Sullivan reports the strength of the enemy at from 5,000 to 10,000 and still crossing. Dodge, however, had a scout among them before they commenced crossing, who estimates their force at about 5,000.

Ingersoll's cavalry watched their movements for the last 25 miles, and yesterday had an engagement with them at Lexington, resulting in a defeat for us, Colonel Ingersoll and two pieces of artillery falling into the hands of the enemy. Last night Sullivan brought them to a halt about 6 miles from Jackson.

I have re-enforced Sullivan to the full extent of the capacity of the road to carry troops, partly from Columbus, partly from Corinth, one brigade from here, and by concentrating of the forces of the District of Jackson. Lowe is also moving from Heiman. I think the enemy must be annihilated, but it may trouble and possibly lead to the necessity of sending further forces from here.

A dispatch from General Halleck, received late last night, directs me to divide my forces into army corps, one of which is to be commanded by Major-General McClernand, he to have the chief command of the Vicksburg expedition, but under my direction. I was in hopes the expedition would be off by this time, and it may be that they are about starting.

We must be ready for any move. I think, however, it will not be a retrograde one.

U. S. GRANT,

Major-General.

OXFORD, MISS., December 19, 1862.

Major General J. B. McPHERSON, Water Valley;

The report of the surgeon is untrue. Order his arrest for being an alarmist. The following dispatch just received from General Sullivan says;

I will move and attack in front this afternoon. The enemy are within 3 miles of this place. They have burned the bridges on the Corinth and Bolivard roads. I have force enough to drive what force they have here to the river. The river is falling very rapidly.

General Dodge moved last night from Corinth with a force of at leaf 2,000, and is probably within striking distance of their flank or rear.

Colonel Lowe was ordered to move with all his available force against the enemy 2 o'clock yesterday afternoon from Fort Henry. The order reached him promptly. He is perhaps near them. One brigade (Colonel Fuller's) was sent from here last night, and it is to be hoped we will not only be able to whip the enemy but will prevent his recrossing the Tennessee.

By order of Major General U. S. Grant:

JNO. A. RAWLINS,

Assistant Adjutant-General.

OXFORD, MISS., December 19, 1862.

Brigadier General GRENVILLE M. DODGE, Corinth, Miss.:

Jackson is now moving north with a cavalry force of about 3,000. He will probably be near the Tallahatchie to-night. I will have him followed with cavalry. Look out for him.

U. S. GRANT,

Major-General.