Official Report of Gen. Vestch.„Hkadquaktkbs, Sr.rond Bhigadk* Fourth Division,'Bolivak, Tknn., Oct. 9, 1862.Captain Binmore, Assistant Adjutant Gen. JFourtk Division—Captain: I have the honor to submit the following report of the battle of Hatchie, so far os the troops under my command were concerned:In obedience to the orders of Gen. Hurlbut, the 2d brigade moved from Bolivar at daylight on the morning of the 4th inst., on the road towards Corinth. At the hour of marching, Colonel 8coit, commanding the 68th Ohio and 12th Michigan, and Major Bicker, commanding a battalion of the 5tb Ohio Cavalry, reported to me for orders, and were attached to my command during the expedition.My force consisted of the 15th Illinois, Lieut. Col. Rodgers; the 25th Indiana, Col.Wm. H. Morgan; the 46th Illinois, Colonel Davis; the 14th Illinois, Colonel Ilall; the 53rd Indiana, Lieut. Col. William Jones; Company L, 2d Regiment Illinois Artillery, : Captain Bolton, and 7th Ohio Battery, Capt. Burnap, with the cavalry under Major Ricker, and two regiments of infantry,under Col. Scott, just mentioned.