Tuesday, December 1.-The Engineer Battalion and contrabands at work. A line of rifle-trenches was located across the gorge between Temperance Hill and Mabry's Hill, and a portion of it completed. The troops on the south side of the river were hard at work with all the tools we had to spare from the north side. During the afternoon large trains belonging to the enemy were seen to move toward the eastward, and the belief began to grow upon us that siege would be raised.

Wednesday, December 2.-The Engineer Battalion and the contrabands were particularly engaged on the rifle-trenches between Temperance Hill and Mabry's Hill, and an epaulement for two guns in the gorge. These were all finished by midnight. Still at work on the large fort at Bell's house and on rifle-trenches on the south side of the river. Everything unusually quiet for the fifteenth day of a siege.

Thursday, December 3.-Still hard at work on both sides of the river. A disposition apparent among the troops to consider the position strong enough to repel any assault the enemy might make, and a consequent indisposition to work. Evidently but a small force of the enemy east of the Tazewell road. The enemy's trains seen moving to the eastward.

Friday, December 4.-Still working a little, but the news of the approaching re-enforcements, and the movements of the enemy's trains led us to believe that he would soon abandon the siege.

Saturday, December 5.-The siege of Knoxville terminated by the retreat of the enemy in the direction of Strawberry Plains (eastward). Heavy re-enforcements for us reached the south side of the river. The enemy's infantry, or at least part of it belonging to Longstreet's own corps, left in a solid body, very deliberately, no signs of haste being apparent. The rear of his column passed the Tazawell road about 7 a.m.

A feeling of intense satisfaction pervaded the whole command, and many persons assured me of their conversion to a belief in "dirt digging." It certainly proved efficient here. Examined the enemy's late position, and was surprised to find so little evidence of good engineering. I saw positions for only seventeen guns on the north side of the river, but could not discover that more than twelve of them had been used. Any other artillery he used was without cover. The cavalry force available was at once sent in pursuit of the retreating enemy, and during the day quite a number of prisoners were brought in.

Monday, December 7.-All the available infantry force of the Army of the Ohio was put in motion toward the enemy and followed him slowly until the 9th, when our forces halted, the cavalry at Bean's Station and the infantry at Rutledge. The enemy had halted at or near Red Bridge, between Bean's Station and Rogersville. No attack was made by us, as the enemy was in vastly superior force.

On the 11th, Major-General Burnside, having been relieved by

Major-General Foster, left for the North, and that properly fixes the close of this report.

To Lieutenant-Colonel Babcock, assistant inspector-general of the Ninth Army Corps, and captain of Engineers, I am under very heavy obligations. He was always ready with the most practical advice, he cheerfully gave it, and it was never passed unheeded.

To Maj. S. S. Lyon, of Fourth Kentucky Cavalry, and assistant