HEADQUARTERS SECOND BRIGADE,
Below Vicksburg, July 6, 1862.

CAPTAIN: The Tennessee left here last evening with the mail, but, hearing the beating of drums at Grand Gulf, preceded no farther, and returned this evening for an additional gunboat to protect her in passing that point. Her return enables me to supply an omission in my report of the 4th instant. It is that the eight long-range rifled guns of Nims' and Everett's batteries, from their position behind the levee at Barney's Point, distant seven-eighths of a mile from the enemy's heaviest batteries on the upper side of Vicksburg, co-operated gallantly and effectively with the fleet in their cannonade and bombardment of the 28th ultimo. Everett's battery lost 1 man and 1 horse killed.

The fall of the water in the river is nearly at a stand-point, and the drift of logs and brush foreshadows what the newspapers promise, viz, a June rise. To-day's work of the negro force on the cut-off, duly organized into squads of 20, with an intelligent non-commissioned officer or private to each, superintended by officers, is highly satisfactory. The flag-officer with his fleet is most sanguine and even enthusiastic. I regard the cut-off to be my best bower. Should it fail me I shall resort to the next best, to seize and hold the enemy's batteries, or at least spike their guns.

Respectfully, your obedient servant,

T. WILLIAMS,

Brigadier-General, Volunteers.

Captain R. S. DAVIS,

Assistant Adjutant-General.

P. S. - The reconnaissance of to-day has shown how we ought not to approach the batteries; that of to-morrow will probably give the affirmative side.

HEADQUARTERS DEPARTMENT OF THE GULF,
New Orleans, July 16, 1862

Brigadier General THOMAS WILLIAMS, Vicksburg:

GENERAL: From all I can learn of operations at Vicksburg your force is at present not so much needed there as it is elsewhere.

The enemy are concentrating some forces in the neighborhood of Baton Rouge, and it is necessary that something be done on the Red River line. Besides, you are in the geographical department of General Halleck. Therefore, if the state of affairs will permit without serious detriment to the public service, you will withdraw your force and return as soon as possible to Baton Rouge.

I remain, &c.,

BENJ. F. BUTLER,

Major-General, Commanding.

HEADQUARTERS SECOND BRIGADE,

Below Vicksburg, Miss., July 17, 1862

On July 11 the general grade of the bottom of the proposed cut-off had been carried about 1 1/2 feet below the then level of the Mississippi, and in some twelve hours would have been ready to let in the water,