BLACK HAWK'S INVASION.- This bold fellow, who has occasioned so much distress on the North Western Frontier, for some months past, has at length paid pretty dearly for his temerity. His whole loss down to the 3rd August, is estimated at 400 killed and 150 prisoners. About 150 were slain in the engagement of the 2d inst. particulars of which were given in our last,-33 to 25 in the affair of the steamboat Warrior, on the 1st inst., 68 in the action with Gen. Dodge and Henry on the Ouisconsin,-and about 200 in various skirmishes previously. The 70 prisoners taken on the engagement of the 2d inst. About 100 horses were also taken. In the affair with the steamboat Warrior, it is estimated that the Indians fired from (sic) 1500 balls. They however proved themselves very bad marksmen, unless the boat was too distant for the range of their muskets, as only 50 of their shots struck her, and only one man on board was wounded. Nothing certain is known of the fate of Black Hawk. A prisoner reports, that the man who stood next to him was killed by the first discharge of the six pounder on board the boat, and that in the midst of the action, he himself retreated up the river.