishment of our posts, and the judicious distribution of our small military force, must we rely, and not upon the disposition of the Indians. The important points of the country are now almost all occupied by our troops, and these points have been selected with great judgment. It is thought by the party, that the erection of a military work at the Saut is essential to our security in that quarter. It is the key of Lake Superior, and the Indians in its vicinity are more disaffected than any others upon the route. Their daily intercourse with Drummond's Island, leaves us no reason to doubt what are the means by which their feelings are excited and continued.