Nashville pike. Cheatham's and Stewart's corps and Johnson's division, of Lee's corps, were all engaged, Clayton's and Stevenson's division, of Lee's corps, and the artillery not having been brought into action.

December 1. - The army moved out from Franklin during the morning on the Franklin and Nashville pike, Lee in front, Stewart next, and Cheatham in the rear, all camping on the pike, a few miles from Franklin. Army headquarters for the night just across Harpeth River from Franklin.

December 2. - The army again in motion on Franklin and Nashville pike, marching in the same order as on yesterday. Late in the evening we took position in front of Nashville, Lee's corps in the center, with its center resting upon the Franklin pike, Stewart's forming on his left and Cheatham's on his right, Forrest's cavalry protecting either flank, our line extending, as near as possible, from the Cumberland, above the city, to the Cumberland, below the city, curving forward from General Lee's center. (See confidential circular of December 2, 1864.) Army headquarters at Mr. Overton's residence, five and a half miles from Nashville, and near the Franklin and Nashville pike.

December 3. - The army occupied the same position as yesterday, with slight alterations. Army headquarters remained at Mr. Overton's.

December 4. - The army occupies the same position around Nashville. The skirmish line was advanced on some portions of the line. The cavalry, under Chalmers, captured two transports seven miles below Nashville, on Cumberland River, and some 300 mules.

December 5. - Army headquarters at Mr. Overton's house. Our line remains pretty much the same. The enemy, in heavy line of battle, drove in General Cheatham's skirmishers across the Nashville and Murfreesborough railroad this morning, but retired without attacking our line. Sears' brigade, of French's division, and Brown's brigade, of Stevenson's division, with a battery each, were sent to the vicinity of Murfreesborough to report to General Forrest. General Bate, with the force under his command, was also directed to report to General Forrest. Circular issued to-day to corps commanders for information of the army announcing the capture of "the block-house and fort at La Vergne, with commissary stores, 100 prisoners, 2 pieces of artillery, 100 small-arms and ammunition, 20 wagons, and some teams by General Forrest, and that General Bate had burned three block-houses." General Hood made a proposition to officer commanding U. S. forces at Nashville for an exchange of the prisoners in his hands for an equal number of Confederate prisoners.

December 6. - General Thomas, commanding U. S. forces, Nashville, in reply to General Hood's proposition for an exchange of prisoners, states "that such an arrangement is impracticable, all Confederate prisoners having been sent North, and consequently placed beyond his control" General Hood asks by telegraph of both Generals Beauregard, commanding department, and Maury, commanding Department of Alabama, Mississippi, and East Louisiana, to have the Memphis and Charleston Railroad repaired to Decatur, Ala., to which point he hopes to have the Tennessee and Alabama Railroad running in a day or two. (See telegrams and letter book, office assistant adjutant-general.) Our lines around Nashville about the same, perhaps with slight alteration by corps commanders, under revision of General Hood. Slight demonstration on our (Cheatham's) right by the enemy. General Forrest's command invests Murfreesborough. The enemy reported very strongly fortified there, and with 6,000 or 8,000 troops in his forts. It is not yet determined whether an assault will be made by our forces.