The rebel pickets discharged their muskets and ran into their rifle-pits; our skirmishers followed on their heels; the line of battle was not far behind; and we saw the gray rebels swarm out of the long line of rifle-pits in numbers which surprised us, and spread over the base of the hill. A few turned and fired their pieces, but the greater number collected into the various roads which creep obliquely up its steep face, and went on the top. Some regiments pressed on and began to swarm up the steep sides of the ridge. Here and there a color was advanced beyond the line. The attempt appeared most dangerous; but the advance was supported and the whole line ordered to storm the heights, upon which not less than forty pieces of artillery, and no one knew how many muskets, stood ready to slaughter the assailants.

With cheers answering to cheers the men swarmed upward. They gathered to the lines of least difficult ascent and the line was broken. Color after color planted on the summit, while musketry and cannon vomited their thunder upon them. A well-directed shot from Orchard Knob exploded a rebel caisson on the summit. A gun was seen galloping to the right, its driver lashing his horses. A party of our soldiers intercepted him, and the gun was captured with cheers.

A fierce musketry fight broke our to the left, where, between Thomas and Sherman, a mile or two of the ridge was still occupied by the rebels. Bragg left the house in which he had held his headquarters and rode to the rear as our troops crowned the hill on each side of him.

General Grant proceeded to the summit, and then only did we knows its height.

Some of the captured artillery was put into position, artillerists were sent for to work the guns, caissons were searched for ammunition. The rebel log breastworks were torn to pieces, and carried to the other side of the ridge and used in forming barricades across it. A strong line of infantry was formed in the rear of Baird's line, hotly engaged in a musketry contest with the rebels to the left, and a secure lodgment was soon effected.

The other assault to the right of our center gained the summit, and the rebels threw down their arms and fled. Hooker, coming in from Rossville, swept the right of the ridge and captured many prisoners.

Bragg's remaining troops left early in the night and the battle of Chattanooga, after three days of maneuvering and fighting, was won. The strength of the rebellion in the center was broken; Burnside relieved from danger in East Tennessee; Kentucky and Tennessee redeemed; Georgia and the Southeast threatened in the rear, and another victory added to the chaplet of Unconditional Surrender Grant.

To-night the estimate of captures is several thousand prisoners and thirty pieces of artillery. Loss for so great a victory not severe. Bragg is firing the railroad as he retreats toward Dalton; Sherman is in hot pursuit.

To-day I visited the battle-field, which extends for 6 miles along Mission Ridge and for several miles on Lookout Mountain.

Probably no so well-directed, so well ordered a battle has been delivered during the war. But one assault was repulsed, but that assault, by calling to that point the reserves, prevented their repulsing any of the others.

A few days since Bragg sent to General Grant a flag of truce to