Well, the enemy has assaulted Lookout Mountain to-day, sure enough, as was intimated in my letter of last night he probably would do. Having accomplished a part of the object of his demonstration yesterday, to wit: the transfer of a portion of our forces on the mountain to the extreme right, he attacked the mountain with a confidence which the sequel will show was not misplaced. The great rise in the Tennessee had carried away his pontoon bridges the night before, but his positions were so well taken, and had been so strongly fortified, that he did not hesitate to make the assault. He opened at 11 o'clock with his batteries in Lookout Valley, directing his fire against our lines along the western side and northern face of the mountain. Our own batteries on the mountain could take no part in the engagement, owing to a dense fog which enveloped Lookout Point and the crest above. At 12 1/2 o'clock the infantry became engaged, and the battle was then fully joined.