ROSECRANS' HEADQUARTERS, July 20, 1862.

Major-General GRANT:

Two dispatches from General Davis, of the Fourth Division, at Jacinto, give contradictory reports. The former that the enemy were moving from Jacinto to Saltillo, the other that they were concentrating at Big Springs Factory, 15 miles southeast of Jacinto, and 25 miles south from Iuka; that they imagine we are reduced in numbers and intend to attack Jacinto and Corinth. They say our guards about Corinth are badly posted and their spies go where they please.

While I do not credit the report of their intended attack on Jacinto I have given Davis orders to be prepared to fight or fall back, sending his baggage before him, on this position.

I have also given Morgan notice of the report. His division will be in Burnsville to-morrow and at Iuka next day and evening. I venture to suggest Davis' division requires caution as to its guard duty.

W. S. ROSECRANS,

Brigadier-General, U. S. Army, Commanding.

HEADQUARTERS ARMY OF THE MISSISSIPPI, July 20, 1862.

Major-General HALLECK, Commander-in-Chief, Washington:

GENERAL: There is little doubt but that so soon as forage for horses can be got from the corn-corps of the rebels we shall have a host of conscript cavalry and mounted infantry to deal with. Now is the time to strain every nerve to meet this contingency effectually.

1st. What we need is that all the cavalry should be promptly and thoroughly armed. There is no excuse for their being in the field without good arms. I have had offers from private arm manufacturers to supply in four weeks all we need in this army. The utter fatuity of not arming them suitably for service may be inferred from the fact that the cost of maintaining a regiment of cavalry in idleness one mouth would arm them with revolving rifles.

2d. The regiments should be filled; to which end, all means, official and unofficial, should be used to induce the Governors of States to fill them up. The cavalry are the eyes of the army. Nay, more; I do not hesitate to say that the time will soon be here when a thousand cavalry will do more damage to the rebels by seizing and destroying their means of subsistence than a brigade of infantry.

Having a position on the front, and, for the first time in this war, given development to the working powers of this arm, I take the liberty of urging the necessity of the steps I have indicated, hoping you will be able to bring about the desired ends in time to meet the emergencies to which I have alluded.

W. S. ROSECRANS,
Brigadier-General, U. S. Army, Commanding.

[Indorsement.]

HEADQUARTERS DISTRICT OF WEST TENNESSEE, Corinth, July 22, 1862.

Respectfully referred to Major General H. W. Halleck, with the request that the within suggestions receive early attention. The proper arming of the cavalry in this department is of vital importance and demands prompt action.

U. S. GRANT,
Major-General.