HDQRS. THIRD DIVISION, TWENTIETH ARMY CORPS,
Cowan, Tenn., July 3, 1863-7 p.m. (Received July 4-1.30 a.m.)

COLONEL: I occupied this place with my division about 3 this p.m., my advance having reached here about an hour before.

The last of Bragg's infantry left here this morning on the cars. There was nothing here on my arrival but a small covering force of cavalry. A number of cavalry prisoners and their horses were captured to-day.

The Second Kentucky and Thirty-ninth Indiana Mounted Infantry made two or three dashes, and were successful whenever they went in with a vim. They lost 4 men wounded and 8 horses killed. The general information says Bragg has gone to Chattanooga.

Colonel Watkins, Sixth Kentucky Cavalry, joined me this afternoon with 1,000 men. I will use them as I find best. My command is in fine spirits, but I must have my haversacks filled before gong farther. I will, perhaps, be able to collect some forage in this section. Shall I fire a Fourth of July salute to-morrow?

I have examined the railroad bridge at this point,a nd find it only slightly damaged. My pioneers can repair it in a short time.

I have the honor to be, colonel, your obedient servant,

P. H. SHERIDAN,

Major-General, Commanding.

Lieutenant Colonel G. P. THRUSTON,

Asst. Adjt. General and Chief of Staff, Twentieth Corps.

HDQRS. THIRD DIVISION, TWENTIETH ARMY CORPS,
Cowan, Tenn., July 7, 1863.

COLONEL: I have the honor to report the following as the operations of my command from June 24 to July 5, 1863:

At daylight on the morning of the 24th of June, I marched with my division and five companies of the Thirty-ninth Indiana Mounted Infantry, Lieutenant-Colonel Jones commanding, on the pike leading from Murfreesborough to Shelbyville. After proceeding about 5 miles from Murfreesborough, I encountered the enemy's pickets, driving them before me, with scarcely any effort, until I reached Christiana, distant from Murfreesborough about 9 miles. At this point I was ordered to halt and cover the road upon my right, upon which the division of General Brannan was to approach Christiana. I was here assailed by the enemy's sharpshooters and one section of artillery, to which I made no reply, my object being to conceal the strength of my force. On the arrival of the head of General Branna's column, I drew off my troops, marched to the left, and encamped at the little town of Millesburg, in the vicinity of Liberty Gap.

On the 25th, I remained in camp at that place.

On the 26th, by direction of the general commanding, I took up my line of march for Hoover's Gap. My march was so impeded by wagon trains in my front that my whole division did not reach Hoover's Gap till the morning of the 27th. Immediately after assembling at Hoover's Gap, I marched on Beech Grove, where I received instructions to continue my march on Manchester, via Fairfield. On reaching Fairfield, about 3 p.m., the head of the column encountered a regiment of rebel infantry and one of cavalry. Colonel Laiboldt, commanding the advanced brigade, immediately deployed his sharpshooters and the Second Missouri Infantry, charged the enemy at double-quick, followed by the