HEADQUARTERS CAVALRY CORPS, Columbia, Tenn., March 20, 1863.

Major K. FALCONER,
Adjutant and Inspector-General, Tullahoma:

SIR: The dispatches of General Bragg, in regard to the information that the enemy were probably falling back from Murfreesborough, and ordering that I should follow them if true, have been received. I have constantly had scouts in every direction, in front and on both flanks, and although rumors have reached me that such a movement was contemplated, no report of any actual movement of the kind has been made. I have been shoeing my horses and resting them after the very trying forced marches I have recently made. I had informed moving back across Duck River as soon as possible. I had ordered the corps over this morning, but owing to reports that the enemy were advancing this way again, I crossed part of the command last night; the balance are going over this morning. I will get in their rear, if possible, and strike at Franklin, Brentwood, and other points in that vicinity. I can only take, however, about 5,000 men. If no movement is made from Franklin, and none against General Bragg's main position, I would respectfully suggest the propriety of allowing my horses rest for a short time, to gain sufficient flesh and strength for the final contest. I am sorry that I must report that they are not in very good condition now; that is, there are many that are not.

Very respectfully, major, I am, your obedient servant,

EARL VAN DORN,

Major-General.

P. S.-Scout just in from College Grove and Eagleville reports Steedman's brigade and two regiments of cavalry at Triune, fortifying hill in vicinity. Some troops came from Nolensville to Triune last Tuesday. The impression among citizens near Murfreesborough is that the enemy intend moving back; part of their army has gone across Stone's River.

A deserter from Franklin yesterday states that he heard officers discussing plans for taking Columbia; they had maps, &c., and that preparations were being made for the movement. The enemy had, he says, 9,000 men.

SPRING HILL, March 20, 1863-9 p. m. (Received March 21, 1863.)

Lieutenant-General POLK:

Enemy have gone to Franklin. All quiet. The following information just received:

Federals were passing through Nashville all day Wednesday in the direction of Kentucky.

I shall move up nearer to-morrow, and will strike if the opportunity affords.

EARL VAN DORN,

Major-General.

HEADQUARTERS WHARTON'S CAVALRY, Unionville, March 20, 1863-12.30 p. m.

Lieutenant-General POLK, Commanding at Shelbyville:

GENERAL: Lieutenant [William L.] Smith, of Company G, and three men from same company, Texas Rangers, with Mr. Lycurgus House, of